Table Of ContentMNRAS000,1–12(2016) Preprint25April2016 CompiledusingMNRASLATEXstylefilev3.0
Low-mass neutron stars: universal relations, the nuclear
symmetry energy and gravitational radiation
Hector O. Silva1⋆, Hajime Sotani2†, Emanuele Berti1,3‡
1Department of Physics and Astronomy, The Universityof Mississippi, University,Mississippi 38677, USA
2Divisionof Theoretical Astronomy, National Astronomical Observatory of Japan, 2-21-1 Osawa, Mitaka, Tokyo 181-8588, Japan
6 3CENTRA, Departamento de F´ısica, InstitutoSuperior T´ecnico, Universidade de Lisboa, AvenidaRovisco Pais 1, 1049 Lisboa, Portugal
1
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2
AcceptedXXX.ReceivedYYY;inoriginalformZZZ
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ABSTRACT
2 Thelowestneutronstarmassescurrentlymeasuredareintherange1.0−1.1M⊙,but
these measurement have either large uncertainties or refer to isolated neutron stars.
2
TherecentclaimofapreciselymeasuredmassM/M⊙ =1.174±0.004(Martinez et al.
] 2015)inadoubleneutronstarsystemsuggeststhatlow-massneutronstarsmaybean
E interesting target for gravitational-wave detectors. Furthermore, Sotani et al. (2014)
H recentlyfoundempiricalformulasrelatingthemassandsurfaceredshiftofnonrotating
. neutronstarstothe star’scentraldensityandto theparameterη ≡(K0L2)1/3,where
h
K0 is the incompressibility of symmetric nuclear matter and L is the slope of the
p
symmetry energyat saturationdensity.Motivated by these considerations,we extend
-
o theworkbySotani et al.(2014)toslowlyrotatingandtidallydeformedneutronstars.
r We compute the moment of inertia, quadrupole moment, quadrupole ellipticity, tidal
t
s and rotational Love number and apsidal constant of slowly rotating neutron stars by
a integrating the Hartle-Thorne equations at second order in rotation, and we fit all of
[
these quantitiesasfunctions ofη andofthe centraldensity.Thesefits may be usedto
2 constrainη,eitherviaobservationsofbinarypulsarsinthe electromagneticspectrum,
v or via near-future observations of inspiralling compact binaries in the gravitational-
7 wave spectrum.
0
4 Key words: stars: neutron – stars: rotation – equation of state
3
0
.
1
0 1 INTRODUCTION ties in the NS mass-radius relation: for a typical NS mass
6 M ∼1.4M⊙, EOSs compatible with our current knowledge
1 The equilibrium of spherically symmetric, nonrotating neu-
of nuclear physics predict radii R ranging between 6 and
: tron stars (NSs) in general relativity is governed by the
v 16 km (Steineret al. 2013).
Tolman-Oppenheimer-Volkoff(TOV)equations,thatfollow
i
X from Einstein’s equations with a perfect-fluid stress energy Unlike black holes, which are vacuum solutions of Ein-
r tensor(seee.g.Stergioulas2003;Shapiro& Teukolsky1983; stein’s equations, NS structure depends on the coupling of
a Friedman & Stergioulas2013).Whensupplementedwithan gravitywithmatter.ThereforeNSscanprobe(andpossibly
equation of state (EOS) relating the density and pressure rule out) theories of gravity that are close to general rela-
of the perfect fluid, the TOV equations form a closed sys- tivity in vacuum, but differ in the description of the cou-
tem of ordinary differential equations, whose solutions are plingbetween matter and gravity in thestrong-field regime
obtained (in general) by numerical integration. The solu- (Berti et al. 2015).Infact, given thestrength of theirgrav-
tions form a single-parameter family, where the parameter itational field, thehigh density of matterat theircores and
canbechosentobethecentraltotalenergy densityρ .De- the existence of pulsars with fast spin and large magnetic
c
spite recent progress, the EOS is still largely unknown at fields, NSs are ideal laboratories to study all fundamental
the energy densities ρ > ρ (where ρ /c2 ≡ 2.68 × 1014 interactions (Lattimer & Prakash 2004, 2007, 2015; Psaltis
0 0
g/cm3 is the nuclear saturation density) characterizing the 2008). However, tests of strong gravity with NSs are made
NScore. Uncertaintiesin theEOS translateintouncertain- more difficult by two fundamental degeneracies: (i) uncer-
tainties in the EOS can mimic the modifications to the
bulkpropertiesofNSsthatmaybeinducedbyhypothetical
strong-field modifications of general relativity; (ii) different
⋆ E-mail:[email protected]
† E-mail:[email protected] theories of gravity can give rise to similar modifications in
‡ E-mail:[email protected] thebulkproperties of NSs(Glampedakiset al. 2015).
(cid:13)c 2016TheAuthors
2 Hector O. Silva et al
These issuesare partially alleviated byrecentlydiscov- an ultra-strippedsupernovais1.1M⊙.Notethatuncertain-
ered“universal”(EOS-independent) relations showing that ties in supernova physics affect all of these bounds, and (if
rotatingNSsare,infact,relativelysimpleobjects.LetM be confirmed)therecentobservationsofMartinez et al.(2015)
the mass of a nonrotating star, J the angular momentum, areonlymarginallycompatiblewiththeironcorebound.To
χ=J/M2 thedimensionless spin, I the moment of inertia, summarize:itiscommonlybelievedthattheminimummass
Qthequadrupolemomentandλ(tid) thetidalLovenumber, of NSs in the universe should be around the minimum ob-
a measure of stellar deformability (here and below we use served mass (∼1M⊙) and that NS masses .1.2M⊙ would
geometricalunits,G=c=1).Workingintheslow-rotation challengetheparadigmofNSformationbygravitationalcol-
approximation,Yagi& Yunes(2013b,a)discoveredthatuni- lapse,buttheseconclusionsareuncertainduetoourlimited
versal (EOS-independent)“I-Love-Q”relations connect the understandingofsupernovaphysics.Thereforethediscovery
three normalized quantities I¯= I/M3, λ¯(tid) = λ(tid)/M5 oflow-massNSsmaygiveusimportantcluesontheirforma-
and Q¯ = −Q(rot)∗/(M3χ2). Subsequent work relaxed the tionmechanism:forexample,observationsofNSswithmass
slow-rotation approximation, showing that the universality M . 1M⊙ could validate the astrophysical viability of the
stillholds(Donevaet al.2013;Pappas & Apostolatos2014; proto-NS fragmentation scenario proposed by Popov et al.
