Table Of ContentNuclear Equation of State and Neutron Star Cooling
Yeunhwan Lima, Chang Ho Hyunb, Chang-Hwan Leec
aRare Isotope Science Project, Institute for Basic Science, Daejeon 305-811, Republic of Korea
bDepartment of Physics Education, Daegu University, Gyeongsan 712-714, Republic of Korea
cDepartment of Physics, Pusan National University, Busan 609-735, Republic of Korea
Abstract
5
1 We investigate the effects of the nuclear equation of state (EoS) to the neutron star cooling. New
0 era for nuclear EoS has begun after the discovery of 2M neutron stars PSR J1614 2230
(cid:12)
2 ∼ −
and PSR J0348+0432 [1, 2]. Also recent works on the mass and radius of neutron stars from
n
low-mass X-ray binaries [3] strongly constrain the EoS of nuclear matter. On the other hand,
a
J observationsoftheneutronstarinCassiopeiaA(CasA)morethan10yearsconfirmedtheexistence
9 of nuclear superfluidity [4, 5]. Nuclear superfluidity reduces the heat capacities as well as neutrino
1
emissivities. With nuclear superfluidity the neutrino emission processes are highly suppressed, and
] the existence of superfluidity makes the cooling path quite different from that of the standard
h
cooling process. Superfluidity also allows new neutrino emission process, which is called ‘Pair
t
- Breaking and Formation’(PBF). PBF is a fast cooling process and can explain the fast cooling rate
l
c
of neutron star in Cas A. Therefore, it is essential to add the superfluidity effect in the neutron star
u
n cooling process. In this work, we simulate neutron star cooling curves using both non-relativistic
[ and relativistic nuclear models. The existence of too early direct Urca process shows that some of
1 nuclear models do not fit for the cooling simulation. After this first selection process, the nuclear
v pairing gaps are searched using the observational neutron star’s age and temperature data.
7
9 Keywords: Neutron star, Nuclear equation of state, Neutron star cooling, Nuclear superfluidity
3
4
0
. 1. Introduction
1
0
5 A neutron star (NS) is born as a result of the core collapse supernova explosion if the initial
1 mass of the main sequence star is around 8 to 20 solar mass (M ). Various EoS, relations between
(cid:12)
:
v thedensityandthepressure,areusedtodescribetheinteriorofneutronstars. Theresultingcentral
i density of a neutron star is expected to reach up to several times of the nuclear saturation density
X
(n 0.16 fm−3). Hence, a neutron star is one of the best astrophysical laboratories to study the
r 0
(cid:39)
a physics of the extremely dense nuclear matter.
Figure 1 shows the structure of neutron stars in the theoretical point of view. Outside the
envelope, there exists a very thin atmosphere which is composed of hydrogen and heavy elements
[6]. Depending on the temperature, its thickness ranges from a few centimeters to a few millimeters
andthethermalradiationisexpectedtooccurinthisregion. Theenvelopeplaysaroleasathermal
insulator [7] bewteen the outer crust and the atmosphere. In the outer crust, heavy nuclei exist
Email addresses: [email protected] (Yeunhwan Lim), [email protected] (Chang Ho Hyun),
[email protected] (Chang-Hwan Lee)
Preprint submitted to Elsevier January 20, 2015
Envelope
Outercrust ρ∼ρs(=108g/cm3)
Z,e
ρ 1010g/cm3 BCClattice
∼
Innercrust ρ∼ρd(=4×1011g/cm3)
n,Z,e n1S superfluidity
0
Outercore ρ∼0.5ρ0(=2×1014g/cm3)
n,p,e,µ Uniformnuclearmatter
8 n3P2,p1S0superfluidity
∼1 Innercore ρ∼2ρ0
5
k
m Hyperons? Bose(K−,π−)condensation
Hyperon1S superfluidity
Quarks? 0
Colorsuperconductivity
Figure 1: (Color online) A schematic of neutron star structure.
with free gas of electrons, and the BCC (Body Centered Cubic) structure of nuclei is expected.
