New Equations of State in Simulations of Core-Collapse Supernovae

Abstract

We discuss three new equations of state (EOS) in core-collapse supernova
simulations. The new EOS are based on the nuclear statistical equilibrium model
of Hempel and Schaffner-Bielich (HS), which includes excluded volume effects
and relativistic mean-field (RMF) interactions. We consider the RMF
parameterizations TM1, TMA, and FSUgold. These EOS are implemented into our
spherically symmetric core-collapse supernova model, which is based on general
relativistic radiation hydrodynamics and three-flavor Boltzmann neutrino
transport. The results obtained for the new EOS are compared with the widely
used EOS of H. Shen et al. and Lattimer & Swesty. The systematic comparison
shows that the model description of inhomogeneous nuclear matter is as
important as the parameterization of the nuclear interactions for the supernova
dynamics and the neutrino signal. Furthermore, several new aspects of nuclear
physics are investigated: the HS EOS contains distributions of nuclei,
including nuclear shell effects. The appearance of light nuclei, e.g.,
deuterium and tritium is also explored, which can become as abundant as alphas
and free protons. In addition, we investigate the black hole formation in
failed core-collapse supernovae, which is mainly determined by the high-density
EOS. We find that temperature effects lead to a systematically faster collapse
for the non-relativistic LS EOS in comparison to the RMF EOS. We deduce a new
correlation for the time until black hole formation, which allows to determine
the maximum mass of proto-neutron stars, if the neutrino signal from such a
failed supernova would be measured in the future. This would give a constraint
for the nuclear EOS at finite entropy, complementary to observations of cold
neutron stars.

Full text

Page 1

arXiv:1108.0848v1 [astro-ph.HE] 3 Aug 2011
Submitted to ApJ
Preprint typeset using LATEX style emulateapj v. 5/2/11
NEW EQUATIONS OF STATE IN SIMULATIONS OF CORE-COLLAPSE SUPERNOVAE
M. Hempel1, T. Fischer2,3, J. Schaffner-Bielich4and M. Liebend¨ orfer1
1: Departement Physik, Universit¨ at Basel, Klingelbergstr. 82, 4056 Basel, Switzerland
2: GSI, Helmholtzzentrum f¨ ur Schwerionenforschung GmbH, Planckstr. 1, 64291 Darmstadt, Germany
3: Technische Universit¨ at Darmstadt, Schlossgartenstr. 9, 64289 Darmstadt, Germany
4: Institut f¨ ur Theoretische Physik, Ruprecht-Karls-Universit¨ at, Philosophenweg 16, 69120 Heidelberg, Germany
Draft version August 4, 2011
ABSTRACT
We discuss core-collapse supernova simulations where three new equations of state (EOS) tables
are applied for the first time. The spherically symmetric core-collapse model is based on general
relativistic radiation hydrodynamics and three-flavor Boltzmann neutrino transport. The new EOS
are calculated with the nuclear statistical equilibrium model of Hempel and Schaffner-Bielich (HS)
which includes excluded volume effects and relativistic mean-field (RMF) interactions. Three different
RMF parameterizations, TM1, TMA, and FSUgold, are considered. The new EOS tables and routines
for the nuclear distributions are available online. We examine the core collapse, bounce, and post-
bounce phases and compare our results with the widely used EOS of H. Shen et al. (1998a) and
Lattimer and Swesty (1991). As the EOS of H. Shen et al. is also based on TM1, this allows a
systematic study of the model description of inhomogeneous nuclear matter, where we find that it
is as important as the nuclear interactions. Several new aspects of nuclear physics are investigated:
The HS EOS contains nuclear distributions, which can become very broad during the collapse phase.
