Microscopic calculation of the equation of state of nuclear matter and neutron star structure

0.23 MB·English

by  S. Gandolfi

Mon.Not.R.Astron.Soc.000,000–000 (0000) Printed19January2010 (MNLATEXstylefilev2.2)
Microscopic calculation of the equation of state of nuclear
matter and neutron star structure
S. Gandolfi1,2,3⋆, A.Yu. Illarionov4,2,3, S. Fantoni2,3,5, J.C. Miller2,3,6,
0 F. Pederiva4,7, K.E. Schmidt8
1
1Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM 87545, USA
0
2International School for Advanced Studies, SISSA, ViaBeirut 2/4 I-34014 Trieste,Italy
2
3INFN, Sezione di Trieste,Trieste,Italy

n 4Dipartimento di Fisica dell’Universita`di Trento, via Sommarive 14, I-38123 Povo, Trento, Italy
a 5INFM DEMOCRITOSNational Simulation Center,Via Beirut 2/4 I-34014 Trieste,Italy
J 6Department of Physics (Astrophysics), University of Oxford, Keble Road, Oxford OX1 3RH, UK
9 7INFN, Gruppo Collegato di Trento, Trento, Italy
1 8Department of Physics, Arizona State University,Tempe, AZ 85287, USA

]
h
t
-
l
c
u ABSTRACT
n Wepresentresultsforneutronstarmodelsconstructedwithanewequationofstatefor
[
nuclearmatteratzerotemperature.ThegroundstateiscomputedusingtheAuxiliary

2 FieldDiffusionMonteCarlo(AFDMC)technique,withnucleonsinteractingviaasemi-
v phenomenological Hamiltonian including a realistic two-body interaction. The effect
7 of many-body forces is included by means of additional density-dependent terms in
8 theHamiltonian.Inthisletterwecomparethepropertiesoftheresultingneutron-star
4 modelswiththoseobtainedusingothernuclearHamiltonians,focusingontherelations
3 betweenmassandradius,andbetweenthegravitationalmassandthebaryonnumber.
.
9
Key words: stars: neutron, equation of state
0
9
0
:
v
i 1 INTRODUCTION (RikovskaStone et al. 2003). Making a microscopic calcu-
X
lation based on a Hamiltonian describing the properties of
Whilerealneutronstarsareverycomplicatedobjects,their
r light nuclei and symmetric nuclear matter (SNM), is both
a main global properties can usually be well-approximated
challenging and of great relevance. Variational approaches
byconsidering simple idealized models consisting of perfect
using correlated basis functions (CBF) and Fermi Hy-
fluidinhydrostaticequilibrium.Ifrotationcanbeneglected
perNettedChaintechniques(FHNC)(Fantoni& Fabrocini
to a first approximation (as is the case for the spin rates
1998) are good candidates for doing this but the strong
of most currently-known pulsars) then the model can be
spin-isospindependenceofthenuclearHamiltonianrequires
taken to be spherical and its structure obtained by solv-
the introduction of approximations into the FHNC scheme
ingtheTolman-Oppenheimer-Volkoff(TOV)equations,en-
whichcannotbefullycontrolled,suchasthesingleoperator
abling one to calculate, for example, the stellar mass as a
chain (SOC) (Pandharipande& Wiringa 1979).
function of radius or of central density. However, this re-
quires specification of an equation of state (EOS) for the
The Auxiliary Field Diffusion Monte Carlo method
neutron-star matter which clearly plays a fundamental role
(AFDMC) (Schmidt& Fantoni 1999) has been employed
in determining the properties of the resulting models. The
for computing the ground state energy of nuclei, giving
EOS needs to take account of the strong spin-isospin cor-
very good agreement with other existing accurate tech-
relations induced by realistic interactions, with particular
niques in few-body physics (Gandolfi et al. 2007a) and
regard to thetensor ones (Raffelt 1996).