As the density increases, neutrons drip out of heavy nuclei and can exist as free gas in the inner
crust. The neutron 1S superfluidity can also exist in the inner crust. In the boundary of the
0
outer core and the inner crust, nuclear pasta phase may appear due to the nature of Coulomb
interaction [8, 9], which favours a special geometrical shape to reduce the free energy. Above half
of the nuclear saturation density uniform nuclear matter exists. Outer and inner cores occupy
roughly 70 % of neutron star’s volume. In the outer core the uniform nuclear matter can exist and
neutron 3P and proton 1S superfluidities are expected. In the inner core exotic states with new
2 0
degrees of freedom, such as kaons, hyperons, and deconfined quarks might appear. However, the
inner structure of neutron star is still unknown due to the theoretical uncertainties, especially at
the densities far beyond the nucleon saturation density.
The mass distributions of observed neutron stars are summarized in Table 1 of Lattimer [10].
NotethatthemassdistributionofNS-NSbinarieshasverysharppeakssincetheerrorbarsofmass
measurements are relatively small, and the well measured neutron stars in NS-NS binaries have
masses < 1.5M . On the other hand, 2M neutron stars have been observed in NS-WD (white
(cid:12) (cid:12)
dwarf) binaries. This implies that the mass distribution may depend on the binary evolution in
addition to the neutron star equation of states [11]. Recent observations of 2M neutron stars
(cid:12)
∼
ruled out many soft equations of states with which the maximum mass of neutron star becomes less
than 2M [1, 2]. Another constraint on the nuclear equation of state comes from the X-ray binary
(cid:12)
observations and their analysis [3]. Mass distribution in X-ray binaries is very broad without sharp
peak indicating that the uncertainties in the mass estimations are very large.
In Table 1 and Fig. 2, we plot the surface temperatures (T∞) and photon luminosities (L∞)
s
of 18 observed neutron stars. Note that three different models are used to estimate the effective
surfacetemperaturesofneutronstarsatinfinity. Forblackbody(BB)model,thephotonluminosity
2
(L∞) and the effective surface temperature (T∞) of a neutron star at infinity are related by
s
L∞ = 4π(R∞)2σ (T∞)4 (1)
SB s
where R∞ is the radius seen by an observer at infinity and σ is the Stefan-Boltzmann constant.
SB
We can determine R∞ by inserting T∞ and L∞ in the Table 1 into Eq. (1).
s
In recent works [12, 13], we have investigated the behavior of nuclear EoS at high densities
by calculating the mass and radius of neutron stars with several Skyrme force models. We could
confirm that the models that satisfy the large mass observation [1, 2] are consistent with the mass-
radius zone in [3]. Moreover, the conclusion doesn’t change even if we include exotic degrees of
freedom such as kaon condensation [12] or hyperons [13]. On the other hand, nuclear EoS is one of
thekeyingredientsthatdeterminethethermalevolutionofneutronstars. Availablenuclearmodels
predict very diverse mass-radius relations, so it is well expected that the cooling behavior will be
sensitive to nuclear models. Therefore, cooling of neutron stars provides a multi-test ground for
the nuclear models; whether a model good for mass-radius relation is consistent with the observed
temperature data or not.