Furthermore, for the infalling matter, effects from neutron magic shells are seen. In the shock heated
matter light nuclei like the deuteron or the triton appear with similar or even larger abundances than
the alpha particle. The presence of additional light nuclei might contribute to the neutrino heating
in the early post-bounce phase, whereas in the later evolution the effect on cooling might be more
important. Apart from the supernova of a 15 solar mass (M⊙) progenitor, we simulate the collapse of a
40 M⊙progenitor to a black hole, and find a correlation between the maximum mass of proto-neutron
stars and the time until black hole formation. If the neutrino signal of such a supernova was measured
in the future, this correlation can be used to deduce important information about the nuclear EOS.
1. INTRODUCTION
Supernova explosions of stars more massive than 8 M⊙
are an active subject of research in astrophysics. The
core-collapse supernova problem is related to the revival
of the stalled bounce shock. It forms when the collaps-
ing stellar core bounces back above normal nuclear mat-
ter density n0
B. Explosions in spherical symmetry have
been obtained only for the low mass 8.8 M⊙O-Ne-Mg-
core from Nomoto (1983, 1984, 1987) by Kitaura et al.
(2006) and Fischer et al. (2010).
model is related to the special structure of the progen-
itor. The explosion is a combination of energy depo-
sition from nuclear burning and neutrino heating. More
massive progenitor stars have large high-density Si-layers
surrounding the central iron core. The post-bounce evo-
lution leads to an extended mass accretion period that
lasts over several 100 milliseconds. For such iron-core
progenitor stars explosions do not occur in spherically
symmetric simulations. The bounce shock continuously
looses energy by dissociation of heavy nuclei and neutrino
heating is not sufficient enough to revive the standing
bounce shock.
In addition to the standard scenario of neutrino-
driven explosions (see Bethe and Wilson (1985)), sev-
eral alternative explosion mechanisms have been pro-
posed. These are the magneto-rotational mechanism
by LeBlanc and Wilson (1970) and the acoustic mech-
anism by Burrows et al. (2006). All of which are work-
ing in multiple spatial dimensions.
The success of this
Multi-dimensional
core-collapse supernova models, based on sophisticated
neutrino transport approximations and general relativ-
ity, have become available only recently. These models
have shown to increase the neutrino heating efficiency
(see, e.g., Miller et al. (1993), Herant et al. (1994),
Janka and Mueller (1996), Burrows et al. (2007)) and
help to aid the understanding of aspherical explosions
(see for example Bruenn et al. (2006), Marek and Janka
(2009), and M¨ uller et al. (2010)).
Apart from the dimensionality, the simulations are also
affected by uncertainties in the nuclear physics involved,
which can be divided in weak processes and the equa-
tion of state (EOS). Changes in the nuclear physics input
can have dramatic consequences and can even generate
explosions in spherically symmetric models as was illus-
trated recently by Sagert et al. (2009) and Fischer et al.
(2011). The authors explored the state of matter around
and above nuclear matter density and investigated the
possibility of the quark-hadron phase transition during
the early post-bounce evolution of low and intermediate
mass iron-core progenitor stars. It was found that the
phase transition leads to the formation of a strong hy-
drodynamic shock wave which triggers the explosion. In
the present article we study new aspects of the hadronic
EOS and their consequences for the supernova dynamics.
Until today, there are only few hadronic EOS avail-
able which cover a sufficiently large domain in density,
temperature, and electron fraction so that they can be
used in simulations of core-collapse supernovae.Usu-

Page 2

2 Hempel et al.
ally, in such simulations the EOS is implemented in form
of a table. The most prominent among them are the
’classic’ EOS from Lattimer and Swesty (1991) (LS) and
H. Shen et al. (1998b,a) (STOS) which are commonly
used in computational astrophysics. However, there has
been a lot of progress for the supernova EOS in the last
years. Recently, for example, the new hadronic EOS
tables of G. Shen et al. (2010, 2011) became available,
which are based on the virial expansion and a relativis-
tic mean-field (RMF) model. In their first article they
applied the NL3 interactions (G. Shen et al. (2010)), fol-
lowed by a table for FSUgold (G. Shen et al. (2011)).
