PHYSICAL REVIEW C 94, 035804 (2016)
Neutron star radii and crusts: Uncertainties and unified equations of state
M. Fortin,
1,*
C. Provid
ˆ
encia,
2
Ad. R. Raduta,
3
F. Gulminelli,
4
J. L. Zdunik,
1
P. Haensel,
1
and M. Bejger
1
1
N. Copernicus Astronomical Center, Polish Academy of Sciences, Bartycka 18, 00-716 Warszawa, Poland
2
CFisUC, Department of Physics, University of Coimbra, P-3004-516 Coimbra, Portugal
3
IFIN-HH, P.O. Box MG6, Bucharest-Magurele, Romania
4
Universit
´
e de Caen Normandie, ENSICAEN, UNICAEN, CNRS/IN2P3, LPC Caen, 14000 Caen, France
(Received 6 April 2016; revised manuscript received 12 July 2016; published 19 September 2016)
The uncertainties in neutron star radii and crust properties due to our limited knowledge of the equation of
state are quantitatively analyzed. We first demonstrate the importance of a unified microscopic description for the
different baryonic densities of the star. If the pressure functional is obtained matching a crust and a core equation
of state based on models with different properties at nuclear matter saturation, the uncertainties can be as large
as ∼30 % for the crust thickness and 4% for the radius. Necessary conditions for causal and thermodynamically
consistent matchings between the core and the crust are formulated and their consequences examined. A large
set of unified equations of state for purely nucleonic matter is obtained based on twenty-four Skyrme interactions
and nine relativistic mean-field nuclear parametrizations. In addition, for relativistic models fifteen equations of
state including a transition to hyperonic matter at high density are presented. All these equations of state have in
common the property of describing a 2M
star and of being causal within stable neutron stars. Spans of ∼3and∼4
km are obtained for the radius of, respectively, 1.0M
and 2.0M
stars. Applying a set of nine further constraints
from experiment and ab initio calculations the uncertainty is reduced to ∼1 and 2 km, respectively. These residual
uncertainties reflect lack of constraints at large densities and insufficient information on the density dependence
of the equation of state near the nuclear matter saturation point. The most important parameter to be constrained
is shown to be the symmetry energy slope L. Indeed, this parameter exhibits a linear correlation with the stellar
radius, which is particularly clear for small mass stars around 1.0M
. The other equation-of-state parameters do
not show clear correlations with the radius, within the present uncertainties. Potential constraints on L, the neutron
star radius, and the equation of state from observations of thermal states of neutron stars are also discussed. The
unified equations of state are made available in the Supplemental Materials and via the CompOSE database.
DOI: 10.1103/PhysRevC.94.035804
I. INTRODUCTION
Simultaneous measurements of the masses and radii of
neutron stars (NS), if sufficiently precise, will impose strong
constraints on the equation of state (EOS) of dense matter
significantly above (standard) nuclear (baryon number) density
n
0
= 0.16 fm
−3
.Thevalueofn
0
is a suitable unit to measure
the baryon (number) density in NS cores. In fact, the two
most massive pulsars PSR J0348+0432 and PSR J1614−2230
alone, with a mass close to 2M
[1–3], already put quite
stringent constraints on the EOS in the 5n
0
–8n
0
density range.
These mass measurements are particularly relevant to assess
the possible existence of exotic phases of dense matter in the
cores of massive NS.
Significant effort has been put into the determination of the
radii of NS, but presently there is still a large uncertainty
associated with this quantity; see the discussion in [4–6].
Particularly interesting is the measurement of radii for the
stellar mass r ange 1.3M
–1.5M
, where on the one hand
many precise NS mass measurements exist, and on the other
hand dense matter theories predict a nearly constant value
of R (albeit different for various dense matter theories). We
expect that up to 2n
0
–3n
0
NS matter involves nucleons only
and therefore that the radius for the “canonical” NS mass
1.4M
, denoted usually as R
1.4
, characterizes the EOS in
*
Corresponding author: fortin@camk.edu.pl
the nucleon segment. Recently, a new constraint has been
added to this discussion. According to Ref. [7], an EOS with
M
max
> 2M
should produce R
1.4
10.7kminordertoavoid
being noncausal at highest NS densities.
