Table Of ContentIsospin asymmetric nuclear matter and properties of axisymmetric neutron stars
Partha Roy Chowdhury1,∗ Abhijit Bhattacharyya1,† and D. N. Basu2‡
1Department of Physics, University of Calcutta, 92 A.P.C. Road, Kolkata-700 009, India
2Variable Energy Cyclotron Centre, 1/AF Bidhan Nagar, Kolkata-700 064, India
(Dated: January 7, 2011)
Pure hadronic compact stars, above a limiting value (≈1.6 M⊙) of their gravitational masses, to
whichpredictionsofmost ofotherequationsofstate(EoSs)arerestricted,canbereachedfrom the
equation of state (EoS) obtained using DDM3Y effective interaction. This effective interaction is
foundtobequitesuccessfulinprovidingunifieddescriptionofelasticandinelasticscattering,various
radioactivitiesandnuclearmatterproperties. Wepresentasystematicstudyofthepropertiesofpure
hadroniccompact stars. Theβ-equilibratedneutronstarmatterusingthisEoS withathincrustis
able to describe highly-massive compact stars, such as PSR B1516+02B with a mass M=1.94+0.17
−0.19
M⊙ and PSR J0751+1807 with a mass M=2.1±0.2 M⊙ to a 1σ confidencelevel.
1
1
PACSnumbers: 21.65.Cd,21.65.Ef,26.60.Kp
0
2
n The theoretical study of the nuclear equation of state Weinvestigatetheimpactsofthecompressionmodulus
a isafieldofresearchwhichties togetherdifferentareasof andsymmetryenergyofnuclearmatteronthemaximum
J physics. Nuclear EoS is of great interest as its features mass of NSs in view of the recent constraints from the
6 control the stability of neutron star (NS), the evolution isospin diffusion in heavy-ion collisions at intermediate
of the universe, supernova explosion, nucleosynthesis as energies [6, 7]. In the present work, the density depen-
]
h wellascentralcollisionsofheavynuclei. Extensivestud- dent M3Y effective interaction (DDM3Y) [8] which pro-
t ies in the past two decades of nuclear matter created videsaunified descriptionofthe elasticandthe inelastic
-
l at subnormal or supernormal density in heavy ion col- scattering [9, 10], various radioactivites [11–15] and nu-
c
lisions have resulted in experimental constraints on the clear matter properties [16–18], is employed to obtain
u
nuclear EoS of symmetric matter. Recent astrophysi- EoS of the β equilibrated NS matter. This EoS is used
n
[ cal observations of massive neutron stars and heavy-ion to carry out a systematic study of the static as well as
data are confronted with our present understanding of rotating NSs in view of the recent observations of the
2
the EoS of dense hadronic matter. We argue that the massive compact stars.
v
data from massive neutron stars and pulsars provide an As mentioned above, the nuclear matter EoS is calcu-
2
0 importantcross-checkbetweenhigh-density astrophysics latedusingtheisoscalarandtheisovectorcomponentsof
9 and heavy-ion physics. The density dependence of nu- M3Yinteractionalongwithdensitydependence. Theen-
1 clear symmetry energy (NSE) obtained by using nuclear ergy per nucleon for isospin asymmetric nuclear matter
. EoS plays an important role for modelling the structure is given by
2
0 of the neutron stars (NSs) and the dynamics of super-
0 nova explosions since a series of observables (e.g. slope 3h¯2k2 ρJ C
1 L of NSE, the value of NSE at nuclear density etc.) can ǫ(ρ,X)=[ F]F(X)+( v )(1−βρn) (1)
v: be determined from the knowledge of symmetry energy. 10m 2
i Thestiffnessofthehigh-densitymattercontrolsthemax- where X=ρn−ρp is the isospin asymmetry with ρ , ρ
X imum mass of compact stars. New measurements of the ρn+ρp n p
and ρ=ρ +ρ being the neutron, proton and nucleon
n p
r propertiesofpulsarspointtowardslargemassesandcor-
a number densities respectively, kF is Fermi momentum
respondingly to a rather stiff EoS [1] characterized by in case of SNM, F(X)=[(1+X)5/3+(1−X)5/3] and J rep-
symmetric nuclear matter (SNM) incompressibility 250- 2 v
resents the volume integrals of the isoscalar and the
270 MeV or more. In a recent analysis of x-ray burster
EXO 0748-676 (M= 2.10±0.28 M ) it is even claimed isovector parts of the M3Y interaction. The details of
⊙ the present methodology may be obtained in Ref.[16].