An important aspect of the supernova EOS is the for-
mation of light nuclei and their properties in the hot
and dense medium. Note that the two classic super-
nova EOS include only alpha particles of all possible
light nuclei, which are implemented with excluded vol-
ume effects. Typel et al. (2010) use a generalized RMF
and a quantum statistical model to study supernova mat-
ter taking into account the most important light nuclei
with mass number A ≤ 4. Significant differences in the
composition were found compared to STOS, similar as
in the recent studies of Horowitz and Schwenk (2006),
Sumiyoshi and R¨ opke (2008), Heckel et al. (2009), and
Hempel and Schaffner-Bielich (2010).
namic variables, like e.g. the symmetry energy, are mod-
ified due to the appearance of light nuclei.
the importance of light nuclei in supernova matter was
also shown by a heavy-ion experiment. In the isoscaling
analysis of Natowitz et al. (2010) the measured symme-
try energy can only be explained, if the formation of
light nuclei is taken properly into account.
markably, the deduced temperatures and densities cor-
respond to conditions which are typical for matter in
core-collapse supernovae.
ment light nuclei can influence the neutrino transport
and consequently the supernova neutrino signal and dy-
namics, see, e.g., O’Connor et al. (2007); Arcones et al.
(2008); Sumiyoshi and R¨ opke (2008); Nakamura et al.
(2009). However, so far there are no investigations of
core-collapse supernovae, which consistently take into ac-
count all light nuclei. With this article we further ad-
vance in this direction, by applying the EOS tables of the
statistical model of Hempel and Schaffner-Bielich (2010)
(HS) to simulations of massive stars.
Statistical models are characterized by having a dis-
tribution of nuclei, in which light nuclei can also be
included. On the other hand, the complexity and
variety of the involved aspects of nuclear physics re-
quire some modeling. Important aspects in the super-
nova context are for example the interactions of un-
bound nucleons, excluded volume effects, excited states,
Coulomb interactions, nuclear binding energies, shell
effects or surface modifications of nuclei, see, e.g.,
Ishizuka et al. (2003); Botvina and Mishustin (2004);
Nadyozhin and Yudin (2004, 2005); Souza et al. (2009);
Blinnikov et al.(2009); Hempel and Schaffner-Bielich
(2010); Botvina and Mishustin (2010); Arcones et al.
(2010); Raduta and Gulminelli (2010); Furusawa et al.
(2011). Interestingly, similar statistical models are also
used for the analysis of multifragmentation experiments
(Koonin and Randrup 1987; Gross 1990; Bondorf et al.
1995). In this article we apply the new EOS from
Hempel and Schaffner-Bielich (2010). The HS EOS uses
Also thermody-
Recently,
Quite re-
In the supernova environ-
nuclear statistical equilibrium (NSE) with excluded vol-
ume effects and RMF interactions of unbound nucleons.
In this model all possible light nuclei (e.g. deuterium and
tritium) and heavy nuclei up to mass number A ∼ 330
from the neutron- to the proton-dripline are included.
The knowledge of the detailed composition makes it pos-
sible to analyze the impact of different baryon contribu-
tions during the different phases of core-collapse super-
novae.
The nuclear distributions are especially relevant for
weak reactions. For example the new treatments of elec-
tron captures on heavy nuclei by Langanke et al. (2003)
and Hix et al. (2003) and inelastic neutrino-nucleon (nu-
clei) scattering by Langanke et al. (2008) are all based
on NSE distributions of nuclei. Idealistically, the nuclear
physics input in the weak reactions should be as con-
sistent as possible with the EOS and therefore the two
parts should be built from the same degrees of freedom.
However, also in the present study we use additional sim-
plifications, which will be specified below.