In this work, we simulate the cooling curve of neutron stars with various EoS obtained from
both relativistic and non-relativistic models. Without the superfluidity effect, the direct Urca
process is a good indicator whether a specific nuclear model is suitable for neutron star cooling
simulation. If the temperature drop is too fast even for the low mass neutron stars ( 1.2 M ),
(cid:12)
∼
such EoS is not consistent with most of the neutron star temperature observations. Even though
the existence of superfluidity highly suppresses neutrino emissivity, once the direct Urca process
is turned on before the critical temperature, the superfluidity cannot slow down the temperature
drop so that the temperature of neutron stars is much below any observational data.Thus, the
study of cooling curve of neutron star can provide hints about the inner structure of neutron stars
and the EoS of dense nuclear matter. In addition to the nuclear EoS, physical conditions such as
the composition of elements in the envelope, and existence of superfluidity in the core play crucial
roles in determining the cooling curve of neutorn stars. In general, chemical composition of the
envelope seldom affects the cooling mechanism, but the surface temperature depends sensitively on
whether the elements in the envelope are light or heavy. On the other hand, superfluidity directly
determines the cooling rate. If nucleons form a cooper pair and transit to a superfluidic state, the
rate of neutrino emissivity is suppressed exponentially. This may lead to a very slow cooling rate.
However, belowacriticaltemperature, creationanddestructionofcooperpairsigniteafastcooling
mechanism, and this can give an abrupt decrease of temperature. Recent literature succeeds to
reproduce the cooling curve of Cas A in terms of PBF [4, 5]. In this work, we incorporate PBF to
various nuclear models and explore the extent to which the models can reconcile with PBF .
The structure of the paper is as follow. In Section 2 we summarize the equation of state of
nuclear matter which we used in the study of thermal evolution of neutron stars. In this work,
we do not consider exotic matter or quark matter in the core of neutron stars but assume that
the core is composed of nucleons (protons and neutrons) and leptons (electrons and muons) in
the form of uniform matter. In Section 3 we present the basic equations for neutron star cooling.
Simple critical temperature formula is proposed and we analyze the pairing effects to neutron star
cooling. In Section 4, we present the cooling curves for the standard cooling in which superfluidity
is neglected with various nuclear models. In Section 5, we discuss the effect of the superfluidity to
the cooling process. In Section 6, we give conclusions from neutron star cooling curves combined
with varios EoS. In Appendix, we summarize numerical methods to solve the diffusion equations
for the thermal evolution of neutron stars.
3
Table 1: Thermal emission from isolated neutron stars. Temperatures were obtained using three
different models; hydrogen atmosphere (HA), magnetized hydrogen atmosphere (mHA), and black-
body (BB) model. For sources with no. 2 7, two different models were used to link the effective
∼
temperature and the photon luminosity. Sources with no. 16 17 have the limited observational
∼
data, thus have only one limit.
No Source Log(t /yr) Log(t /yr) Log(T∞/K) Log(L∞/erg s−1) Model Ref.
sd kin s
1 PSR J1119-6127 3.20 - 6.08+0.09 32.88 - 33.66 mHA [14]
−0.07
6.24+0.04 33.85 - 34.00 HA