that soft nuclear EoS are ruled out [2]. In addition, it is
However, for solving the Einstein’s equations for stellar
arguedinRef.[2]thatcondensatesanddeconfinedquarks
structure, we need to consider the total energy density
[3] may not exist in the cores of NSs. Recently, new ob-
including mass (also called, the mass-energy density) ε
servations for the mass and the radius of compact stars
which is related to the ǫ and baryon number density ρ
have been obtained which provide stringent constraints
as ε = ρ(ǫ+m) where m (∼ 938.919 MeV) is the aver-
on the EoS of strongly interacting matter at high densi-
age of the neutron and proton masses in MeV unit. As
ties [4, 5].
the saturationenergy per nucleon ǫ =−15.26MeV and
0
the saturation density ρ =0.1533 fm−3 are used in this
0
work, the corresponding total energy density at satura-
∗ [email protected] tionisε0=141.597MeV/fm3=2.524×1014gm/cm3. Ob-
† [email protected] viously, as these two parameters ǫ and ρ are extracted
0 0
‡ [email protected] on the basis of information on finite nuclei, they put the
2
400
DD-F
DD-F
200
KVOR
300 KVOR
V) DBHF V) DBHF
e This Work e
M M This Work
100 200
(m A (
y
Es E/ 100
0
-50 0
0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1
ρ -3 ρ -3
(fm ) (fm )
FIG. 1. (Color online) (a) (Left) Density dependent behavior of symmetry energy (Esym vs. ρ) determined by different
approaches. (b) (Right) The EoS determined by different models (E/A vs. ρ) are shown for beta equilibrated charge neutral
neutron star matter.
two constraints to the high density behaviour of any nu- DBHFandourEoSare0.1469,0.1600,0.1810and0.1533
clear matter EoS. The pressure-density relationship for fm−3 respectively. SotheDBHFusesconsiderablylarger
the present EoS is consistent with the experimental flow density whereas the ρ used by other two models are
0
data[19]confirmingits highdensitybehaviour. The val- slightlydifferentfromtheexperimentallyextractedvalue
ues of other important quantities, L, K , K and of 0.1533 fm−3. The values of NSE at saturation den-
sym asy
K , defined and calculated in Refs.[17, 18], also agree sitycalculatedbyDD-F,KVOR,DBHFandourEoSare
τ
extremely well with those extracted from experiments. 31.6, 32.9, 34.4 and 30.71 MeV respectively. It is clear
thatDBHFslightlyoverestimatesthevalueofNSEatρ
The NSE given by Esym(ρ) = ǫ(ρ,1)− ǫ(ρ,0) has a which is roughly around 30 MeV. 0
value of 30.71±0.26 MeV at the saturation density ob-
AnegativeNSEathighdensitiesimpliesthatthepure
tained from this calculation which satisfies one of the
neutron matter becomes the most stable state. Conse-
constraints on the high density EoS. At higher densi-
quently, pure neutron matter exists near the core [20] of
ties the NSE (see Fig.1(a)) using DDM3Y interaction
peaks at ρ ≈ 1.95ρ and becomes negative at ρ ≈ 4.7ρ . the NSs. Although the present EoS is ‘stiff’ since the
0 0 SNM incompressibility K = 274.7±7.4 MeV, but the
In Fig.1 the symmetry energy and energy per nucleon ∞
NSE is ‘super-soft’because it increasesinitially with nu-
(E/A) are plotted as a functions of the baryon density.
cleonicdensityuptoabouttwotimesthenormalnuclear
Theequationofstatefortheβ-stablechargeneutralneu-
density and then decreases monotonically (hence ‘soft’)
tron star matter (see Fig.1(b)) is calculated numerically
andbecomes negative (hence ‘super-soft’)athigher den-
using Eq.(1) with β-equilibrated proton fraction deter-
sities[16,17]. Thisisconsistentwiththerecentevidence
mined from the NSE. These plots compare the symme-
fora soft NSE atsuprasaturationdensities [21] andwith
try energy functions and the EoSs determined by the
thefactthatthesuper-soft[22]nuclearsymmetryenergy
present calculation and several relativistic models (e.g.
is preferred by the FOPI/GSI experimental data on the
DD-F, KVOR, DBHF) [4] for the neutron star matter.