To classify the characteristic features of HS, we take
the two aforementioned classic supernova EOS, i.e. LS
and STOS, as standard references. These two EOS both
use the single nucleus approximation (SNA), i.e. the dis-
tribution of heavy nuclei is represented by only one sin-
gle nucleus. There are further limitations of the de-
scription of nuclei in the two models: STOS is based
on the Thomas-Fermi and local-density approximation
and a minimization of the free energy for parameter-
ized nucleon-density profiles within a RMF model. LS
is based on a non-relativistic liquid drop description in-
cluding surface effects. Both models do not include shell
effects and only give an approximate description of the
iron-group nuclei, which appear at low temperatures.
This leads to artificial shifts in the EOS at the transition
to non-NSE. Contrary, due to the use of nuclear masses,
the HS model reproduces the thermodynamic state of the
ideal gas of e.g.56Fe or56Ni (depending on the proton-
to-baryon ratio) at low temperatures by construction and
naturally includes shell effects. Thus, it represents a ma-
jor improvement to other commonly used EOS as it gives
a more detailed description of the chemical composition
and the nuclear effects involved at low densities below
∼ 1012g/cm3.
However, with statistical models it is difficult to
describe the transition to uniform nuclear matter
which occurs around ρ0/2 ∼ 1014g/cm3.
scopic SNA models allow a more detailed descrip-
tion of the effects which occur at such large den-
sities, like the formation of non-spherical inhomoge-
neous structures, the so-called nuclear pasta phases, see
e.g. Ravenhall et al. (1983); Newton and Stone (2009).
In Hempel and Schaffner-Bielich (2010) it was shown
that the thermodynamic variables in the transition re-
gion calculated with the HS model are in satisfactory
agreement with the results of STOS and LS. From this
point of view, the present investigation serves as a prac-
tical test whether the HS EOS is suitable to be used for
all possible conditions in core-collapse supernova simula-
tions.
Recently, four new supernova EOS tables were cal-
culated with the HS model for different nuclear inter-
Micro-

Page 3

New equations of state in core-collapse supernova simulations3
actions and are available online1. One of the new HS
EOS tables is based on the same parameterization TM1
(Sugahara and Toki 1994) of RMF interactions of the nu-
cleons as STOS. This allows to investigate the impact of
the model assumptions for the description of inhomoge-
neous nuclear matter below saturation density, which is
one of the main topics of the present study. In addition
to TM1 and the general features of the HS model, we also
investigate two other RMF parameterizations of the nu-
clear interactions, TMA (Toki et al. (1995)), and FSUg-
old (Todd-Rutel and Piekarewicz (2005); Piekarewicz
(2007)). We note that these two interactions have not
been applied in supernova-simulations so far. The differ-
ent nuclear interactions become significant at large den-
sities and we analyze to which extent they can influence
the supernova dynamics and the observable neutrino sig-
nal.
The manuscript is organized as follows.
present our core-collapse model, including the new HS
EOS. The results from simulations of the collapse,
bounce and the early post-bounce phases of a 15 M⊙
progenitor are presented in §3. We compare HS (TM1)
with the LS and STOS EOS and also analyze the neu-
trino signal. In §4 we discuss the neutrino signal of a
40 M⊙ progenitor by applying four different EOS: LS,
STOS, HS (FSUgold), and HS (TMA). For this massive
progenitor we focus on the time until black hole forma-
tion, and investigate its relation to characteristic proper-
ties of the different nuclear interactions, before we close
with a summary in §5.
In §2 we
2. THEORETICAL AND NUMERICAL SETUP
2.1. Core-collapse supernova model
Our core-collapse model was originally constructed
based on Newtonian radiation hydrodynamics and three
flavor Boltzmann neutrino transport (for details, see
Mezzacappa and Bruenn 1993a,b,c)).
to solve the general relativistic equations for both hydro-
dynamics and neutrino transport in Liebend¨ orfer et al.