We expect that future simultaneous determinations of the
mass and radius of a NS with a 5% precision will be possible
through the analysis of the x-ray emission of NS, thanks to
the forthcoming NICER [8], Athena [9] and LOFT-like [10]
missions. It is therefore important to be able to quantify the
uncertainties introduced in the NS mass and radius calculations
simultaneously by the approximations used when constructing
the complete EOS for stellar matter, by the scarcely available
constraints on the EOS at high densities and large isospin
asymmetries, and by the lack of information about the possible
exotic states of the matter existing in the interior of a NS.
In the present work we aim to understand how the
calculation of the NS radii are affected by the EOS of the
crust, having in mind that the EOS constructed to describe NS
matter are typically non-unified, i.e., built piecewise starting
from different models for each sector of NS matter. This is
to be contrasted with unified EOS, where all segments (outer
crust, inner crust, liquid core) are calculated starting from the
same nuclear interaction. In practice, for a NS crust with n
10
−4
n
0
one uses experimental nuclear masses. For higher crust
densities, where the relevant experimental nuclear masses are
not available, they should be calculated theoretically. Usually,
one employs an effective nuclear Hamiltonian (or Lagrangian)
2469-9985/2016/94(3)/035804(21) 035804-1 ©2016 American Physical Society
M. FORTIN et al. PHYSICAL REVIEW C 94, 035804 (2016)
and a many-body method that makes the calculation feasible
(e.g., the Thomas-Fermi approximation or the compressible
liquid-drop model). It should be mentioned that some minor
matching problems exist already at the transition between the
experimentally based low-density segment of the EOS and
that obtained with an effective nuclear interaction, if the latter
does not fit perfectly experimental nuclear masses. However,
examples in the present paper show that resulting uncertainty in
R is very small. The calculated EOS for the crust will depend
on the assumed effective nuclear interaction, but the phase
transition between the inner crust (including a possibility of a
bottom layer with nuclear pastas) and the liquid core will be
described correctly. The EOS is then continuous through the
whole NS core, and yields a unique R(M) for each effective
interaction, with negligible residual model dependence.
In contrast, in the standard case of a non-unified EOS model,
the resulting R(M) depends on the procedure of matching the
crust and core EOS segments. As an example, in [11]itis
proposed that the Baym-Pethick-Sutherland (BPS) EOS [12]
be chosen to describe the crust and a matching of the crust
EOS to the core one is performed at 0.01 fm
−3
, while the core
is described within a relativistic mean field (RMF) approach
allowing for fitting several parameters of nuclear matter at
saturation. Similarly a parametrization of the high-density
equation of state based on piecewise polytropes is presented
in [13] and allows one to systematically study the effect of
observational constraints on the EOS of cold stellar matter.
Although for the high density range several models have
been considered, for low densities a single EOS, the one
of Douchin and Haensel [14] based on a specific Skyrme
interaction, namely SLy4 [15], is used. In an equivalent way,
the authors of [5] have studied constraints on the NS structure
by considering two classes of EOS models, and in both the
BPS EOS was taken for the low density EOS, alone or
supplemented by the Negele-Vautherin EOS [16]. Both of
these models are based on old energy functionals which do
not fulfill present experimental nuclear physics constraints. In
all these examples, one can wonder by how much the simplified
choice for the subnuclear density EOS affects the conclusions
obtained from experimental and observational constraints on
theEOS.Infact,in[17] it has been argued that, depending
on the assumed properties of the low density EOS, it is
possible to obtain pressures at the crust-core transition large
enough to explain the large Vela glitches, even considering the
entrainment effect. This indicates that a proper description of
the crust and the crust-core transition as well as a sensitivity
study and a systematic uncertainty evaluation are required.
In the present paper, we will first study how the matching of
the crust EOS with the core one affects the NS radius and the
crust thickness, when models that describe the crust and the
core EOS are not the same. In order to reduce the uncertainties
introduced on the calculation of the star structure, some general
indications will be presented on how to build a non-unified
EOS.
Next we will take a set of unified EOS obtained in the
framework of the RMF models and Skyrme interactions. For
both frameworks we restrict ourselves to EOS that are able to
describe a 2M
star and remain causal; a nontrivial condition
for the second set of nonrelativistic models.