Chakrabarti et al. 2014; Yagi et al. 2014a). (2007).
Most investigations of relativistic stellar structure fo- Another key motivation for this work is that the low-
cus on NSs with the “canonical” 1.4M⊙ mass or higher. mass regime is sensitive to – and carries information on
From a nuclear physics standpoint, recent measurements – the isospin dependence of nuclear forces, and in partic-
of masses M & 2M⊙ have ruled out EOS models that are ular on the nuclear symmetry energy (Steineret al. 2005;
unable to support such high masses (Demorest et al. 2010; Tsang et al.2012;Li& Han2013;Lattimer & Steiner2014;
Antoniadiset al.2013)1.Large-massNSsaremorecompact, Li et al. 2014; Newton et al. 2014). Sotaniet al. (2014) re-
and therefore more interesting for tests of strong gravity. centlycomputedthestructureoflow-mass nonrotatingNSs
From an astrophysical point of view, the large-mass regime for a wide range of EOSs. They found that their mass M
is also interesting to improve our understanding of core- and surface redshift z can be fitted by simple functions of
collapse physics. Observations of NS binaries (particularly thecentral densityρ and of thedimensionful parameter
c
via radio pulsars) and black hole X-ray binaries indicate
η≡ K L2 1/3 , (1)
thattheremaybeamassgapbetweenthetwopopulations: 0
thehighestmeasuredNSmassesjustexceed2M⊙ (Lattimer wher(cid:0)eK0is(cid:1)theincompressibilityofsymmetricnuclearmat-
2012),whileblackholemassesmayonlystartat∼4–5.5M⊙ terandListheslopeofthesymmetryenergyatsaturation
(Ozel et al. 2010; Farr et al. 2011), depending on the as- density (note that K , L and η all have units of energy).
0
sumed shape of the distribution (but see Kreidberget al. Therefore, at least in principle, measurements of M and z
2012, who point out that selection biases could yield couldbeusedtoconstrainη;infact,theNSradiusishighly
lowerblackholemasses).Gravitational-waveobservationsof correlated with theNSmatterpressureat densitiesclose to
mergingcompactbinarieswillofferauniqueopportunityto nuclear saturation density. A practical complication is that
probetheexistenceofthis“massgap”(Dominik et al.2015; the determination of z and of the stellar radius, e.g. via
Mandel et al. 2015; Littenberget al. 2015; Stevensonet al. photosphericradiusexpansionburstsandthermalemissions
2015; Belczynski et al. 2015; Chatziioannou et al. 2015a). fromquiescentlow-massX-raybinaries,ismodel-dependent
Our focus here is instead on low-mass NSs. There are and affected by systematic errors. Therefore, at present,no
observational and theoretical reasons why this regime is in- individualobservationcandetermineNSradiitobetterthan
teresting. The lowest well-constrained NS masses currently ∼20%accuracy.Thistranslatesintonearlya100% errorin
measuredareintherange1.0–1.1 M⊙ (Lattimer2012).Re- the determination of L, since L ∼ R4 (Lattimer& Steiner
centlyMartinez et al.(2015)claimedaprecisemeasurement 2014).
of M/M⊙ = 1.174 ± 0.004 in a double NS system with A possible way to circumvent this problem is
large mass asymmetry. While the minimum mass of a star to rely on the fact that all NSs in nature are
constructed from a cold dense matter EOS is quite small spinning. Considering rotating NSs is of interest be-
(<0.1M⊙),theminimummassofahotprotoneutronstaris cause near-future experiments in the electromagnetic
considerably larger, in therange 0.89–1.13M⊙ for themod- spectrum – such as NICER (Gendreau et al. 2012),
els considered byStrobelet al.(1999).Thisminimum mass LOFT (Feroci et al. 2012), Astro-H (Takahashi et al. 2012)
providesapracticallowerboundonNSmassesformedfrom and SKA (Wattset al. 2015) – or in the gravitational-
supernovae, unless lower-mass stars form by fragmentation wavespectrum–suchasAdvancedLIGO(Aasi et al.2015),
(seee.g.thespeculativescenarioofPopov et al.2007).Esti- Advanced Virgo (Acerneseet al. 2015), KAGRA (Somiya
matesbasedonthebaryonicmassoftheironcoreofthesu- 2012) and the Einstein Telescope (Sathyaprakashet al.
pernovaprogenitorgiveaminimummassof∼1.15–1.2M⊙, 2012) – could measure or constrain the additional mul-
as discussed in Sec. 3.3 of Lattimer (2012). Tauris et al. tipoles that determine the structure of rotating NSs or
(2015) estimate that the minimum mass of a NS formed in other properties (such as the “Love numbers”) that are
related to their deformability. Spin-orbit coupling in bi-
nary pulsars may allow us to measure the moment of in-
ertia (Damour & Schaefer 1988; Lattimer & Schutz 2005;
1 Here we will consider some EOS models that do not respect
Bejger et al. 2005; Kramer& Wex 2009) and gravitational-
this constraint. This is because we are primarily interested in
densities ρ ∼ ρ0, and (conservatively) we assume no correlation wave observations may be used to infer the tidal Love
between the EOS near the saturation density and the EOS at numbers, as well as additional information on the EOS
higherdensities(cf.Steineretal.2015). (Mora & Will 2004; Berti et al. 2008; Flanagan & Hinderer
MNRAS000,1–12(2016)
Low-mass neutron stars 3
2008; Read et al. 2009; Hindereret al. 2010; Vineset al.