(2001). Special attention has been devoted to accu-
rately conserve energy, momentum and lepton number
in Liebend¨ orfer et al. (2004). The standard weak inter-
actions included in the Boltzmann neutrino transport
are listed in Table 1. The emission of (µ/τ)-neutrino
pairs by the annihilation of trapped (νe, ¯ νe) has been
included by Fischer et al. (2009). In Liebend¨ orfer et al.
(2004), the authors compare the model with S. Bruenn’s
multi-group flux limited diffusion neutrino transport ap-
proximation. A similar comparison with the Boltzmann
transport used by the Garching-grouphas been published
in Liebend¨ orfer et al. (2005). These two studies show a
good qualitative agreement of the spherically symmetric
models used by the different groups.
It was extended
2.2. Use of the EOS
The EOS in supernova simulations is part of the nu-
clear physics input, next to weak interactions in the
neutrino transport. There are two intrinsically different
regimes. In nuclear statistical equilibrium (NSE), the
destruction and production of nuclei is in thermal and
chemical equilibrium regarding strong and electromag-
netic interactions. The conditions for NSE are achieved
1http://phys-merger.physik.unibas.ch/~hempel/eos.html
TABLE 1
Neutrino reactions considered, including references.
Weak processa
νe+ n ⇄ p + e−
References
Bruenn (1985)
¯ νe+ p ⇄ n + e+
Bruenn (1985)
νe+ ?A? ⇄ ?A? + e−
Bruenn (1985)
ν + N/?A?/α ⇄ ν′+ N/?A?/α Bruenn (1985),
Mezzacappa and Bruenn (1993a)
ν + e±⇄ ν′+ e±
Bruenn (1985),
Mezzacappa and Bruenn (1993c)
ν + ¯ ν ⇄ e−+ e+
Bruenn (1985),
Mezzacappa and Messer (1999)
ν + ¯ ν + N + N ⇄ N + NO. E. B. Messerb
νe+ ¯ νe⇄ νµ/τ+ ¯ νµ/τ
Buras et al. (2003),
Fischer et al. (2009)
Notes:
aν = {νe, ¯ νe,νµ/τ, ¯ νµ/τ} and N = {n,p}
bprivate communications, based on Hannestad and Raffelt (1998)
typically above a temperature of T ≃ 0.5 MeV2. For tem-
peratures below T = 0.5 MeV, where NSE cannot be ap-
plied, we use an ideal Si-gas for the reference simulations
with the LS EOS. In our new simulations with the STOS
and HS EOS this simplification has been replaced by a
nuclear reaction network (see Thielemann et al. (2004)).
The conditions where the network is used in the simula-
tions correspond to the Si-S-, as well as C-O- and even
up to the He-layer, depending on the overlap between the
computational domain and the progenitor. Our nuclear
reaction network calculates the time evolution of the set
of abundances, given initially by the progenitor model,
based on tabulated reaction rates. Due to computational
limitations we consider only 20 nuclei, starting from neu-
trons and protons,3He,4He up to56Ni (for details, see
Fischer et al. (2010)). The baryon EOS in that regime is
calculated as an ideal Maxwell-Boltzmann gas with the
measured nuclear binding energies from the compilation
of Audi et al. (2003).
For the NSE regime with temperatures of T
0.5 MeV, we use the HS, LS, and STOS EOS in tab-
ular form for the baryon component.
approximation to treat electrons (and positrons) as a
rigid uniform background.
as a non-interacting ideal Fermi-Dirac gas based on
Timmes and Arnett (1999). The trivial black-body con-
tribution of photons is also taken into account.
baryon EOS are calculated for given temperature T,
baryon number density nB and total proton fraction
Yp= ntot
of charge neutrality, in the absence of muons the number
density of electrons neis given by: ne= YpnBwhich con-
nects the baryon and the electron EOS. Accordingly, Yp
is equivalent to the electron fraction Ye= ne/nB= Yp.