2 RX J0822-4247† 3.90 3.57+0.04 −0.04 [15, 16]
−0.04 6.65+0.04 33.60 - 33.90 BB
−0.04
6.21+0.07 33.27 - 33.74 HA
3 1E 1207.4-5209 5.53+0.44 3.85+0.48 −0.07 [17, 16]
−0.19 −0.48 6.48+0.01 33.27 - 33.74 BB
−0.01
5.88+0.04 32.46 - 32.80 mHA
4 PSR J1357-6429 3.86 - −0.04 [18]
6.23+0.05 32.35 - 32.76 BB
−0.05
6.03+0.03 33.08 - 33.33 HA
5 RX J0002+6246 - 3.96+0.08 −0.03 [19, 16]
−0.08 6.15+0.11 32.18 - 32.81 BB
−0.11
5.83+0.02 32.41 - 32.70 mHA
6 PSR B0833-45† 4.05 4.26+0.17 −0.02 [20, 16]
−0.31 6.18+0.02 32.04 - 32.32 BB
−0.02
5.80+0.13 31.81 - 32.93 mHA
7 PSR B1706-44 4.24 - −0.13 [21, 16]
6.22+0.04 32.48 - 33.08 BB
−0.04
8 PSR J0538+2817 4.47+0.05 - 5.94+0.08 32.32 - 33.33 mHA [22]
−0.06 −0.08
9 PSR B2334+61 4.61 - 5.84+0.08 31.93 - 32.96 mHA [23]
−0.08
10 PSR B0656+14 5.04 - 5.71+0.03 32.18 - 32.97 BB [24, 16]
−0.04
11 PSR B0633+1748† 5.53 - 5.75+0.04 30.85 - 31.51 BB [25, 16]
−0.05
12 RX J1856.4-3754 - 5.70+0.05 5.63+0.08 31.32 - 32.35 mHA [26, 16]
−0.25 −0.08
13 PSR B1055-52 5.73 - 5.88+0.08 32.05 - 33.08 BB [27, 16]
−0.08
14 PSR J0243+2740 6.08 - 5.64+0.08 29.10 - 30.13 mHA [23]
−0.08
15 RX J0720.4-3125 6.11 - 5.70+0.08 31.37 - 32.40 HA [28]
−0.08
16 PSR J0205+6449†† - 2.91 <6.01 <33.29 BB [29]
17 PSR B0531+21†† - 3.0 <6.30 <34.45 BB [30]
18 RX J0007.0+7303†† - 4.0-4.4 <5.82 <32.54 BB [31]
† Alternative names: Puppis A (PSR B0822-4247), Vela (PSR B0833-45), Geminga (PSR B0633+1748).
†† PSR J0205+6449 is a pulsar in supernova remnant 3C 58, PSR B0531+21 is in SN 1054 in Crab Nebula,
and RX J0007.0+7303 is in the CTA1.
4
36
6.6 1. PSR J1119-6127 (mHA) 1. PSR J1119-6127 (mHA)
2. RX J0822-4247 (HA) 35 2. RX J0822-4247 (HA)
3. 1E 1207.4-5209 (HA) 3. 1E 1207.4-5209 (HA)
6.4 4. RX J1357-6429 (mHA) 4. RX J1357-6429 (mHA)
56.. RPSXR J 0B000823+36-4254 6(m (HHAA)) 17 2 3 1)−34 56.. RPSXR J 0B000823+36-4254 6(m (HHAA)) 16 17 2
Log(/K)T∞s656...082 7891111...0123 PPP.... SSSPPRPRRRSSSXRRR BJBJ0 1BBB12587300135036608664535+.-+635442++--64538111 271( 47 5m7(( 4m4B ((H8B BmH( ABm()AHB)))HBAA))) 1 16 4 5 8 9 13 15 g(/erg sL∞33 7891111...0123 PPP.... SSSPPRPRRRSSSXRRR BJBJ0 1BBB12587300135036608664535+.-+635442++--64538111 271( 47 5m7(( 4m4B ((H8B BmH( ABm()AHB)))HBAA))) 1 4 3 8 10 13 15
14. PSR J0243+2740 (mHA) o32 14. PSR J0243+2740 (mHA)
5.6 111567... RPPSSXRR J 0JB0702250035.14++-36214124 59(B ((BHB)AB)) 7 6 18 1011 12 14 L31 111567... RPPSSXRR J 0JB0702250035.14++-36214124 59(B ((BHB)AB)) 75618 12
5.4 18. RX J0007.0+7303 (BB) 18. RX J0007.0+7303 (BB) 11
14
5.2 30
0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7
Log(t/yr) Log(t/yr)
Figure 2: (Color online) The effective temperature and the photon luminosity for observed neutron
stars. Left panel : the effective temperature at infinity. Right panel : photon luminosity seen by
the observer at infinity. Note that three different models are used to link the effective temperature
and the luminosity as in Table 1.
2. Neutron Star Equation of State
Nuclear matter properties beyond nuclear saturation density are still unknown and different
EoSs give quite different masses and radii of neutron stars. In order to investigate the properties of
neutron star matter, we first consider both relativistic and non-relativistic models for the neutron
star core, which are consistent with 2.0M neutron stars [1, 2]. We consider the crust of neutron
(cid:12)
star separately because heavy nuclei can exist in the crust. The properties of neutron star crust is
very important in understanding the neutron star properties in low-mass X-ray binaries.