Table1.ParametersoftheEOSsusedinthiswork.EOSswhich
2011;Damour et al.2012;DelPozzo et al.2013;Read et al. result in NSs with a maximum mass larger than 2M⊙ are indi-
2013; Favata 2014; Yagi& Yunes 2014; Lackey et al. catedbyanasterisk.
2014; Chatziioannou et al. 2015b; Yagi & Yunes 2015;
Dietrich et al.2015).Quiteremarkably,measurementsofthe
EOS K0 (MeV) L(MeV) η (MeV)
momentofinertiawithinanaccuracy∼10%alonecanyield
tight constraints on the pressure over a range of densities OI180/220 180 52.2 78.9
OI230/350 230 42.6 74.7
(Steineret al. 2015). The correlation between the moment
OI230/220 230 73.4 107
ofinertiaandthetidaldeformability(the“I-Love”relation)
OI280/350 280 54.9 94.5
is very tight for massive NSs (Yagi& Yunes 2013b,a), but OI280/220∗ 280 97.5 139
not so much in thelow-mass regime: see e.g. Fig. 3 below. OI360/350∗ 360 76.4 128
One of the main results of the present work is that all OI360/220∗ 360 146 197
of the properties of rotating and tidally deformed stars can
beexpressedassimplefunctionsofρcandη.Thereforemea- Shen∗ 281 114 154
surementsofanytwobulkpropertiesofalow-massNS–for Miyatsu 274 77.1 118
example,themassM andthemoment ofinertiaI –can be
used to determine a region in the ρ -η plane, permitting to FPS 261 34.9 68.2
c SLy4∗ 230 45.9 78.5
estimate η.
BSk19 237 31.9 62.3
The plan of the paper is as follows. In Section 2 we
BSk20∗ 241 37.9 69.6
discusstheEOSmodelsusedinthispaperandtheproperties BSk21∗ 246 46.6 81.1
of nuclear matter that are relevant in the low-mass regime.
In Section 3 we present our numerical results for the bulk
properties of nonrotating and slowly rotating NSs, and we
fitthepropertiesof slowly rotating NSsby nearly universal
functions of the central density ρ and of the parameter η.
c order to derive empirical formulas expressing the proper-
IntheconcludingSection4wediscusspossibleobservational
ties of low-mass NSs that do not rely on specific EOSs, fol-
applications and future extensionsof our work.
lowing Sotani et al. (2014) we adoptseveral tabulated EOS
models that can beseparated intothree categories:
2 LOW-MASS NEUTRON STAR PROPERTIES (1) The phenomenological EOS model constructed by
AND THE NUCLEAR SYMMETRY ENERGY Oyamatsu and Iida (Oyamatsu & Iida 2003). The bulk en-
ergyw(n ,α)isconstructedtoreproduceEq.(2)inthelimit
Inthissectionweintroducesomenotationfortheproperties b
wheren →n andα→0,andtheoptimalvaluesofw ,n ,
of uniform nuclear matter near saturation density, and we b 0 0 0
and S are determined by requiringthat thedensity profile
0
describe theEOS models used in ournumerical work.
of stable nuclei (determined within the extended Thomas-
FermitheoryforgivenvaluesofLandK )reproduceexper-
0
2.1 Properties of uniform nuclear matter imental nuclear data. The EOSsin this category will be la-
beled as OI K/Y, where K =K and Y =−K S /(3n L).
0 0 0 0
The energy of uniform nuclear matter at zero temperature At variance with Sotaniet al. (2014) we will omit the OI
can be expanded around the saturation point of symmetric 180/350EOS,becausetheassociatedvaluesofK andLare
0
nuclearmatter(i.e.,mattercomposedofanequalnumberof ruledoutbycurrentnuclearphysicsconstraints(cf.Sec.2.2).
neutrons and protons). If nb is the nucleon numberdensity (2) TwoEOSmodelsbasedontherelativisticframework.
andα≡(nn−np)/nb,wherenn(np)istheneutron(proton) One of these models (Shen) is constructed within relativis-
number density, the bulk energy per nucleon w of uniform ticmeanfieldtheorytogetherwiththeTM1nuclearinterac-
nuclearmatter can bewritten as tion(Shenet al.1998);thesecond(Miyatsu)isbasedonthe
K L relativisticHartree-Focktheorywiththechiralquark-meson
w=w + 0 (n −n )2+ S + (n −n ) α2, (2)
0 18n20 b 0 (cid:20) 0 3n0 b 0 (cid:21) coupling model (Miyatsu et al. 2013). The spherical nuclei
in thecrust region are determined using theThomas-Fermi
where w , n andK are thesaturation energy, thesatura-
0 0 0 theory.
tion density andtheincompressibility of symmetricnuclear
(3) Five EOS models based on the Skyrme-type
matter, while S and L are associated with the symmetry
0 effective interactions: FPS, SLy4, BSk19, BSk20,
energy coefficient S(n ):
b and BSk21 (Lorenz et al. 1993; Douchin & Haensel
dS 2001; Goriely et al. 2010; Pearson et al. 2011, 2012;
S =S(n ), L=3n . (3)
0 0 0 (cid:18)dn (cid:19)(cid:12) Potekhin et al. 2013).
b (cid:12)nb=n0
(cid:12)
Theparametersw ,n andS c(cid:12)anberelativelyeasilydeter- All of these models are unified EOSs, i.e., both the crust
0 0 0
minedfromempiricaldataforthemassesandradiiofstable and core regions can be described with the same EOS with
nuclei.TheparametersK andL,whichdeterminethestiff- specific values of K and L. From the EOS tables we can
0 0
nessofneutron-richnuclearmatter,aremoredifficulttofix, computeK , Land theauxiliary dimensionfulparameter η
0
and they affect thestructureof low-mass NSs. introduced in Eq. (1) above, with the results listed in Ta-
Different EOS models are based on different theoret- ble 1. The mass-radius relations predicted by these EOS
ical and computational approaches in nuclear physics. In models for nonrotating low-mass NSsare shown in Fig. 1.