In astrophysics, very often instead of the baryon num-
>
It is a good
Thus they can be added
The
p/nBand are stored as a separate table. Because
2Throughout the article we use natural units, i.e. ? = c = kB=
1, for quantities where it is appropriate.

Page 4

4 Hempel et al.
ber density nBa baryon mass density ρ is used. This is
completely equivalent, if ρ is defined as ρ = nBm, with
an arbitrary but constant mass m. We will also use such
a ’baryon mass density’ with the usual convention ρ =
nBmu, with the atomic mass unit mu= 931.49432 MeV.
However, we want to stress that ρ defined in such a
way is not the rest-mass density, because e.g. the par-
ticle species in NSE will change with any change of the
thermodynamic state of the system. Furthermore, in a
relativistic description there is no conservation of rest
mass, but only of the total energy and baryon number.
Analogously to the baryon mass density one also defines
a baryonic mass M = NBmu which is just a redefini-
tion of the total baryon number NB of the investigated
system.
The HS EOS contains the full distribution of all avail-
able nuclei, as will be discussed in more detail in the next
subsection. However, regarding weak processes we only
consider an average heavy nucleus and an average light
nucleus. We separate the distribution of all nuclei into
light nuclei and heavy nuclei by the charge number six,
i.e. carbon:
?
A≥2,Z≤5
?
A≥2,Z≥6
Xa=
A nA,Z/nB
(1)
XA=
A nA,Z/nB, (2)
with nA,Zdenoting the number density of a nucleus with
mass number A and charge number Z, so that the sum
of the mass fractions of neutrons Xn, protons Xp, light
nuclei Xa and heavy nuclei XA is unity. The average
mass and charge of the light and heavy nuclei are:
?a?=
?
?
A≥2,Z≤5
A nA,Z/
?
?
A≥2,Z≤5
nA,Z,
?z?=
A≥2,Z≤5
Z nA,Z/
A≥2,Z≤5
nA,Z,
?A?=
?
?
A≥2,Z≥6
A nA,Z/
?
?
A≥2,Z≥6
nA,Z,
?Z?=
A≥2,Z≥6
Z nA,Z/
A≥2,Z≥6
nA,Z.
We further simplify the neutrino reactions with light
nuclei by treating all of them as alpha particles. Note
that for the alpha particles, and thus for all light nuclei,
in the present investigation only neutral current reactions
are considered, as listed in Table 1. Actually, charged-
current and other break-up reactions of the weakly bound
light nuclei should be taken into account, as the cross-
sections can be large, see e.g. Nakamura et al. (2009) for
neutrino deuteron reactions. However, in this article we
first want to investigate their possible appearance and
their contribution to the EOS.
2.3. Model of the HS EOS
The HS model consists of an ensemble of nucleons
and nuclei in NSE, whereas interactions of the nucle-
ons and excluded volume corrections for the nuclei are
implemented. Note that in the present version of the
HS EOS tables, the formation of nuclei is neglected for
T > 20 MeV for simplicity, i.e. matter is assumed to
be uniform. At such large temperatures there is only
a limited density region slightly below n0
appear at all. Still it would be better to avoid such a
hard switch in the EOS, which we plan to do in future
releases of the EOS tables. In the following, we give a
brief summary of the HS model, all details can be found
in Hempel and Schaffner-Bielich (2010).
For the unbound interacting nucleons (neutrons and
protons) a relativistic mean-field (RMF) model is ap-
plied. Its Lagrangian is based on the exchange of the
isoscalar scalar σ-, the isoscalar vector ω- and the isovec-
tor vector ρ-mesons between nucleons. In comparison
with more sophisticated theoretical approaches, based on
nucleon-nucleon scattering data, it has been recognized
that non-linear σ- and ω-self-coupling terms are neces-
sary to achieve a reasonable description of the EOS at
large densities and the properties of nuclei at the same
time, see e.g. Sugahara and Toki (1994).
tive approach, which is not used in the present inves-
tigation, are RMF models with density-dependent cou-
plings, see e.g. Typel (2005). The free parameters of the
Lagrangians, the masses of the nucleons and the mesons
and their coupling strengths, have to be determined by
fits to experimental data.