highlighting important limitations of other many-body
The EOS can in principle be computed by means of
theories used for calculations of both nuclear structure
many-body theories using effective density-dependent in-
(Gandolfi et al. 2007b) and neutron matter (Gandolfi et al.
teractions, such as those given by Skyrme forces but the
2009). In this letter we present results from an AFDMC
phenomenological nature of these can be a disadvantage
calculation of the EOS of β-equilibrium matter (relevant
in making reliable calculations of neutron-star properties
for neutron stars), based on a new class of non-relativistic
Hamiltonians using nuclear potentials with a realistic two-
⋆ E-mail:[email protected] body interaction, taken from the Urbana-Argonne scheme
2 S. Gandolfi, A. Yu. Illarionov, S. Fantoni, J.C. Miller, F. Pederiva, K.E. Schmidt
(Wiringa et al. 1995) and incorporating many-body inter- properties and to give the best fit to NN scattering data
actions viadensity-dependentterms. (Wiringa & Pieper 2002).
Modern nuclear Hamiltonians are usually constrained Themany-bodyinteractionsarerepresentedbydensity-
so as to reproduce properties of light nuclei as measured dependent factors of the structural form given by the LP
inlaboratoryexperiments(Pieper et al.2001),buttheden- model. The resulting potential, denoted as DD6′, is given
sities involved there are much lower than those found in bythe following six two-body components
neutron-star cores for which it becomes necessary to in-
troduce a three-body potential, such as UIX or IL1-IL5 vDpD6′ = vOpPEP +vIpe−γ1ρ+vSp +TNA(ρ),
(Pieper et al. 2001). However, these give very large and 2 ρ −ρ 2
very different energy contributions to the EOS of pure TNA(ρ) = 3γ2ρ2e−γ3ρ 1− 3 ρn+ρp (1)
neutron matter (PNM) at high density (Sarsa et al. 2003; (cid:18) n p(cid:19) !
Fantoniet al. 2007; Gandolfi 2007). Also, recent AFDMC withγ ,γ andγ beingfixedsoastoreproducetheexper-
1 2 3
results by Gandolfi et al. (2007b) indicate that SNM is not imentalvaluesofthesaturationdensityρ =0.16fm−3,the
0
wellreproducedbytheFHNC/SOCtechniquesusedtocon- binding energy per particle E = −16 MeV and the com-
0
straintheseHamiltonians,whicharethereforenotverysuit- pressibility K = 9ρ2 ∂2E(ρ)/∂ρ2 ≈ 240 MeV. (Note
able for calculating neutron-starmodels. 0 ρ0
that in this paper we follow the nuclear-physics convention
Because of this, we have proceeded here in a different of using ρ to refer to a(cid:0)particle num(cid:1)berdensity rather than
way,usingdensity-dependenttermstosimulate many-body a mass density.)
forces with theforms used coming from explicit integration
Many nuclear matter calculations of the 1980s were
of theseforces overthe variables of particle 3 for thethree- made with the LP model interaction, and gave very good
bodyforce,overthoseofparticles3and4forthefour-body agreement with data for large nuclei, as discussed by
force, and so on. In bulk matter, the neutron and proton
Benhar & Pandharipande (1993) and Pandharipande et al.
densities ρn and ρp can be taken to be constant quantities, (1997).Afterwards,mainlybecauseofthedifficultyoftreat-
whereasinconfinedsystems,likenuclei,theyareoperators. ingthedensityasanoperatorwhendealingwithnuclei,this
As a first step in this direction, we have re- model was forgotten and attention was switched to the de-
visited the old three-parameter density-dependent velopment of increasingly sophisticated three-body poten-
LP model of Lagaris & Pandharipande (1981) and tials. On the basis of the results obtained with AFDMC
Friedman & Pandharipande (1981), with the values of quantumsimulations,webelievethatweshouldinsteadnow
the free parameters fixed so as to reproduce the saturation proceedinthedirectionofconstructingincreasingly sophis-
point and compressibility of SNM. The LP calculations ticated density-dependent two-body effective interactions,
were performed using the FHNC/SOC approximation whichmayalsoaccountforN-bodyinteractionswithN >3.