MNRAS000,1–12(2016)
4 Hector O. Silva et al
2.7 Lattimer & Steiner 2014; Newton et al. 2014). One possi-
OI180/220 ble reason for this discrepancy may be that the constraints
2.4 OI230/220 from nuclear experiments were obtained from almost sta-
2.1 OI230/350 ble nuclei, whose neutron excess α in Eq. (2) is very small,
OI280/220 while NS matter deviates significantly from symmetric nu-
1.8 OI280/350 clearmatter.Indeed,somenuclearexperimentswith unsta-
⊙ 1.5 OI360/220 ble nuclei suggest the possibility that L may have larger
M OI360/350 values(Tsang et al. 2009; Yasudaet al. 2013).
M/ 1.2 J0453+1559 FPS If we adopt as fiducial values 40 MeV< L <
SLy4
0.9 Miyatsu 80 MeV – recalling that higher values cannot be ex-
Shen cluded (Newton et al. 2014) – and K0 = 230 ± 40 MeV
0.6
BSk19 (Khan& Margueron 2013), respectively, we can conclude
0.3 BSk20 thataplausiblerangeforηis67<η<120,andthathigher
BSk21 valuesofη maybepossible. Thisshouldbekeptinmindin
0.0
8 10 12 14 16 18 20 Section 3 below, where we discuss how the bulk properties
R[km] of NSsdependon η.
Figure1.Mass-radiusrelationsfortheEOSsadoptedinthispa-
per,anddiscussedinSec.2.ThecurvesspanNSmodelsstarting
3 NEUTRON STAR STRUCTURE
from uc = ρc/ρ0 = 0.9 up to the value of uc resulting in the
maximummassallowedbyeachEOS.Thesolidhorizontal band 3.1 Nonrotating neutron stars
correspondstothelowesthigh-precisionNSmassmeasurementof
M/M⊙=1.174 ± 0.004(fromMartinezetal.2015).Thedotted For nonrotating NSs, the effect of the nuclear symmetry
horizontal lineindicates aconservative lowerboundonthemass parameters introduced above on the mass-radius relation
of M/M⊙ =0.89 (see e.g. Strobeletal. 1999). Symbols on each M(R) was investigated by Sotani et al. (2014). The main
linecorrespondtouc=1.5anduc=2.0,respectively. findingof theirwork was that theNS mass M and thesur-
face gravitational redshift z, definedas
2.2 Experimental constraints 2M −1/2
z= 1− −1, (4)
Anextensivediscussionoftheoreticalandexperimentalcon- (cid:18) R (cid:19)
straints on the properties of uniform nuclear matter can be can be expressed as smooth functionsof η of theform
found in Li & Han (2013), Lattimer& Steiner (2014) and
η
NtheewrtaonngeetKal.(=2021340).±G4e0neMraelVly(aKccheapnte&dvMalauregsuoefroKn02a0r1e3i)n. y=c0+c1(cid:16)100MeV(cid:17) , (5)
0
whereycollectivelydenoteseitherM orz.Wealsonotethat
The current consensus in the nuclear physics community is
thatvaluesofLintherangeL=60±20MeVareplausible2 these relations can be combined to write the radius R as a
function of η.
buthigher valuesare not excluded(Newton et al. 2014).
Thecoefficientsc dependontheratiou ≡ρ /ρ which
Thereareproposalstoconstrainnuclearsaturationpa- i c c 0
specifies the central density of the stellar model. Follow-
rameters, and in particular the value of L, via astronomi-
ing Sotani et al. (2014), we will fit these coefficient using
calobservations.Thisapproachiscompletelydifferentfrom
a quadraticpolynomial in u :
constraintsbasedonnuclearphysicsexperiments.Underthe c
assumption that the observed frequencies of quasiperiodic c =c +c u +c u2. (6)
i i,0 i,1 c i,2 c
oscillations (QPOs) in NSs are related to crustal torsional
oscillations, Sotani et al. (2013) found theconstraint 101< Therefore each of ourempirical formulas will dependon six
L < 131 MeV when all the observed frequencies are inter- constant parameters ci,j.
preted as torsional oscillations, or 58 < L < 85 MeV when In Fig. 2 we confirm the main results of Sotani et al.
the second lowest frequency is assumed to have a different (2014). The left (right) panel shows that, quite indepen-
origin.Theinclusionofelectronscreeningeffectscanmodify dently of the chosen EOS, the mass M (the redshift z, re-
theformer constraint on L totherange 97<L<127 MeV spectively) is indeedwell fittedby alinear function of η for
(Sotaniet al. 2016). Furthermore, in some cases having in- any fixed value of the central density uc: the plots show
formationonthemassandradiusoflow-massNSsmayallow this explicitly in the three cases uc = 1, uc = 1.5 and
us to constrain EOS parameters. For example, theobserva- uc = 2. The bottom insets show that the fractional dif-
tion of the X-ray burster 4U 1724-307 by Suleimanov et al. ferences δy ≡ |ydata−yfit|/ydata for M and z are typically
(2011) allowed Sotaniet al. (2015) to set the constraint below ∼10% (with theexception of EOS BSk21) whenever
η & 130 MeV. These astrophysical constraints seem in- η&67 MeV.
compatible with constraints on L obtained from the ter- The values of the fittingconstants are listed in the top
restrial nuclear experiments quoted above (Li & Han 2013; two rows of Table 2. To quantify the accuracy of these fits,
Table 2 also lists therms relative error
2 Lattimer&Steiner (2014) suggest a tighter plausible range of σ≡v 1 N 1− yifit 2, (7)
44 MeV< L < 66 MeV (see the discussion of their Figure 1), uN (cid:18) ydata(cid:19)
butwewillfollowNewtonetal.(2014)inanattempttobemore ut Xi=1 i
conservative. where the sum runs over all stellar models i = 1, ...,N.