2.1. Non-relativistic nuclear force model : energy density functional
For the non-relativistic nuclear force model, we use Skyrme force model to obtain the equation
of state for nuclear matter [32]. In Skyrme force model, the interaction between two nucleons has
the form of
t (cid:104) (cid:105)
vˆ (r ,r ) = t (1+x Pˆ )δ(r r )+ 1(1+x Pˆ ) δ(r r )kˆ2+kˆ†2δ(r r )
SF i j 0 0 σ i j 1 σ i j i j
− 2 − −
1
+t (1+x Pˆ )kˆ† δ(r r )kˆ + t (1+x Pˆ )nαδ(r r )
2 2 σ i j 3 3 σ i j
· − 6 −
+iW kˆ†δ(r r ) kˆ (σˆ +σˆ ), (2)
0 i j i j
− × ·
where Pˆ = 1(1+σˆ σˆ ) is the spin-exchange operator, t , x and α are the parameters of the
σ 2 i · j i i
interactions, and kˆ is defined as
1
kˆ = ( ). (3)
i j
2i ∇ −∇
Note that the interaction contains terms up to quadratic in derivatives, t -term is added to include
3
many body effect beyond quadratic in density n, and W -term gives the spin-orbit interaction
0
which is important to explain the nuclear structure. Parameters used for Skyrme force models are
summarized in Table 2.
5
Table 2: Parameters used for Skyrme force models.
SLy4 SkI4 SGI SV TOV min LS 220
t (MeV fm3) -2488.913 -1855.83 -1603.0 -1248.3 -2129.735 -2182.73
0
·
t (MeV fm5) 486.818 473.829 515.9 970.6 305.398 0
1
·
t (MeV fm5) -546.395 1006.855 84.5 107.22 362.532 0
2
·
t (MeV fm3+3α) 13777.0 9703.607 8000.0 0.0 13957.064 14960.296
3
·
x 0.8340 0.405082 -0.02 -0.17 0.169949 -0.3121
0
x -0.3438 -2.889148 -0.5 0.0 -3.399480 0
1
x - 1.0 -1.325150 -1.731 0.0 -1.782177 0
2
x 1.3540 1.145203 0.1381 1.0 0.402634 -0.5
3
α 1/6 0.25 1/3 1 0.2504 0.2602
(cid:126)2
20.7525 20.7525 20.7525 20.7525 20.7498207 20.7484
2mp
(cid:126)2
20.7525 20.7525 20.7525 20.7525 20.7212601 20.7484
2mn
In Hartree-Fock level, the total energy can be expressed as
(cid:88) 1 (cid:88)
E = i tˆj ρ + v¯ ρ ρ (4)
ji ijkl ki lj
(cid:104) | | (cid:105) 2
ij ijkl
where tˆis the kinetic energy operator and
v¯ = ij vˆ(1 Pˆ Pˆ Pˆ ) kl . (5)
ijkl σ r τ
(cid:104) | − | (cid:105)
Here Pˆ is the parity operator and Pˆ = 1(1+τˆ τˆ ) is the iso-spin exchange operator. Due to the
r τ 2 i· j
zero range property of Skyrme force model, the total energy can be easily obtained as
(cid:90) (cid:90)
E = d3r = d3r( + + + ) (6)
B C g J
E E E E E
where is the bulk part contribution, is the Coulomb contribution, is the contribution
B C g
E E E