In the case of the RMF models one can consider also
their extensions allowing for the presence of hyperons. Vector-
meson couplings to hyperons are obtained assuming the SU(6)
symmetry. Repulsion in the hyperon sector associated with
their coupling to a hidden-strangeness vector-isoscalar meson
φ allows for M>2M
. We also study how adding the hidden-
strangeness scalar-isoscalar meson σ
to get a weak
attraction softens the EOS. In principle the same exercise could
be done for the nonrelativistic models. However, the present
uncertainties in the hyperon-nucleon and hyperon-hyperon
interactions are such that the introduction of hyperon degrees
of freedom is still extremely model dependent. In particular,
the most sophisticated many-body approaches available in
the literature [18] either did not yet succeeded in producing
2M
stars, or cannot deal with the full baryonic octet [19].
However even in the case of RMF, strong uncertainties are
associated to the couplings. We make all the EOS used here
available in the Supplemental Material and via the CompOSE
database [20].
Within our large set of unified EOS we will study the
dependence of the NS radius and the thickness of the crust
on the mass in order to pin down the residual uncertainties
due to our imperfect knowledge of the EOS parameters. As
we remind in Sec. II, the EOS of nuclear matter near n
0
and for small neutron excess is constrained by the semi-
empirical evaluations of nuclear matter parameters extracted
from nuclear physics data. We will seek the correlations
between theoretically calculated nuclear matter parameters
near n
0
and NS structure. We will specifically show that the
best correlation is obtained between the radius of light NS
with M 1.4M
and the symmetry energy slope L.This
confirms that indeed the L parameter is the most important
one to be constrained from laboratory experiments and/or ab
initio calculations. A most crucial constraint could potentially
come from the threshold density above which the direct Urca
(DUrca) process operates. Indeed the interval of L which is
compatible with terrestrial constraints largely overlaps with the
one for which the nucleonic DUrca process does not operate
in massive NS. In turn, the presence of nucleonic DUrca
appears to be needed in order to explain the thermal states
of accreting neutron stars [21]. This means that combining
radii measurements with observations of thermal states of NS
might constitute a very stringent test for the EOS.
The plan of the paper is as follows. In Sec. II we
give a very general overview of nuclear matter in NS. We
also establish notations for nuclear matter and its relation
to the semi-empirical nuclear-matter parameters. Section III
describes the different techniques that are used to match the
crust and core EOS, and the resulting uncertainty associated
with the star radius and the crust thickness. The relativistic
and nonrelativistic unified EOS employed for this work are
described in Sec. IV, and the corresponding M(R) relations
are given. Section V contains the main results of this work.
The predictions for the radius and crust thickness are given,
the correlation between the radius and the EOS parameters
is discussed, and the different unified EOS are compared
to the terrestrial constraints. Potential constraints from the
necessity of DUrca processes to explain low-luminosity NS
are presented. Finally Sec. VI concludes the paper.
035804-2
NEUTRON STAR RADII AND CRUSTS: UNCERTAINTIES . . . PHYSICAL REVIEW C 94, 035804 (2016)
II. NUCLEAR MATTER IN NEUTRON STARS
AND SEMI-EMPIRICAL PARAMETERS
Consider the NS interior from the very basic point of view
of nuclear matter states relevant for each main NS layer. The
T = 0 approximation can be used since the Fermi energy of
the nucleons is much larger than the thermal energy associated
with the temperatures of ∼10
7
–10
9
K expected inside NS.
The outer core consists of a lattice of nuclear-matter droplets
permeated by an electron gas. The inner crust is made of a
lattice of nuclear-matter droplets coexisting with a neutron gas.
With increasing pressure, droplets can become unstable with
respect to merging into infinite nuclear matter structures (rods,
plates) immersed in a neutron gas. The plates of nuclear matter
then glue together leaving tubes filled with neutron gas, then
the tubes break into bubbles of neutron gas in nuclear matter.
Both the inner crust and the (possible) mantle of nuclear pastas
form inhomogeneous two-phase states of nucleon matter.
At the edge of the outer core, inhomogeneous nucleon
matter coexists with uniform homogeneous nuclear matter.