MNRAS000,1–12(2016)
Low-mass neutron stars 5
2.0 0.25
OI180/220 OI180/220
2.0ρ0 OI230/220 2.0ρ0 OI230/220
OI230/350 0.20 OI230/350
1.5 OI280/220 OI280/220
1.5ρ0 OOII238600//325200 0.15 1.5ρ0 OOII238600//325200
]⊙
M 1.0 OI360/350 z OI360/350
M[ FSLPyS4 0.10 FSLPyS4
1.0ρ0 Miyatsu 1.0ρ0 Miyatsu
0.5 Shen Shen
BSk19 0.05 BSk19
BSk20 BSk20
BSk21 BSk21
0.0 0.00
100 100
10−1 1.0ρ0 10−1 1.0ρ0
δM10−2 21..05ρρ00 δz10−2 21..05ρρ00
10−3 10−3
50 100 150 200 50 100 150 200
η[MeV] η[MeV]
Figure 2.Left:Dependence oftheNSmassM foranonrotatingstarontheparameterη introducedinEq.(1)atthreegivenvaluesof
thecentraldensity:uc≡ρc/ρ0=1.0,1.5,2.0.Right:same,butforthesurfacegravitationalredshiftz definedinEq.(4).Inbothcases,
the solidlines represent the fit given inEq. (5) usingthe fitting parameters listedinTable 2. The lower panels show the relative error
ofthefitwithrespecttothenumericaldata,|ydata−yfit|/ydata,asafunctionofη.Theshadedareacorrespondstothemostplausible
range of values for η, namely 67 < η < 120 (see Sec. 2.2). This plot reproduces and extends Fig. 2 of Sotanietal. (2014). Horizontal
dashedlinescorrespondtoaNSwithM/M⊙=0.89(thevalueofz wascomputedusingtheShenEOS).
Table 2.Numericalvalues oftheconstant inthefittingexpressionsandofthermspercentage errorσ (lastcolumn), computed with/
withoutEOSBSk21.
UsingEqs.(5)and(6)–QuantitiesatorderO(ǫ0)inrotation
Quantity c0,0 c0,1 c0,2 c1,0 c1,1 c1,2 σ
M 0.34626 -0.82183 0.29265 -0.60780 1.2996 -0.25890 0.086/0.076
z=(1−2M/R)−1/2−1 0.0040470 -0.059527 0.026591 -0.042719 0.10673 -0.014208 0.100/0.089
UsingEqs.(9)and(6)–Quantities atorderO(ǫ1)andO(ǫ2)inrotation
Quantity c0,0 c0,1 c0,2 c1,0 c1,1 c1,2 σ
I¯=I/M3 4.1429 -2.2458 0.46120 -0.23654 -0.26292 0.083322 0.122/0.083
Q¯=−Q(rot)∗/(M3χ2) 2.9160 -1.3835 0.24677 -0.25594 -0.093784 0.015956 0.120/0.114
λ¯(rot)=λ(rot)/M5 11.203 -5.8769 1.1697 -0.24302 -0.21457 0.055320 0.424/0.237
k(rot)=(3/2)λ(rot)/R5 -3.9878 3.3914 -0.91026 -3.4378 2.6267 -0.53179 0.273/0.272
2
e∗ =−Q(rot)∗/I -2.1203 1.8784 -0.52335 -4.0307 3.0883 -0.55984 0.129/0.126
Q
UsingEqs.(9)and(6)–Tidallydeformedstar
Quantity c0,0 c0,1 c0,2 c1,0 c1,1 c1,2 σ
λ¯(tid)=λ(tid)/M5 11.238 -5.9413 1.1450 -0.21434 -0.25432 0.052281 0.462/0.263
Two EOS models, namely BSk20 and BSk21 (and particu- 3.2 Slowly rotating and tidally deformed neutron
larly the latter), deviate more from our best-fit function as stars
ρ increases.AspointedoutbySotaniet al.(2014),thesede-
c Ingeneral,rotatingstellarmodelsingeneralrelativitymust
viationsare of theorder oftheuncertantieson themass M
beconstructednumericallybysolvingacomplicatedsystem
due to three-neutron interactions obtained from the quan-
of partial differential equations. These numerical calcula-
tumMonteCarlo evaluations(Gandolfi et al.2012).There-
tions(reviewedinStergioulas2003;Friedman & Stergioulas
foreinTable2welistthevaluesofσ obtainedeitherinclud-
2013) suggest that uniformly rotating NSs with phys-
ing or omitting EOS BSk21, the EOS for which the errors
ically motivated EOSs have dimensionless angular mo-
are larger.
mentum χ . 0.7 (Cook et al. 1994; Berti & Stergioulas
2004; Lo& Lin 2011), but the spin magnitudes of NSs
MNRAS000,1–12(2016)
6 Hector O. Silva et al
in binary systems observable by Advanced LIGO are At second order in rotation, i.e. O(ǫ2), the star de-
likely to be much smaller than this theoretical upper viates from its spherical shape and acquires a rotational
bound(Mandel& O’Shaughnessy2010;Brown et al.2012). quadrupolemoment Q(rot)∗. For convenience, we define the
The spin period of isolated NSs at birth should be in the dimensionless rotation-induced quadrupole moment Q¯ ≡
range 10-140 ms (or χ . 0.04, Lorimer 2001). Accretion −Q(rot)∗/ M3χ2 . The ℓ=2 rotational Love numberλ(rot)
fromabinarycompanioncanspinupNSs,butitisunlikely can be de(cid:0)fined in(cid:1)terms of Q(rot)∗ as λ(rot) ≡ −Q(rot)∗/Ω2,
toproduceperiodslessthan1ms(i.e.χ.0.4,Chakrabarty and it can be made dimensionless by definingλ¯(rot) ≡I¯2Q¯.