π+/π− ratio in relativistic heavy-ion reactions for the
InFig.1(a),theNSEcalculatedbythephenomenological
stability of NSs.
RMF models using density dependent masses and cou-
pling constants (e.g. DD-F, KVOR) goes on increasing Theβ equilibriumprotonfractionx [23]ofaNScon-
β
withdensityandneverbecomenegative. Intherelativis- sisting of neutrons (n), protons (p) and electrons (e) is
ticDirac-Brueckner-Hartree-Fock(DBHF)approach,the completely governed by the density dependent behavior
NSE increases more rapidly with density indicating very of NSE. Contrary to the relativistic models like DD-F,
large proton fraction at higher density. This shows an KVOR, DBHF etc. this work does not support the fast
opposite trendto the NSE function determinedfromour cooling via direct nucleon URCA process as the maxi-
EoS. The saturation density (ρ ) used in DD-F, KVOR, mum x is 4.4% only. Recently it has been concluded
0 β
3
2
) 2
s )
s s
a s
m a
m
r Static
a r
ol 1 Rotating a Static
l
S o 1
( DD-F S Rotating
M KVOR ( DD-F
M
DBHF KVOR
0 DBHF
0 0.5 1 1.5 0
ρ -3 0 4 8 12 16 20
(fm )
c
R (km)
FIG. 2. (Color online) (a) (Left) The variation of NS masses (M) is shown with central density (ρc). The results of this
calculation are denoted as ‘Static’ and ’Rotating’ for non-rotating and rotating NSs at the Keplerian limit. (b) (Right) Mass-
Radius relationship is given for static and rotating stars at Keplerian frequency using this EoS. The other three plots (DD-F,
KVOR,DBHF) present thesame relationship for static star only.
theoretically that an acceptable EoS of asymmetric nu- rotatingstellarstructurearesolvedbyusingourEoSfol-
clear matter shall not allow the direct URCA process lowingtheprocedureadoptedbyKomatsu,Eriguchiand
to occur in NSs with masses below 1.5 solar masses [4]. Hachisu [27, 30]. We choose the ‘rns’ code written by N.
However, the possibility of fast cooling [24, 25] via di- Stergioulas [31] in calculating rotating as well as static
rect hyperon URCA or any other processes enhancing NS properties.
neutrino emissivities like π− and K− condensates may
In Fig.2(a), we have shown the mass of the stars as a
not be completely ruled out. Also a recent experimen-
function of central baryon density (ρ ). Our results are
c
tal observation suggests [26] high heat conductivity and
plottedforthestaticandKeplerianlimit. Thisisobvious
enhanced core cooling process indicating the enhanced
fromthe Fig. 2(a)thatfor the samemasscomparatively
level of neutrino emission, which may be due to Cooper
less central density appears for the rotating stars due
pairing. Further theoretical studies and sufficient obser-
to centrifugal action. It may be noted that as angular
vationaldataareneededtoshedsomelightonthecooling
frequency (Ω) becomes greater, the structure of NS [32]
phenomenon of NS.
gets changed not only because of centrifugal flattening,
Letus nowexplorethe variouspropertiesofstatic and
butalsobecauseitistakingplaceagainstthebackground
rotating NSs using the proposed EoS. To study the ro-
of a radially dependent frame dragging frequency. For
tating stars the following metric is used:
comparisonwehavealsoplottedtheresultsofthreeother
EoSs as mentioned earlier in the text. The maximum
ds2 =−e(γ+ρ)dt2+e2α(dr2+r2dθ2) (2) mass for the static case is about 1.92 M⊙ with radius
∼ 9.7 km and for the rotating case it is about 2.27 M
+e(γ−ρ)r2sin2θ(dφ−ωdt)2 with radius ∼ 13.1 km. So a mass higher than 1.92 M⊙
⊙
wouldruleoutastaticstarasfarasthisEoSisconcerned.
wherethe gravitationalpotentialsγ,ρ,αandω arefunc-
The phenomenological RMF models DD-F and KVOR
tions of polar coordinates r and θ only. The Einstein’s
predictmaximummass aroundtwice solarmass for non-
field equations for the three potentials γ,ρ,α have been
rotating star. The relativistic DBHF model calculates
solved using Green’s function technique [27–29] and the
themaximummass∼2.33M andtherefore,theDBHF
fourth potential ω has been determined from other po- ⊙
predictsmassiveNSevenforstaticcase. Thisisalsoclear
tentials. All the physical quantities may then be deter-
fromtheFig.2(b)wherethemass-radiusrelationshipsfor
mined from these potentials [30]. The matter inside the
all the above EoSs are shown.