To model it, we consider a mixture of strongly interacting
neutrons and protons, with Coulomb interactions switched
off. Let us define the baryon number density n = n
n
+ n
p
and
the neutron excess parameter δ = (n
n
− n
p
)/n. The energy
per nucleon (excluding the nucleon rest energy) is E
NM
(n,δ).
Theoretical models of nuclear matter give E
NM
(n,δ) and yield a
set of parameters that characterize the EOS near the saturation
point (minimum of E
NM
) and for small δ. For a given model,
the minimum of energy per nucleon, E
s
, is reached at the
saturation density n = n
s
and for δ = 0.
The difference between the calculated values for the
saturation density n
s
and the commonly used normal nuclear
density n
0
defined in the first sentence of Sec. I deserves a
comment. The values of n
s
are model dependent and vary
between 0.146 and 0.154 fm
−3
for the RMF models (Table II)
and between 0.151 and 0.165 fm
−3
for the Skyrme models
(Table IV). The use of a precise value of n
s
is crucial for the
correct calculation of the EOS. In contrast, n
0
is just a chosen
baryon number density unit.
Let us define the so-called symmetry energy,
E
sym
(n) =
1
2
∂
2
E
NM
∂δ
2
δ=0
, (1)
and its value at saturation,
J = E
sym
(
n
s
)
. (2)
Two additional parameters related to the first and second
derivatives of the symmetry energy at the saturation point are,
respectively, the symmetry-energy slope parameter L,
L = 3n
s
dE
sym
dn
n
s
, (3)
and the symmetry incompressibility K
sym
,
K
sym
= 9n
2
s
d
2
E
sym
d
2
n
n
s
. (4)
Finally, the incompressibility at saturation K is
K = 9n
2
s
∂
2
E
NM
∂n
2
n
s
,δ=0
. (5)
Knowledge of parameters {n
s
,E
s
,K,J, . . . ,} is sufficient
to reproduce theoretical EOS of nuclear matter near the
saturation point, a situation characteristic of laboratory nuclei.
However, after being fine-tuned at the saturation point, the
energy-density functionals are actually extrapolated up to n ∼
8n
s
8n
0
and δ 1, characteristic of the cores of massive
NS. Therefore, making {n
s
,E
s
,K,J, . . . ,} consistent with the
semi-empirical evaluations of these parameters obtained, using
a wealth of experimental data on atomic nuclei, yields con-
straints on the corresponding EOS of NS, and consequently,
NS models, and in particular NS radii.
III. NON-UNIFIED EQUATIONS OF STATE
AND CORE-CRUST MATCHING
In the present section we will discuss the problem of the
core-crust matching of the EOS when a non-unified EOS is
used to describe stellar matter. The use of a non-unified EOS
will be shown not to affect the determination of the NS mass
but to have a significant influence on the radius calculation.
A. Different procedures for core-crust matching
The determination of the mass and radius of a NS is
possible from t he integration of the Tolman-Oppenheimer-
Volkoff (TOV) equations for spherical and static relativistic
stars [22], given the EOS of stellar matter P (ρ), where P is
the pressure and ρ the energy density. The EOS for the whole
NS is generally obtained by the matching of three different
segments: the first one for the outer crust, the second one for
the inner crust, and the last one for the core. The EOS for
the outer crust, which extends from the surface to the neutron
drip density, requires knowledge of the masses of neutron-rich
nuclei [12,23–25]. This information comes from experiments
or, when no information exists, from some energy-density
functional calculations. The inner crust corresponds to a
nonhomogeneous region between the neutron drip and the
crust-core transition. This region may include nonspherical
nuclear clusters, generally known as pasta phases [26], and has
been described within several approaches [27–34]. Finally the
core formed by a homogeneous liquid composed of neutrons,
protons, electrons, muons, and possibly exotic matter, in β
equilibrium, extends from the crust-core t ransition to the center
of the star. It should be pointed out, however, that in addition
to exotic phases which can possibly appear at high densities,
matter may also be nonhomogeneous in the core, e.g., in the
form of a mixed hadron-quark phase [35]. In the present work
we consider a homogeneous core.