and we then re-fitted the density-dependent term using A particular limitation of DD6′ is the exclusion of the
the AFDMC calculations. By considering the chemical non-localcomponentsofAV18.Themostimportantofthese
potentials, we have constructed an EOS for a mixture of isthespin-orbitterm,particularlywhendealingwithnuclei.
protons, electrons and muons in β-equilibrium and have We estimate that the missing non-local components con-
then used the resulting EOS for constructing neutron-star tribute to the EOS of neutron matter by no more than 5%
models. We stress that (i) the ground states of nuclear (Gandolfi et al. 2009). Work towards introducing the spin-
matterasafunctionofρp andρn havebeencalculatedwith orbit components is in progress. A second important point
the same AFDMC method which has been shown to be concernsthestructuralformofthedensity-dependentterms
very accurate for calculating the ground states of various and one may want to increase the numberof Feynman dia-
nucleonic systems, and (ii) the many-body part of the gramsincludedintheconstructionofUIXandcontributing
nuclearinteractionhasbeenself-consistently determinedby to themany-bodyinteraction.
these solutions.
3 CALCULATION OF THE EQUATION OF
STATE OF NUCLEAR MATTER
2 THE MODEL
We computed the ground state of the system using the
We model the EOS by simulating nuclear matter using a AFDMC method (for details, see Gandolfi 2007 and refer-
finitenumberofinteractingnucleonsinaperiodicbox.The ences therein), simulating SNM with A = 28 nucleons in a
number of particles is chosen from among the magic num- periodic box, as described by Gandolfi et al. (2007b). The
bers giving a rotationally invariant wave function for the next magic number of nucleons providing small finite-size
correspondingnon-interactingsystemandthevolumeofthe correctionsis132,asshownbyGandolfi et al.(2009),which
boxisfixedbythedensity.Forthetwo-bodyinteraction,we requires an unjustified computational effort given the re-
take the Argonne AV6′ potential (Wiringa & Pieper 2002), striction to local components in the DD6′ model. Table 1
whichincludesthefourcentralspin-isospincomponentsand givesthevalues of thefree parameters of DD6′ correspond-
the two tensor ones. The six components of the long range ingtothebestfittoexperimentaldataforρ ,E andK,and
0 0
OPEPpotentialarefullyincluded,whereasonlythefirstsix compares these with thevalues for theoriginal LP model.
of the 18 components of the intermediate and short range The AFDMC density-dependent terms give more at-
parts of theAV18 interaction (Wiringa et al. 1995), vp(r ) traction than in the original LP model, consistent with the
I ij
andvp(r ),arekept.ThecorrespondingamplitudesI and fact that FHNC/SOC overbinds SNM. The TNA term is
S ij p
S are re-fitted so as to correctly reproduce the deuteron ∼30%largeratρ andmorethanthedoubleat5ρ ,giving
p 0 0
Microscopic calculation of the equation of state of nuclear matter and neutron star structure 3
Table1.TheAFDMCresultsforthefreeparametersoftheDD6′
interactionascomparedwiththeoriginalvaluesoftheLPmodel 175
E(ρ,x=0.5)
calculatedwithintheFHNC/SOCapproximation. E(ρ,x=0.0)
150
AFDMC, PNM
0.1 AFDMC, SNM
parameter FHNC/SOC AFDMC V]125 xp E(ρ,x), n+p+e
Me E(ρ,x), n+p+e+µ
γ1 0.15fm3 0.10fm3 n [100
γγ23 -1730.60ffmm36 -1735.90ffmm36 Nucleo 75 00 0.4 ρ [0f.m8-3] 1.2 1.6
er
y p 50
g
er
n 25
E
moreattraction,andexp(γ1ρ)−1is∼30%smalleroverthe 0
whole range (ρ ,5ρ ), giving less repulsion.