We apply three different RMF parameterizations:
TM1 by Sugahara and Toki (1994), TMA by Toki et al.
(1995) and FSUgold by Todd-Rutel and Piekarewicz
(2005). The Lagrangians of TM1 and TMA have the
same form and include non-linear terms of the σ- and ω-
mesons. TM1 was developed together with TM2, which
were fitted to binding energies and charge radii of light
(TM2) and heavy nuclei (TM1). TMA is based on an in-
terpolation of these two parameter sets. The coupling pa-
rameters giof the set TMA are chosen to be mass-number
dependent of the form gi= ai+ bi/A0.4, with aiand bi
being constants, to have a good description of nuclei over
the entire range of mass numbers. For uniform nuclear
matter the couplings become constants and are given by
ai. TMA was also used in Hempel and Schaffner-Bielich
(2010), where the HS model was introduced. FSUgold
includes the coupling between the ω- and the ρ-meson in
addition. This leads to a better description of nuclear
collective modes, the EOS of asymmetric nuclear mat-
ter and a different density dependence of the symmetry
energy (Piekarewicz 2007) which is very low at large den-
sities. The coupling constants of FSUgold are fitted to
binding energies and charge radii of a selection of magic
nuclei.
The HS EOS tables take into account the experimen-
tal data on nuclear masses from Audi et al. (2003). For
the masses of the experimentally unknown nuclei differ-
ent theoretical nuclear structure calculations in form of
nuclear mass tables are used.
are combined with the mass table of Geng et al. (2005),
which is also calculated with the RMF model TMA. Thus
all nuclear interactions are consistent. This mass table
lists 6969 even-even, even-odd and odd-odd nuclei, ex-
tending from16
100Fm from slightly above the pro-
ton to slightly below the neutron drip line. The nuclear
binding energies are calculated under consideration of
axial deformations and the pairing is included with a
BCS-type δ-force. For the parametrization TM1 we do
not have a suitable mass table at hand, thus we cannot
B, where nuclei
An alterna-
The TMA interactions
8O to331

Page 5

New equations of state in core-collapse supernova simulations5
avoid the minor inconsistency to also use the table of
Geng et al. (2005), which is based on the TMA parame-
terization. For FSUgold we take a mass table which was
calculated by X. Roca-Maza, which was also applied in
Roca-Maza and Piekarewicz (2008). This table contains
1512 even-even nuclei, from the proton to the neutron
drip, with 14 ≤ A ≤ 348 and 8 ≤ Z ≤ 100. Odd nuclei
are not included in this table. The nuclei were calcu-
lated only with spherical symmetry and the pairing is
introduced through a BCS approach with constant ma-
trix elements. The constant matrix element for neutrons
has been fitted to reproduce the experimental binding in
the tin isotopic chain and the constant matrix element
for protons to the experimental binding in the N = 82
isotonic chain.
To describe nuclei in the supernova environment, we
not only need binding energies, but have to account for
medium and temperature effects. For the screening of the
Coulomb field of the nuclei in the uniform background of
electrons we use the most basic expression: for each nu-
cleus we assume a spherical Wigner-Seitz (WS) cell at
zero temperature. More elaborated approaches for the
Coulomb energy of a multi-component plasma at finite
temperature can e.g. be found in Nadyozhin and Yudin
(2005); Potekhin et al. (2009); Potekhin and Chabrier
(2010). However, we leave this for future studies as the
Coulomb energy becomes only important at low temper-
atures so that the simplest expression is sufficient for our
purposes.