Since the core accounts for most of the mass and radius
of the star, authors frequently work with a non-unified EOS,
and match the core EOS to one for the crust, in particular the
Baym-Pethick-Sutherland (BPS) [12] together with the Baym-
Bethe-Pethick (BBP) [36], the Negele-Vautherin (NV) [16],
or the Douchin-Haensel (DH) [14]. The matching is generally
done so that the pressure is an increasing function of the energy
035804-3
M. FORTIN et al. PHYSICAL REVIEW C 94, 035804 (2016)
FIG. 1. Mass versus radius (left) and crust thickness l
cr
versus
mass (right) for the relativistic mean field model GM1, using different
matching procedures (see text).
density. This condition still leaves a quite large freedom in
the matching procedure. In principle the matching procedures
done at a specific density should be performed using a Maxwell
construction, i.e., at constant baryonic chemical potential, so
that the pressure is an increasing function of both the density
and the chemical potential.
In the following a non-unified EOS is built from two
different EOS. The one for the crust defined by P
cr
, ρ
cr
, n
cr
is
used up to P
1
, ρ
1
, n
1
, while another one for the core, P
co
, ρ
co
,
n
co
, is considered above P
2
, ρ
2
, n
2
. The matching is performed
in the region of pressure P
1
P P
2
, and if P
1
= P
2
a linear
interpolation between (P
1
,ρ
1
) and (P
2
,ρ
2
) is considered. The
pressures P
1
and P
2
are generally defined at a reference density
such as the neutron drip density n
d
, the crust-core transition
density n
t
, the saturation density n
0
, and the density n
c
where
the two EOS cross.
In Fig. 1 we plot the radius-mass curves (left) and the crust
thickness (right) versus the star mass obtained with the GM1
parametrization [37] with a purely nucleonic core obtained for
different gluing procedures:
(1) Unified: by unified we mean an EOS built with the
DH EOS for the outer crust (n 0.002 fm
−3
) and the
inner crust and core obtained within the same model,
here GM1. The inner crust was calculated within a
Thomas-Fermi calculation of the pasta phase [38] and
the core EOS matches the inner crust at the crust-core
transition density n
t
;
(2) n
1
= 0.01 fm
−3
: the crust BPS+BBP EOS is glued to
the core EOS at 0.01 fm
−3
as indicated in [11];
(3) n
1
= n
c
: the gluing is done at the density where the
DH EOS and the core EOS cross as in [13];
(4) n
1
= n
t
: the DH EOS is considered for the crust and
homogeneous matter EOS for n>n
t
.
(5) n
1
= n
0
:theDHEOSisusedforn<n
0
and the core
EOS above the saturation density n
0
.
(6) n
1
= 0.5n
0
, n
2
= n
0
: DH EOS is used for n<0.5n
0
,
the homogeneous matter EOS is used above n
0
.
TABLE I. NS radii R
1
and R
1.4
(in km) and crust thicknesses
l
cr
1
and l
cr
1.4
(in km) for masses of 1.0M
and 1.4M
for different
matchings between the core and the crust. x (in %) for a given
quantity x corresponds to the relative difference between the value of
x for unified EOS and the one for a given matching. Three functionals
are considered: NL3, NL3ωρ , and GM1.
R
1
R
1
R
1.4
R
1.4
l
cr
1
l
cr
1
l
cr
1.4
l
cr
1.4
GM1
unified 13.71 13.76 1.62 1.09
n = 0.01 13.86 1.09 13.86 0.73 1.78 9.88 1.19 9.17
n
t
14.12 2.99 13.92 1.16 1.64 1.23 1.10 0.92
n
0
13.61 −0.73 13.70 −0.44 2.04 25.93 1.36 24.77
0.5n
0
–n
0
13.96 1.82 13.92 1.16 2.00 23.46 1.33 22.02
0.1n
0
–n
t
14.27 4.08 14.12 2.62 2.18 34.57 1.44 32.11
Max. diff. 0.66 0.42 0.56 0.35
NL3
unified 14.54 14.63 1.91 1.30
n = 0.01 14.78 1.65 14.78 1.03 2.15 12.57 1.45 11.54
n
c
14.97 2.96 14.91 1.91 2.35 23.04 1.58 21.54
n
t
14.96 2.89 14.90 1.85 2.34 22.51 1.57 20.77
n
0
14.00 −3.71 14.26 −2.53 2.02 5.76 1.42 9.23
0.5n
0
–n
0
14.47 −0.48 14.57 −0.41 2.17 13.61 1.50 15.38
0.1n
0
–n
t
15.09 3.78 14.97 2.32 2.46 28.80 1.65 26.92
Max. diff. 1.09 0.71 0.55 0.35
NL3ωρ
unified 13.42 13.75 2.02 1.43
n = 0.01 13.51 0.67 13.81 0.44 2.11 4.46 1.49 4.20
n
c
13.5 1.12 13.85 0.73 2.18 7.92 1.53 6.99
n
t
13.5 0.60 13.8 0.36 2.1 3.96 1.48 3.50
n
0
13.49 0.52 13.8 0.36 2.1 3.96 1.48 3.50
0.5n
0
–n
0
13.51 0.67 13.81 0.44 2.11 4.46 1.49 4.20
0.1n
0
–n
t
13.49 0.52 13.8 0.36 2.1 3.96 1.48 3.50
Max. diff. 0.15 0.10 0.16 0.10
(7) n
1
= 0.1n
0
, n
2
= n
t
: a low matching of the EOS is
considered. The DH EOS is used for n<0.1n
0
and
the core EOS above n
t
.