NS is approximated as a perfect fluid. Solution of the
potentials and hence the calculation of physical quan- To summarise, we have presented a nuclear EoS at
tities can be done numerically. The field equations for supersaturation densities which satisfies both the con-
4
straints from NS and heavy ion collision phenomenol- to disfavour too stiff behavior of the EoS. The data
ogy. Ourresultsshowthatwiththestellarconfiguration, from massive NSs and pulsars may provide an impor-
whichcontainalargefractionofβequilibratedNSmatter tant cross-check between high-density astrophysics and
with a thin crust is able to describe highly-massivecom- heavy-ionphysics. Thevariationofpressurewithdensity
pact stars, such as the one associated to the millisecond for the present EoS is consistent with the experimental
pulsars PSR B1516+02B with a mass M=1.94+0.17 M flow dataconfirming its highdensity behaviour. We find
−0.19 ⊙
(1σ)[33]andPSRJ0751+1807,witha massM=2.1±0.2 that the large values of gravitational masses (≃2.0 M )
⊙
M to a 1σ confidence level (and 2.1+0.4 M to a 2σ for the NSs are possible with the present EoS with the
⊙ −0.5 ⊙
confidence level) [34]. In the case of PSR J1748-2021B, SNM incompressibility K =274.7±7.4 MeV, which is
∞
a millisecond pulsar in the Globular Cluster NGC 6440, rather ‘stiff’ enough at high densities to allow compact
the measured mass is M=2.74+0.41 M (2σ) [35]. There stars with large values of gravitational masses ∼ 2 M
−0.51 ⊙ ⊙
are few other EoSs which can explain such a high mass while the corresponding symmetry energy is ‘super-soft’
forstaticcase,however,they failtoexplaintheexpected as preferred by FOPI/GSI experimental data. Thus the
behavior of the NSE. We would like to mention at this DDM3Y effective interaction which is found to provide
stage thata starmay not rotateas fast asKeplerianfre- unifieddescriptionofelasticandinelasticscattering,var-
quency due to r-mode instability [36]. There have been ious radioactivites and nuclear matter properties, also
suggestions that r-mode may limit the time period to provides excellent description of β equilibrated NS mat-
1.5 ms [37]. However, pulsar rotating faster (e.g. PSR ter to allow the recent observations of the massive com-
J17482446ad)than this limit has already been observed pact stars.
[38]. Furtherobservationsandabetterr-modemodelling The research work of P. Roy Chowdhury is sponsored
may shed more light on this issue. by the UGC (No.F.4-2/2006(BSR)/13-224/2008(BSR))
Modern constraints from the mass and mass-radius- underDr. D.S.KothariPostdoctoralFellowshipScheme.
relation measurements require stiff EoS at high densi- The work of A. Bhattacharyya is partially supported by
ties, whereas flow data from heavy-ion collisions seem UGC (UPE & DRS) and CSIR.
[1] J. M. Lattimer and M. Prakash, Phys. Rept. 442, 109 Rev. C 80, 011305(R) (2009).
(2007). [18] D. N. Basu, P. Roy Chowdhury and C. Samanta, Phys.
[2] F. O¨zel, Nature441, 1115 (2006). Rev. C 80, 057304 (2009).
[3] V. S. Uma Maheswari, D. N. Basu, J. N. De and S.K. [19] P. Danielewicz, R.Lacey and W.G. Lynch,Science298,
Samaddar, Nucl.Phys. A615, 516 (1997). 1592 (2002).
[4] T. Kla¨hn et al.,Phys.Rev. C 74, 035802 (2006). [20] I. N. Mishustin, M. Hanauske, A. Bhattacharyya, L. M.
[5] H. Heiselberg and M. Hjorth-Jensen, Phys. Rept. 328, Satarov,H.Stoecker,andW.Greiner,Phys.Lett.B552,
237 (2000). 1 (2003).