2008). The fastest spinning observed pulsar, PSR J1748- A quantity closely related with λ(rot) is the ℓ = 2 ap-
2446ad,hasaperiodof1.4ms(χ∼0.3,Hessels et al.2006); sidal constant, defined as k(rot) ≡ (3/2)λ(rot)/R5; note
2
the fastest known pulsar in a NS-NS system, J0737-3039A, that Berti et al. (2008) used a different definition. Follow-
has a period of 22.70 ms (χ∼0.02, Burgay et al. 2003). ingColaiuda et al.(2008),wecanalsodefinethequadrupo-
The perturbative formalism to construct slowly rotat- lar rotational ellipticity3 e∗ ≡ −Q(rot)∗/I. Note that all of
Q
ingNSmodelswasdevelopedintheseminalworksbyHartle the barred quantities defined above (as well as k(rot)) are
2
(1967)andHartle& Thorne(1968).Theformalismbasically
independentoftheactualvalueoftherotationparameterǫ.
consists of an expansion in terms of the small parameter
ǫ ≡ Ω/Ω∗ ≪ 1, where Ω is the stellar angular velocity
and Ω∗ ≡ M/R3 is a characteristic rotation frequency, 3.3 Tidally deformed neutron stars
comparablepinorderofmagnitudetothemass-sheddingfre-
We will also be interested in the tidal deformation of a NS
quencyofthestar.Subsequentworkextendedtheformalism
due the presence of an orbiting companion, e.g. in a bi-
uptofourthorderinǫ,showingthattheequilibriumproper-
narysystem.Theresponsetotidaldeformationsis encoded
tiesof slowly rotatingsolutions comparefavorably with nu-
in the so-called ℓ = 2 tidal Love number λ(tid) (Hinderer
merical codesfor arbitrary rotation rates (Berti et al. 2005;
2008, 2009; Damour& Nagar 2009; Binnington & Poisson
Benharet al. 2005; Yagi et al. 2014a) even for the fastest
2009; Vineset al. 2011), which is potentially measurable
known milisecond pulsar PSR J1748-2446ad (Hessels et al.
byadvancedgravitational-wave interferometers. This quan-
2006).Thispulsarspinswellbelowtheestimatedǫ≈0.5for
tity is in general spin-dependent (Landry& Poisson 2015a;
which theHartle-Thorneapproximativeschemeagrees very
Pani et al. 2015a,b; Landry & Poisson 2015b), but for sim-
well with full numerical calculations.
plicitywewillassumethatthetidallydeformedNSisnonro-
Theslow-rotation approximationisbasically anexpan-
sionintermsofthesmallparameterǫ≡Ω/Ω∗ ≪1,whereΩ tating.ThetidalLovenumbercanbeputinadimensionless
form bydefiningλ¯(tid) ≡λ(tid)/M5.Wecalculated thetidal
is the stellar angular velocity and Ω∗ ≡ M/R3 is a char-
Love numberusing the structureequations as presented by
acteristic rotation frequency, comparablepin order of mag-
Postnikov et al. (2010), and validated our results by com-
nitude to the mass-shedding frequency of the star. Even
parison against Yagi & Yunes(2013b).
the fastest known millisecond pulsar (Hessels et al. 2006)
spinswellbelowtheestimatedǫ≈0.5forwhichtheHartle-
Thorneapproximativeschemeagreesverywellwithfullnu- 3.4 Empirical relations for slowly rotating and
merical calculations (Berti et al. 2005). Therefore the slow- tidally deformed neutron stars
rotationapproximationismorethanadequatetoextendthe
work on low-mass NSsby Sotaniet al. (2014). We constructed NS models for all of the 14 EOS mod-
The formalism to construct slowly rotating NS mod- els listed in Table 1. We integrated the structure equa-
els was developed in the seminal works by Hartle (1967) tions for central total energy densities within the range
andHartle& Thorne(1968).Subsequentworkextendedthe uc ≡ ρc/ρ0 ∈[1.0,2.0] in increments ∆uc =0.1, for a total
formalism uptofourth orderinǫ,showing thattheequilib- of154stellarmodels. Weverifiedthatthenormalized bind-
rium properties of slowly rotating solutions compare favor- ing energy Mb/M −1 (where Mb is the baryonic mass) is
ably with numerical codes for arbitrary rotation rates even positive,sothatallofthesestellarconfigurationsarebound.
forPSRJ1748-2446ad(Berti et al.2005;Benharet al.2005; Our results for the I-Love-Q relations are shown in
Yagi et al. 2014a). Fig. 3, which confirms the main findings of Yagi& Yunes
We use the stellar structure equations as presented by (2013b):theuniversality holdswithin afew percent,except
Sumiyoshiet al. (1999), correcting the misprints listed by for very low-mass stars. This breakdown of the I-Love-Q
Berti et al. (2005). Ournumericalresults were validated by relations was already visible e.g. in Fig. 9 of Yagi& Yunes
comparisonagainstthetablesbyBerti et al.(2005).Forthe (2013b), but it is much more noticeable in the low-mass
dimensionless bulk properties we follow the definitions of range explored in this work. Yagiet al. (2014b) suggested
Yagi & Yunes (2013b). The explicit form of the structure that the I-Love-Q relations hold because of an approxi-
equations,theirderivationanddetailsoftheintegrationpro- mate self-similarity in the star’s isodensity contours. This
cedurecan befound in thesereferences. approximate symmetry only holds for compact stars, but
At order O(ǫ0) in the perturbative expansion, a static it is broken in low-mass NSs, white dwarfs and ordinary
nonrotatingstarischaracterizedbyitsgravitationalmassM
and radius R. Sometimes it is useful to replace the radius
3 This quantity is different from the surface ellipticity es ≡
bythe surface redshift z definedin Eq.(4).