0 0
WefindthattheAFDMCresultsforthebindingenergy -250 0.1 0.2 0.3 0.4 0.5 0.6 0.7
pernucleonofSNMatdensitieslargerthan∼0.08fm−3can ρ [fm-3]
bevery well described by:
E (ρ)=E +a(ρ−ρ )2+b(ρ−ρ )3eγ(ρ−ρ0), (2)
SNM 0 0 0 Figure 1. The equation of state for symmetric nuclear mat-
where E = −16.0 MeV, ρ = 0.16 fm−3, a = 520.0 MeV ter (dashed line), pure neutron matter (dot-dashed line) and β-
0 0
fm6, b = −1297.4 MeV fm9 and γ = −2.213 fm3. This equilibrium nuclear matter with both electrons and muons (full
line) and with electrons only (dotted line). The points show the
parametrizationwaschosentorepresenttheEOSofnuclear
AFDMC results for symmetric nuclear matter (SNM, squares)
matter,reproducingpropertiesconstrainedbyterrestrialex-
andpureneutronmatter(PNM,circles)whichhavebeenusedto
periments on nuclei (Danielewicz et al. 2002). fittheEOS.Intheinset,weshowtheprotonfractionxp plotted
The DD6′ Hamiltonian was then used to compute the as a function of the total nucleon density, computed consider-
EOS of PNM, by making a simulation with 66 neutrons in ingthe presence of just electrons as negatively-charged particles
aperiodicbox.TheEOSfornuclearmatterasafunctionof (dotted line)andwithbothelectronsandmuons(fullline).
theproton fraction x =ρ /ρ is then parametrized as
p p
γs
ρ
E(ρ,x )=E (ρ)+C (1−2x )2. (3)
p SNM s ρ0 p their contribution must then also be considered. For our
(cid:18) (cid:19) EOS,muons begin to appear at ρ≈0.133 fm−3.
Thetwoextraparametersofthesymmetryenergyterm,C
s TheEOSscalculatedforSNM,PNMandβ-equilibrium
andγ ,wereobtainedbyfittingE(ρ,x =0)totheAFDMC
s p matter are shown in Fig. 1, with the energy per nucleon
result for PNM. This gives C = 31.3 MeV and γ = 0.64.
s s being plotted against the nucleon number density ρ; in the
Typical values for these parameters have been quoted as
inset, we show the proton fraction x plotted as a function
C ≈ 31−33 MeV and γ ≈ 0.55−0.69 by Shettyet al. p
s s of ρ, computed considering both electrons and muons (full
(2007) and as C = 31.6 MeV and γ ≈ 0.69 − 1.05 by
s s line) and only electrons (dotted line). TheEOS for PNM is
Worley et al. (2008). It should be noted that usually the softerthanthatcalculatedwiththeAV8′+UIXpotentialby
symmetry energy is constrained over a range of densities
Gandolfi et al. (2009) using the same AFDMC many-body
typical of nuclei, whereas we have here fitted the parame-
method and this then produces a similar behaviour for the
ters over a very wide density range. This means that the
β-equilibrium matter. The main reason for this comes from
parametrization of eq. (3) should be accurate up to very
thedifferent treatment of many-bodyeffects.
high densities.