from the density gradient term, and is the contribution from the spin-orbit term. For a uniform
J
E
matter in the neutron star core, is dominant. Hence the energy density can be approximated
B
E
as [33]
(cid:126)2 (cid:126)2 (cid:20)t (cid:16) x (cid:17) t (cid:16) x (cid:17)(cid:21)
1 1 2 2
= τ + τ +n(τ +τ ) 1+ + 1+
B n p n p
E (cid:39) E 2m 2m 4 2 4 2
n p
(cid:20) (cid:21)
t (cid:16)1 (cid:17) t (cid:16)1 (cid:17)
2 1
+(τ n +τ n ) +x +x
n n p p 2 1
4 2 − 4 2
(cid:20) (cid:21)
t (cid:16) x (cid:17) (cid:16)1 (cid:17)
+ 0 1+ 0 n2 +x (n2 +n2)
2 2 − 2 0 n p
(cid:20) (cid:21)
t (cid:16) x (cid:17) (cid:16)1 (cid:17)
+ 3 1+ 3 n2 +x (n2 +n2) nα, (7)
12 2 − 2 3 n p
6
Table 3: Nuclear matter properties at the saturation density (n ). Upper 6 models correspond
0
to non-relativistic Skyrme force models and lower 6 models correspond to relativistic mean field
models. B is the binding energy of the symmetric nuclear matter, S is the symmetry energy, L
v
is the slope of the symmetry energy, K is the compression modulus, and m(cid:63) is Landau effective
N
nucleon mass (effective chemical potential).
Model n (fm−3) B (MeV) S (MeV) L (MeV) K (MeV) m(cid:63) /m Ref
0 v N N
SLy4 0.160 16.0 32.0 45.9 230 0.694 [35]
SkI4 0.160 16.0 29.5 60.4 248 0.649 [36]
SGI 0.155 15.9 28.3 63.9 262 0.608 [37]
SV 0.155 16.1 32.8 96.1 306 0.383 [38]
TOV-min 0.161 15.9 32.3 76.2 222 0.934 [39]
LS220 0.155 16.0 28.6 73.1 220 1.000 [40]
IU FSU 0.155 16.4 31.3 47.2 231 0.669 [41]
DD-MEδ 0.152 16.1 32.4 52.9 219 0.668 [42]
SFHo 0.158 16.2 31.6 47.1 245 0.810 [43]
NLρ 0.160 16.1 30.4 84.6 241 0.800 [44]
TMA 0.147 16.0 30.7 90.1 318 0.691 [45]
NL3 0.148 16.2 37.3 118 272 0.655 [46]
where m and m are neutron and proton masses, n and n are the densities of neutrons and
n p n p
protons, the total baryon density n = n + n , and τ and τ are kinetic energy densities of
n p n p
neutrons and protons, respectively. The pressure can be obtained by taking a density derivative of
energy per baryon,
∂( /n)
P = n2 E . (8)
∂n
In the upper part of Table 3, we summarize the basic properties of nuclear matter for Skyrme force
models which are used in this work. In the upper part of Figure 3, we plot the pressure for Skyrme
force models for both symmetric and pure neutron matter. In the left panel of Figure 4, masses
and radii of neutron stars are summarized for Skyrme force models.