re/rp −1, where re and rp are the equatorial and polar radii
At first order in rotation, i.e. O(ǫ1), the star is also oftheoblaterotatingstar,respectively. Thesurfaceellipticityis
characterized by its moment of inertia I. Given I, we can related to the so-called“eccentricity”e ≡ (cid:2)(re/rp)2−1(cid:3)1/2 (see
define a dimensionless moment of inertia I¯≡I/M3 as well e.g. Bertietal. 2005) by e2 =e2s +2es, and it describes the ge-
as a spin parameter χ≡IΩ/M2 (Yagi& Yunes2013b). ometryofthestar.
MNRAS000,1–12(2016)
Low-mass neutron stars 7
103 103 103
OI180/220
OI230/220
OI230/350
OI280/220
102 OI280/350
OI360/220
¯I102 ¯I102 ¯Q OI360/350
FPS
SLy4
101
Miyatsu
Shen
BSk19
BSk20
BSk21
101 101 100
100 100 100
10−1 10−1 10−1
¯I10−2 ¯I10−2 ¯Q 10−2
δ10−3 δ10−3 δ 10−3
10−4 10−4 10−4
1 10 100 1000 102 103 104 105 106 107 108 109 102 103 104 105 106 107 108 109
Q¯ λ¯(tid) λ¯(tid)
Figure3.TheEOS-independentI-Love-Qrelations(Yagi&Yunes2013a,b)inthelow-massregime.ThedifferentpanelsshowtheI-Q
(left),I-Love(center)andLove-Q(right)relationswithintherangeofcentralenergydensitiesconsideredhere.Forreference,thevertical
dashed line corresponds to the values of Q¯ and λ¯(tid) of a NS model using the Shen EOS with M/M⊙ =0.89. The lower panels show
thatthefractionaldeviationsintheI-Love-Qrelationsincreaseforverylowmass(i.e.,largervaluesofQ¯ andλ¯(tid)).Nevertheless,near
andabovetheminimummassvalue0.89M⊙ therelationsholdwithinanaccuracy <2%.Theexplicitfunctional formoftheI-Love-Q
relationscanbefoundinYagi&Yunes(2013b,Eq.(54)andTable1).Observethatevenforverylow-massNSstheuniversalityremains,
althoughitisnotcapturedbytheI-Love-Qrelations.
stars.Indeed,theI-Love-QrelationspresentedinYagi et al.
1010
OI180/220 (2014b) were obtained by fitting data in the range Q¯ < 20
109 OI230/220 and λ¯(tid)<2×104 (Yagi 2016).
108 OI230/350 In Fig. 4 we show that a universal “Love-Love” rela-
OI280/220
tion also holds between the tidal and rotational Love num-
107 OI280/350 bers.Awell-knownresultinNewtoniangravityisthattidal
OI360/220
¯(rot)λ 106 OFPIS360/350 2a0n0d4)r.oTtahtiisoneqaluaLliotvyenonulomnbgeerrshaorldestthreuesfaomrerel(aMtiovrisati&csWtairlsl
105 SLy4 (Berti et al. 2008; Yagi & Yunes 2013b). Therefore we pro-
Miyatsu
104 Shen pose a differentfit, namely:
BSk19
103 BSk20 4 j
BSk21 lnλ¯(rot) = kj lnλ¯(tid) , (8)
102 Xj=0 (cid:16) (cid:17)
100
¯(rot)δλ111000−−−321 wk3h=ere−k80.9=8921.1×08190,−k41a=nd6.k5408=4×1.31908−51,×k120=−52..4T6h8i8s×fit10u−se2s,
10−4 datainthecentraldensityrangeuc ∈[0.9, 2.0],anditworks
102 103 104 105 106 107 108 109
accurately in therange of masses of our interest.
λ¯(tid)
Using these numerical calculations we then fitted the
various bulk properties of NSs as functions of η and of the
Figure4.TheEOSindependentLove-Loverelationsinthelow- central density.The quantitiescharacterizing rotating stars
mass regime. The Love-Love relation between λ¯(tid) and λ¯(rot) – namely I¯, Q¯, the ℓ = 2 rotational Love number λ¯(rot) ≡
becomes an equality in the Newtonian – i.e., small-M – limit I¯2Q¯, the ℓ =2 apsidal constant k(rot) and the quadrupolar
(Mora&Will2004),anddeviatesfromunity(solidline)formore ellipticity e∗ – are well fittedby fu2nctionsof theform
relativisticstars.Tomakeourfitsofλ¯(tid)andλ¯(rot)withrespect Q
toηusefulincombinationwiththeLove-Loverelation,wederived η c1
the improved fit of Eq. (8), corresponding to the dash-dotted log10y=c0(cid:16)100MeV(cid:17) . (9)
line. This fit is accurate within < 1% for M/M⊙ ≥ 0.89 (lower
panel). The vertical dashed line corresponds to NS model with JustasforthenonrotatingbulkpropertiesofNSs,thecoeffi-
massM/M⊙=0.89usingtheShenEOS. cientsci dependontheratiouc ≡ρc/ρ0,whichspecifiesthe
central density of the stellar model. Following Sotani et al.