If the matching is defined at a given density n
m
= n
1
,the
gluing is done imposing P
2
= P
1
. The curves do not coincide
because the matching has been performed in different ways.
While the maximum mass allowed for a stable star is not
affected by the chosen crust-core matching, the same is not
true for the radius and crust thickness of stars with a standard
mass of ∼1.4M
. The two EOS considered in this example
for the crust and the core have quite different properties at
saturation density, in particular for the density dependence
of the symmetry energy; see Table II. This situation is,
however, common in the literature. In fact, the GM1 EOS was
parametrized to describe both nuclear saturation properties and
neutron star properties.
In Table I, the radius and crust thickness of 1.0 and 1.4M
NS are given for three models, GM1, NL3 [ 39], and NL3ωρ
[40], and several matching schemes, together with relative
differences with respect to the value for the unified EOS.
As expected the crust-core matching affects more strongly
the less massive stars. Depending on the matching procedure,
035804-4
NEUTRON STAR RADII AND CRUSTS: UNCERTAINTIES . . . PHYSICAL REVIEW C 94, 035804 (2016)
the differences in the radius and the crust thickness for a 1.0M
star can be as large as ∼1 and ∼0.5 km, respectively. This
corresponds to relative differences as large as ∼4% for the
radius and ∼30% for the crust thickness. This is to be compared
with the expected precision of ∼5% on the radius measurement
from future x-ray telescopes (NICER, Athena, . . . ). Similarly
the crust thickness differs by ∼0.5 km depending on the gluing.
This quantity is particularly important for the study of the ther-
mal relaxation of accreting NS [41,42], the glitch phenomenon
[17,43,44], the torsional crustal vibrations, and the maximum
quadrupole ellipticity sustainable by the crust [45].
The differences between matchings are much smaller if
the NL3ωρ core EOS is considered, because this model has
nuclear matter saturation properties similar to the ones of the
SLy4 parametrization [15]usedintheDHEOS.
B. Matching and thermodynamic inconsistency
Two basic methods can be used in order to match two EOS
for the crust and the core: the first based on the P (n) relation
and the second on the P (ρ) function.
The first method consists of treating the baryon number
density as an independent variable. Consider an EOS for the
crust, P
cr
(n) and ρ
cr
(n), and another one for the core, P
co
(n)
and ρ
co
(n).
Let us assume that the matching region lies between two
densities, n
1
and n
2
>n
1
. First let us build the matched P (n)
function. For n<n
1
, P (n) = P
cr
(n) and for n>n
2
, P (n) =
P
co
(n). In the matching region, one can assume a form (usually
linear or logarithmic) for the function P (n) such that P (n
1
) =
P
1
= P
cr
(n
1
) and P (n
2
) = P
2
= P
co
(n
2
).