[6] Lie-WenChen,CheMingKoandBao-an Li,Phys.Rev. [21] Zhigang Xiao, Bao-An Li, Lie-Wen Chen, Gao-Chan
Lett.94, 032701 (2005). Yong and Ming Zhang, Phys. Rev. Lett. 102, 062502
[7] B. A. Li, L. W. Chen, and C. M. Ko, Phys. Rep. 464, (2009).
113 (2008). [22] De-HuaWen,Bao-An Liand Lie-WenChen, Phys.Rev.
[8] G.Bertsch, J.Borysowicz, H.McManus, W.G.Love, Nucl. Lett. 103, 211102 (2009).
Phys.A284,399 (1977); G.R.SatchlerandW.G.Love, [23] J.M.Lattimer,C.J.Pethick,M.PrakashandP.Haensel,
Phys.Reports 55, 183 (1979). Phys. Rev.Lett. 66, 2701 (1991).
[9] D.GuptaandD.N.Basu,Nucl.Phys.A748,402(2005). [24] D. G. Yakovlev, A. D. Kaminker, P. Haensel and O. Y.
[10] D. Gupta, E. Khan and Y. Blumenfeld, Nucl. Phys. Gnedin, Astron. Astrophys.389, L24 (2002).
A773, 230 (2006). [25] C. O. Heinke, P. G. Jonker, R. Wijnands, C. J. Deloye
[11] D. N. Basu, P. Roy Chowdhury and C. Samanta, Phys. and R.E. Taam, Astrophys.J. 691, 1035 (2009).
Rev.C 72, 051601 (R) (2005). [26] E.M.Cackett,etal.,Mon.Not.R.Astron.Soc.372,479
[12] P. Roy Chowdhury, C. Samanta and D. N. Basu, Phys. (2006).
Rev.C 73, 014612 (2006); ibidPhys. Rev.C 77, 044603 [27] H. Komatsu, Y. Eriguchi and I. Hachisu, Mon. Not. R.
(2008); ibid Atomic Data and Nuclear Data Tables 94, Astro. Soc. 237, 355 (1989).
781 (2008). [28] Abhijit Bhattacharyya, Igor N Mishustin and Walter
[13] P. Roy Chowdhury, D.N. Basu and C. Samanta, Phys. Greiner, J.Phys. G 37, 025201 (2010).
Rev.C 75, 047306 (2007). [29] Abhijit Bhattacharyya, Sanjay K. Ghosh, Matthias
[14] C. Samanta, P. Roy Chowdhury and D.N. Basu, Nucl. Hanauske, and Sibaji Raha, Phys. Rev. C 71, 048801
Phys.A789, 142 (2007). (2005).
[15] D.N. Basu, Phys. Rev. C 66, 027601 (2002): ibid Phys. [30] G.B.Cook, S.ShapiroandS.Teukolosky,Astrophys.J.
Lett.B566, 90 (2003). 422, 227 (1994).
[16] D. N. Basu, P. Roy Chowdhury and C. Samanta, Nucl. [31] N. Stergioulas and J. L. Friedman, Astrophys. J. 444,
Phys.A811, 140 (2008). 306 (1995).
[17] P. Roy Chowdhury, D. N. Basu and C. Samanta, Phys. [32] Norman K. Glendenning, Compact Stars: Nuclear
5
Physics, Particle Physics, and General Relativity, 2nd [35] P. C. C. Freire et al., Astrophys. J. 675, 670 (2008);
Edition (NewYork: Springer-Verlag) (2000). arXiv:0711.0925.
[33] P. C. C. Freire, A. Wolszczan, M. van den Berg and [36] Mohit NayyarandBenjamin J. Owen,Phys.Rev.D73,
J. W. T. Hessels, Astrophys. J. 679, 1433 (2008); 084001 (2006).
arXiv:0712.3826. [37] Nikolaos Stergioulas, Living Rev. Rel.6:3,(2003),
[34] D. J. Nice, E. M. Splaver, I. H. Stairs, O. L¨ohmer, A. arXiv:gr-qc/0302034v1.
Jessner,M.KramerandJ.M.Cordes,Astrophys.J.634, [38] Jason W. T. Hessels et al., Science 311, 1901 (2006);
1242 (2005). arXiv:astro-ph/0601337v1.