Finite temperature leads to the population of excited
states of the nuclei. Here we use the temperature depen-
dent degeneracy function of F´ ai and Randrup (1982). It
is the same analytic expression as in the original reference
of the HS model (Hempel and Schaffner-Bielich 2010),
but now we consider only excitation energies below the
binding energy of the corresponding nucleus, in order to
represent that the excited states still have to be bound
(see, e.g., R¨ opke (1984)). We note that the inclusion of
excited states up to infinite energies had only a minor
influence on the composition but would lead to an un-
physically large contribution of the excited states to the
energy density and entropy at very large temperatures.
We describe nuclear matter as a chemical mixture of
the different nuclear species and nucleons. As we distin-
guish between nuclei and the surrounding interacting nu-
cleons we still have to specify how the system is composed
of the different particles. Our thermodynamic model is
built on two main assumptions: First, we assume for un-
bound nucleons that they are not allowed to be situated
inside of nuclei, whereas nuclei are described as uniform
hard spheres at saturation density n0
clei (with mass number A ≥ 2) we assume that they must
not overlap with any other baryon in the system (nuclei
or unbound nucleons). Thus we take the volume which
is available for the nucleons to be the part of the total
volume of the system which is not excluded by nuclei.
This is described by the filling factor of the nucleons
B. Second, for nu-
ξ = 1 −
?
A,Z
A nA,Z/n0
B, (3)
(here and in the following, we mean A ≥ 2). The free
volume in which a nucleus can move is the total volume
minus the volume filled by nuclei and nucleons. This is
incorporated via the free volume fraction
κ=1 − nB/n0
B,(4)
with the total baryon number density nB, which includes
the contributions of unbound neutrons and protons:
nB=nn+ np+
?
A,Z
A nA,Z. (5)
Based on these two main assumptions, the EOS is
derived in a consistent way, using the non-relativistic
Maxwell-Boltzmann description for nuclei and the full
Fermi-Dirac integrals for nucleons (solved with the rou-
tines from Aparicio (1998) and Gong et al. (2001)). We
obtain modifications of all thermodynamic quantities due
to the excluded volume. Here we give the thermody-
namic potential, the free energy density f, as an exam-
ple:
f =
?
A,Z
f0
A,Z(T,nA,Z) +
?
A,Z
fCoul
A,Z
−T
?
A,Z
nA,Zln(κ)
+ξf0
RMF(T,nn/ξ,np/ξ) ,(6)
The first term in Eq. (6) is the summed ideal gas ex-
pression of the nuclei. The Coulomb free energy of the
nuclei appears in addition. The second line in Eq. (6) is
the direct contribution from the excluded volume. Be-
cause of this term, as long as nuclei are present, the free
energy density goes to infinity when approaching satu-
ration density, because the free volume of nuclei goes to
zero, κ → 0. Thus, nuclei will always disappear before
saturation density is reached. The RMF contribution of
the nucleons f0
RMFis weighted with their filling factor ξ,
as the free energy is an extensive quantity. If nuclei are
absent, ξ = 1, and we get the unmodified RMF descrip-
tion, as it should be. The excluded volume correction for
the nuclei represents a hard-core repulsion of the nuclei
at large densities close to saturation density. Instead the
modification of the free energy of the unbound nucleons
is purely geometric and just describes that the nucleons
fill only a fraction of the total volume. In this sense, the
two aforementioned model assumptions for the excluded
volume are essential, as they lead to the desired limiting
behavior of the EOS.
2.4. EOS characteristics & constraints
Table 2 lists some characteristic saturation properties
of uniform bulk nuclear matter for the three different
RMF parameterizations. We also include the LS EOS
with the compressibility of K = 180 MeV in the table.
The quantities shown in Table 2 correspond to the co-
efficients of the following power-series expansion of the
binding energy per baryon at T = 0 around the satura-
tion point:
E(x,β)=−E0+
1
18Kx2+
J +1
3Lx + ...
1
162K′x3+ ...
?
+β2
?
+ ... , (7)
with x = nB/n0
the saturation density, and the asymmetry parameter β
B−1 denoting the relative deviation from