2.2. Relativistic mean field model
Relativistic mean field models have been very successful in explaining the nuclear properties,
such as binding energy, density profile, root mean square radius, etc. Based on these successes,
many efforts have been given to build RMF models which are suitable for both finite nuclei and
bulk nuclear matter properties. The typical RMF models are described by the Lagrangian [34],
(cid:20) (cid:21)
= ψ¯ i∂/ g ω/ 1g (cid:126)τ (cid:126)/b+g (cid:126)δ (cid:126)τ m +g σ 1e(1+τ )A/ ψ 1F Fµν
ω ρ δ N σ 3 µν
L − − 2 · · − − 2 − 4
1 1 1 1 1 1
+ ∂ σ∂µσ m2σ2 Ω Ωµν + m2ωµω R(cid:126) R(cid:126)µν + m2(cid:126)bµ (cid:126)b
2 µ − 2 σ − 4 µν 2 ω µ− 4 µν 2 ρ · µ
1 1
+ ∂ (cid:126)δ ∂µ(cid:126)δ m2(cid:126)δ2 V (σ,ωµω ,(cid:126)bµ (cid:126)b ) (9)
2 µ · − 2 δ − eff µ · µ
7
SymmetricMatter PureNeutronMatter
102 102
) )
3 3
m m
/f /f
V SLy4 V SLy4
e e
M SkI4 M SkI4
( SGI ( SGI
P101 P101
SV SV
TOVmin TOVmin
LS220 LS220
100 100
0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.2 0.3 0.4 0.5 0.6 0.7 0.8
n(fm−3) n(fm−3)
SymmetricMatter PureNeutronMatter
102 102
) )
3 3
m m
/f /f
V IU-FSU V IU-FSU
e e
M DD-MEδ M DD-MEδ
( SHFo ( SHFo
P101 P101
NLρ NLρ
TMA TMA
NL3 NL3
100 100
0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.2 0.3 0.4 0.5 0.6 0.7 0.8
n(fm−3) n(fm−3)
Figure 3: (Color online) Pressure of symmetric nuclear matter and pure neutron matter from
both non-relativistic Skyrme force models (upper panels) and relativistic mean field models (lower
panels). The shaded area is the result from Ref. [47]. For the pure neutron matter (right panels),
the upper (lower) shaded area in each plot represents the stiff (soft) equation of state.
8
2.5 2.5
2.0 2.0
)(cid:12) )(cid:12)
M1.5 M1.5
( (
M SLy4 M IU-FSU
SkI4 DD-MEδ
1.0 1.0
SGI SFHo
SV NLρ
0.5 TOVmin 0.5 TMA
LS220 NL3
0.0 0.0
8 9 10 11 12 13 14 15 16 8 9 10 11 12 13 14 15 16
R(km) R(km)
Figure 4: (Color online) Neutron star’s mass and radius relation from non-relativistic Skyrme force
models (left) and relativistic mean field models (right). Thick horizontal lines indicate the masses
of PSR J1614 2230 and PSR J0348+0432 [1, 2]. The shaded area is the most probable mass and
−
radius, 1σ and 2σ region, from the analysis of Steiner et al.[3].
where σ is the scalar field, ωµ is the vector-isoscalar field, (cid:126)bµ is the vector-isovector field, A(cid:126) is
the photon field, (cid:126)δ is the scalar-isovector field, F = ∂ A ∂ A , Ω = ∂ ω ∂ ω , R(cid:126) =
µν µ ν ν µ µν µ ν ν µ µν
− −
∂ (cid:126)b ∂ (cid:126)b , and V is the general effective potential for meson fields. 1 The equations of motion
µ ν ν µ eff
−
for meson fields can be obtained using the Euler-Lagrange equation
(cid:18) (cid:19)
∂ ∂
∂ L L = 0, (10)
µ
∂(∂ φ) − ∂φ
µ
where φ = σ,ωµ,bµ,δ . By taking expectation value of each field, one can define the scaled meson
i i
mean fields Φ g σ , W g ω0 , and R g b0 . Then the equations of motion for the uniform
≡ σ(cid:104) (cid:105) ≡ ω(cid:104) (cid:105) ≡ ρ(cid:104) 3(cid:105)
nuclear matter become
1 ∂V (Φ,W,R)
eff
n = Φ+ , (11)
s c2 ∂Φ
σ
1 ∂V (Φ,W,R)
eff
n = W , (12)
c2 − ∂W
ω
1 1 ∂V (Φ,W,R)
eff
n = R , (13)
2 3 c2 − ∂R
ρ
where the scaled coupling c = g /m , the baryon scalar density n = ψ¯ψ , baryon density n =
i i i s
ψ†ψ = (k3 +k3 )/(3π2), and baryon isovector density n = ψ¯τ γ0ψ (cid:104)= (k(cid:105)3 k3 )/(3π2). The
(cid:104) (cid:105) Fp Fn 3 (cid:104) 3 (cid:105) Fp − Fn
pressure and energy density for nuclear matter can be obtained as
1 (cid:88)(cid:90) kF k4 1 1 1
P = dk Φ2+ W2+ R2 V (Φ,W,R),
3π2 n,p 0 (cid:113)k2+m∗2 − 2c2σ 2c2ω 2c2ρ − eff
N
1In some literature, meson mass terms are also included in the effective potential. However, in this work, mass
termsareexplicitlyspecifiedandtheeffectivepotentialshaveonlyhigherorderinteractiontermsbeyondmassterm.