At very high densities, the chemical potential of the
In high-density matter, neutrons can produce protons
nucleonicmatterbecomes larger thanthethreshold forcre-
and electrons by β decay and so the equilibrium configura-
ationofheavierparticles.Suchstatesareduetotheupand
tion can have a non-zero proton/neutron ratio, modifying
down quarks transforming to strange quarks, so that parti-
theEOSawayfromthatforPNM.Theequilibriumconcen-
cleswithstrangeness(hyperons)starttoappear.Arealistic
tration of protonsx can becomputed by imposing
p EOS should include these when they appear and this can
µ =µ +µ , (4) seriously modify the structure of the star. We do not in-
n p e
clude this in our present calculations (although we make
where µi is the chemical potential (for neutrons, protons some comment about its likely effects in the next section);
and electrons respectively). For doing this, we consider the we leave inclusion of thisuntil a subsequentpaper.
electrons as comprising a relativistic Fermi-gas:
µ =[m2+h¯2(3πρ )2/3]1/2, (5)
e e e
and charge neutrality imposes that ρ = x ρ, where ρ is
e p 4 RESULTING NEUTRON STAR MODELS
the total nucleon density. The chemical potentials of the
neutrons and protons are derived from equation (3), and When the EOS of the neutron-star matter has been speci-
equation (4) is then solved to give x as a function of ρ. fied,thestructureofanidealizedspherically-symmetricneu-
p
Anotherconsideration isthat when µ becomes larger than tronstarmodelcanbecalculatedbyintegratingtheTolman-
e
the muon mass, the production of muons is favoured, and Oppenheimer-Volkoff(TOV) equations:
4 S. Gandolfi, A. Yu. Illarionov, S. Fantoni, J.C. Miller, F. Pederiva, K.E. Schmidt
dP G[m(r)+4πr3P/c2][ǫ+P/c2]
=− , (6)
dr r[r−2Gm(r)/c2] 3
dm(r) PNM, DD6’
=4πǫr2, (7) 2.5 PNM, AV6’
dr PNM, AV8’+UIX
β-equilibrium, n+p+e+µ
where P =ρ2(∂E/∂ρ) and ǫ=ρ(E+mN) are thepressure 2
and the energy density, m is the average nucleon mass,
N
m(r) is the gravitational mass enclosed within a radius r, MΘ1.5
andGisthegravitationalconstant.ThesolutionoftheTOV M/
equations for a given central density gives the profiles of ρ,
1
ǫ and P as functions of radius r, and also the total radius
R and mass M =m(R). A sequence of models can be gen-
0.5
erated by specifying a succession of values for the central
density. In Fig. 2 we plot the mass M (measured in so-
lar masses M⊙) as a function of the radius R (measured in 06 8 10 12 14 16 18 20
km), as obtained from calculations with four different pre- R [km]
scriptions for the EOS: the β-equilibrium and PNM EOSs
discussed in Sections 2 and 3, the equivalent one for PNM
withjusttwo-bodyinteractions(usingAV6′),andaprevious Figure 2.Predicted neutron-star masses (inunits of M⊙)plot-
ted as a function of stellar radius (in km). Four different equa-
one from Gandolfi et al. (2009), for PNM with three-body
tions of state are considered: those discussed in this paper for
interactions (using AV8′+UIX). Models to the right of the
β-equilibriummatter (fullline)andpureneutronmatter (PNM,
maximum of each curve are stable to radial perturbations DD6′, dashed line), the equivalent one for PNM with just two-
and these are the ones of interest for astrophysical neutron bodyinteractions(PNM,AV6′,dot-dash-dashedline),andapre-
stars. The maximum mass obtained with the two-body in- vious one from Gandolfietal. (2009), for PNMwith three-body
teraction AV8′ is verysimilar to that for AV6′. interactions(PNM,AV8′+UIX,dot-dashedline).