Then one needs to build the f unction ρ(n). For n<n
1
,
ρ(n) = ρ
cr
(n). Let us define the chemical potential at the den-
sity n
1
: μ
1
= [P
1
+ ρ
cr
(n
1
)]/n
1
. By imposing thermodynamic
consistency, the value of chemical potential μ at a density n in
the matching region can be derived using the P (n) relation:
μ(n) = μ
1
+
n
n
1
dP( n)
n
. (6)
The matched energy density equals
ρ(n) = nμ(n) − P (n). (7)
However this technique generally leads to thermodynamic
inconsistency with the core EOS: the value of chemical
potential μ
2
at the density n
2
obtained from Eq. (6) differs
from μ
co
(n
2
) = [P
2
+ ρ
co
(n
2
)]/n
2
. As a consequence ρ(n
2
)
given by Eq. (7) is different from ρ
co
(n
2
). In order to get a
thermodynamically consistent EOS for n>n
2
one has to add
a constant value (an energy shift):
μ = μ(n
2
) − μ
co
(n
2
)(8)
to the chemical potential in the core. Then the energy density
ρ(n)forn>n
2
is
ρ(n) = ρ
co
(n) + nμ. (9)
Of course, such a procedure affects the whole EOS for the
core, but the main effect on the M(R) relation is for NS with
a central pressure close to P
2
.
The second method considers the energy density ρ as an
independent variable. This can be motivated by the TOV
equations since this quantity and the function P (ρ) actually
enter the stress-energy tensor in the Einstein equations. Thus
the EOS can be written in the form P
cr
(ρ) and n
cr
(ρ)forthe
crust and P
co
(ρ) and n
co
(ρ) for the core. The matching region
is defined such that ρ
1
<ρ<ρ
2
.
The first step consists of obtaining the function P (ρ).
For ρ<ρ
1
, P (ρ) = P
cr
(ρ) and for ρ>ρ
2
, P (ρ) = P
co
(ρ).
Similarly to the first method one can assume a form for the
function P (ρ) in the matching region such that P (ρ
1
) = P
1
=
P
cr
(ρ
1
), P (ρ
2
) = P
2
= P
co
(ρ
2
).
Then one wants to derive the relation n(ρ). For ρ<ρ
1
one has n(ρ) = n
cr
(ρ). Let us define n
1
= n
cr
(ρ
1
). Assuming
thermodynamic consistency, in the matching region, i.e., ρ
1
ρ<ρ
2
, one gets
n(ρ) = n
1
exp
ρ
ρ
1
dρ
P (ρ) + ρ
. (10)
However, as for the first method, this construction does not
ensure that n(ρ
2
) obtained from the previous formula is equal to
n
co
(ρ
2
). A similar conclusion can be reached for the chemical
potential at ρ
2
. Thus one has to modify the n(ρ) dependence for
the core EOS in order to ensure thermodynamic consistency.
For ρ>ρ
2
, t he matched EOS is
n(ρ) = n
co
(ρ)
n(ρ
2
)
n
co
(ρ
2
)
. (11)
This approach does not affect the P (ρ) relation (nor the
gravitational mass and the radius), but strictly speaking the
microscopic model of dense matter is changed since it is
the baryon number density which is the basic quantity for
the theoretical calculations, within the many-body theory, of
dense matter properties. Of course the accepted procedure
givenbyEq.(11) also influences the value of a baryon chemical
potential (dividing it by the same factor).
C. Thermodynamic consistency and causality
In principle, when gluing two EOS one should match all
thermodynamic quantities: the pressure P , the energy density
ρ, and the baryonic density n. In other words, a pair of
functions for the pressure and the energy density should be
constructed so that thermodynamic consistency is fulfilled.
Let us consider the EOS for the core and the crust, this
time in terms of the chemical potential μ, P
cr
(μ) in the crust
and P
co
(μ) in the core. The matching region is defined by
μ
1
<μ<μ
2
. Let us define P
cr
(μ
1
) = P
1
and P
co
(μ
2
) = P
2
.
The function P (μ) in the matching region and its first
derivative, which is the baryon number density n, should fulfill
the conditions of continuity given by
P (μ
1
) = P
1
,P(μ
2
) = P
2
. (12)
Thermodynamic consistency and causality imply that the
following conditions on the derivatives must be fulfilled in the
matching region:
(1) n is an increasing function of P , i.e., P (μ) is increasing
and convex;
035804-5
![FIG. 11. Energy per particle of neutron matter as a function of density n for Skyrme models and constraints by [93] and [94].](/figures/fig-11-energy-per-particle-of-neutron-matter-as-a-function-1kcgb6n5.webp)