9
1 (cid:88)(cid:90) kF (cid:113) 1 1 1
= k2 k2+m∗2dk+ Φ2 W2 R2+V (Φ,W,R)
E π2 N 2c2 − 2c2 − 2c2 eff
n,p 0 σ ω ρ
1
+Wn+ Rρ , (14)
3
2
wherem∗ = m g σ g δ (+: proton, : neutron). Variousformsofmesoneffectivepotentials
N N− σ ± δ 3 −
V (Φ,W,R) are used in the literature,
eft
κ λ ζ ξ
V (σ,ωµω ,(cid:126)bµ(cid:126)b ) = (g σ)3+ (g σ)4 g4(ωµω )2 g4((cid:126)bµ(cid:126)b )2 f(σ,ωµω )g2(cid:126)bµ(cid:126)b , (15)
eff µ · µ 3! σ 24 σ −4! ω µ −4! ρ · µ − µ ρ · µ
with
6 3
(cid:88) (cid:88)
f(σ,ωµω ) = a σi+ b (ωµω )j. (16)
µ i j µ
i=1 j=1
ThevaluesoftheparametersinvariousmodelsaresummarizedinTable4. Inthedensitydependent
coupling constant model (DD-MEδ), the coupling constant has the form of
g (n) = g (n )s (x), (17)
λ λ 0 λ
where x = n/n , λ = σ, ω, ρ, δ and
0
1+b (x+d )
λ λ
s (x) = a . (18)
λ λ
1+c i(x+e )
λ λ
Numerical values of parameters are summarized in Table 5. Note that g ’s in Table 4 are equal to
λ
g (n ) in Eq. (17). In the lower part of Table 3, we summarize the basic nuclear matter properties
λ 0
of RMF models selected in this work. In the lower part of Figure 3, we plot the pressure for various
RMF models for both symmetric and pure neutron matter. In the right panel of Figure 4, masses
and radii of neutron stars are summarized for RMF models.
2.3. Neutron star crust
In the crust of neutron stars, heavy nuclei are expected to exist with free gas of neutrons and
electrons. A simple but appropriate description of this situation is completed using liquid droplet
formalism [12, 40]. The total energy density (without electron contribution) is given by
3s(u) 4π
F = un f + [σ(x)+µ ν ]+ (r n x e)2c(u)+(1 u)n f , (19)
i i s n N i i no o
r 5 −
N
where u is the volume fraction of heavy nuclei to Wigner-Seitz cell, n is the density inside of heavy
i
nuclei, f is the energy per baryon of the heavy nuclei, s(u) is the surface shape factor, r is the
i N
radius of heavy nuclei, σ(x) is a surface tension as a function of proton fraction x, µ is the neutron
s
chemical potential on the surface, ν is the areal neutron density on the surface, x is the proton
n i
fraction of heavy nuclei, c(u) is the Coulomb shape function, n is the neutron density outside
no
of heavy nuclei, and f is the energy per baryon outside of the heavy nuclei. Minimizing energy
o
density, we have four equations to solve
(cid:18) (cid:19)
2
P P β (cid:48) D = 0,
i o
− − D − 3u
un x nY = 0,
i i p
− (20)
2β
un + ν +(1 u)n n = 0,
i n no
3σD − −
µ µ = 0,
ni no
−
10