It is interesting to make a comparison between these
curvessoastoseethechangescausedbyintroductionofthe
variousdifferentfeatures.Thesolidcurve(β-equilibrium)is
ourbestproposalfortheneutron-starEOSbutitcanbeseen massforβ-equilibriummatterisρc ≈1.2fm−3 whichiswell
that it differs only verylittle from thepureneutron matter within the range where these changes are likely to have oc-
EOS(wheretheradiiforagivenmassarejustslightlylarger curred,andsoweexpectthatthemaximummasswouldbe
within the main range of interest). There is a considerable slightlylowerthanthatshowninFig.2.Thisbringsuswell
difference,however,withrespecttothepreviousAV8′+UIX within theexpected range.
curveforpureneutronmatter,withthemaximummassbe- Observationalconstraintsfortheradiusaremoreprob-
ing reduced from ∼ 2.5 M⊙ to ∼ 2.2 M⊙ and the radii in lematic, but one of the best of these seems to be the indi-
themainregionofinterestalsobeingsubstantiallyreduced. rect constraint suggested by Podsiadlowski et al. (2005) in
This reflects the effective softening of the EOS caused by thecaseof thelessmassive componentofthedoublepulsar
thedifferent treatment of many-bodyeffects. PSR J0737-3059. If, as seems likely, this neutron star was
An objective of this type of work is to attempt to the product of an electron-capture supernova, then the to-
constrain microphysical models for neutron-star matter by tal pre-collapse baryon number of the stellar core is rather
making comparison with astronomical observations (see precisely known from model calculations and, since only a
Lattimer & Prakash 2007). This is just starting to be pos- very small loss of material is expected to have occurred in
sible now and further progress is anticipated within the the subsequent collapse, the baryon number of the neutron
nextfewyears.Atpresent,theonlyneutronstarsforwhich star is itself also well-known. Together with the very accu-
massesareaccuratelyknownarethecomponentsofthebest- ratevalueforthegravitationalmass(calculatedfrompulsar
observedbinarypulsars,forwhichtimingmeasurementsgive timing),thiscanbeusedtoplaceaquitestringentconstraint
results correct to many significant figures. The maximum ontheEOS.ThebaryonnumberAofaneutron-starmodel
mass for any of these is the 1.441 M⊙ for the Hulse/Taylor can bereadily calculated from
bHionwareyvepr,ultshaerrePSisRac1c9u1m3+ul1a6tin(sgeeevWideeinsbceerfgor&hTigahyelrorm2a0s0s5es),. dA(r) =4πρr2 1− 2Gm −12 , (8)
dr rc2
particularly for neutron stars in binary systems together
(cid:16) (cid:17)
withwhitedwarfs(seeRansom et al.2005)andthereisnow which needs to be solved together with equations (6) and
a widespread belief that the maximum should probably be (7) (we recall that ρ is here the baryon number density).
in the range 1.8 M⊙ −2.1 M⊙ (at least when the rotation Inpractice, it is convenienttotalk in termsof thebaryonic
is sufficiently slow, as is the case for almost all pulsars so mass, defined as M = m A(R), rather than A(R) itself:
0 N
far observed). At high enough densities, it is expected that the difference between the baryonic mass and the gravita-
the composition of the matter would change because of the tionalmassdependsonthecompactnessoftheneutronstar,
appearance of either hyperons or deconfined quarks, both and hence indirectly on the radius. If M is plotted against
of which are likely to decrease the maximum mass (see, for M for a given EOS, then the curve needs to pass through
0
example, Schulzeet al. 2006 for the case of hyperons, and a certain error box in order to be consistent with the ob-
Akmalet al. 1998 for an example of the inclusion of quark servations for PSR J0737-3059, subject to the assumptions
matter).Thecentraldensitycorrespondingtoourmaximum being made in theanalysis. This plot is shown in Fig. 3 for
Microscopic calculation of the equation of state of nuclear matter and neutron star structure 5
Foundation;KESthanksSISSAforkindhospitalityandac-
1.28 knowledges support from NSF grant PHY-0757703. Calcu-
lations were performed using the HPC facility “BEN” at
β-equilibrium, n+p+e+µ
PNM, DD6’ ECT⋆inTrento,underagrantforSupercomputingProjects,
PNM, AV6’
andusingtheHPCfacilityofSISSA/DemocritosinTrieste.
PNM, AV8’+UIX
1.26
Θ
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