(2014), we fitted the c ’s by quadratic polynomials of the
i
MNRAS000,1–12(2016)
8 Hector O. Silva et al
metric nuclear matter and L is the slope of the symmetry
−0.10 1.2
energyatsaturationdensity.Thecoefficientsofthesefitting
−0.15 1.0
0.8 relations are summarized in Table 2.
c0 −0.20 0.6c1 The main conclusion of this work is that the measure-
−0.25 M 0.4 mentofanytwoofthesebulkpropertiesinlow-massNSscan
−0.30 0.2 beused–atleastinprinciple–toinferthevaluesofρc and
3.0 −0.40 η, providing important information on the EOS. However
2.5 I¯ −0.42 thereare some important practical caveats.
−0.44 Firstandforemost–asshowninthelowerpanelsofFig-
c0 2.0 −0.46c1 ures6–8–thefittingrelationsareapproximate,withrelative
1.5 −0.48 errorsthattypicallygetlargerforthelowestplausiblevalues
1.0 −0.50 ofη.Furthermore,asdiscussedintheliterature,constraints
2.0 −0.35 onthebulkpropertiesofNSsandontheEOSrequireMonte
1.8 Q¯ −0.40 Carlo simulations or dedicated Bayesian studies which are
1.6 −0.45
c0 1.4 −0.50c1 beyond the scope of this paper (Steineret al. 2010, 2013,
2015; Lattimer & Steiner2014).
1.2 c0 c1 −0.55 Even assuming accurate measurements of two of the
1.0 −0.60
1.0 1.2 1.4 1.6 1.8 2.0 bulk properties of a given NS – to be concrete, say M and
u
c I – aconceptual limitation is that not all EOSs predict the
existenceof NSswith“realistic”masses (say,M >0.89M⊙)
Figure 5. Illustration of the behavior of c0 (circles) and c1 intherange1≤uc ≤2whereourfittingrelationshavebeen
(squares), appearing in the fitting expressions (5) and (9), as derived(cf.Fig.1).Thisproblemcaninprinciplebecircum-
functions of uc. Theci’sareshownforthreerepresentative bulk vented,becausethemeasurementofM andI canbeusedto
properties of NSs: the mass M (top) fitted using Eqs. (5) and inferbothu andη,andthustoverifywhethertheNSreally
(6); the dimensionless moment of inertia I¯(center) fitted using c
has central densities in the range of interest. However it is
Eqs. (9) and (6); and the dimensionless quadrupole moment Q¯
possiblethatsystematicerrorscouldspoiltheseconsistency
(bottom)fittedusingEqs.(9)and(6).
tests. For example one could imagine a situation where the
“true”central density corresponds to (for example) u = 3,
c
form (6),sothateach ofourempirical formulasdependson butsinceweareapplyingthefittingrelationsoutsideoftheir
six constant parameters ci,j. region of validity, we recover values of uc ∈[1, 2] and get a
The quality of the fits is shown in the three panels of wrongestimateforη(Yagi2016).Thesedataanalysisissues
Fig. 5 for three representative bulk NS properties, namely deservefurtherstudy.
themassM,thedimensionlessmomentofinertiaI¯andthe The recent discovery of universal I-Love-Q relations is
dimensionlessquadrupolemomentQ¯.Insummary:thenon- also helpful. For example we can imagine measuring (say)
rotatingbulkpropertiesofNSscanbeexpressedasfunctions M, Rand I¯and gettinginformation on theremaining bulk
of η of the form (5) combined with Eq. (6); the bulk prop- propertiesbyexploitingtheI-Love-Qrelations. Ameasure-
erties of rotatingand tidally deformed NSscan befittedby ment of multiple parameters for the same astrophysical NS
functions of the form (9) combined with Eq. (6). The fit- canbecombinedwithourfittingformulaseithertocheckthe
tingcoefficients arelisted in Table2, thatcollects themain consistency of the inferred values of uc and η, or to reduce
results of our investigation. statistical and/or systematic errors.
The fits and their accuracy are also presented graphi- Weconcludebyspeculatingonsomeobservationalpos-
callyinFigures6–8.Inthelowerpanelofeachfigureweplot sibilities to implement thisprogram.
the fractional differences δy. As in Fig. 2, the shaded area Perhaps the most promising avenue in the near fu-
corresponds to the most plausible range of values for η as ture is the measurement of mass and moment of inertia
discussedinSec.2.2.Thesefiguresshowthatingeneralthe through relativistic spin-orbit coupling in systems such as
fitsworkquitewellinourfiducialrange,i.e.forη>67MeV, the“double pulsar”PSR J0737-3039 (Burgay et al. 2003),
with larger errors for small η. especially considering that a 10% measurement of the mo-
ment of inertia alone can yield tight constraints on the
pressure over a range of densities to within 50 − 60%
(Steineret al. 2015). This possibility was discussed by var-
4 CONCLUSIONS AND OUTLOOK
ious authors (Damour& Schaefer 1988; Lattimer& Schutz
In this paper we have integrated the Hartle-Thorne equa- 2005;Bejger et al.2005).Theexperimentalchallengesasso-
tionsforanextensivesetofEOSmodels.Wehavecomputed ciated with these measurements in the case of the double
thebulkpropertiesofnonrotating(massM andradiusR,or pulsar are reviewed in (e.g.) in Section 6 of Kramer & Wex
equivalently mass M and surface redshift z), rotating (mo- (2009).
ment of inertia I, quadrupole moment Q, quadrupole ellip- Inthenearfutureitmayalsobepossibletoconstrainη
ticity eQ, rotational Love number λ(rot), ℓ= 2 apsidal con- by gravitational-wave observations. Low-mass isolated NSs
stant k(rot)) and tidally deformed (ℓ=2 tidal Love number are relatively promising gravitational-wave sources because
2
λ(tid)) low-mass NSs. All of these bulk NS properties can they are more deformable and their crusts can support
be fitted by relatively simple functions of the central den- larger ellipticities, generating stronger gravitational-wave
sity (more precisely, of u ≡ ρ /ρ ) and of the parameter signals (Horowitz 2010; Johnson-McDaniel & Owen 2013;
c c 0
η ≡ (K L2)1/3, where K is the incompressibility of sym- Johnson-McDaniel2013,cf.alsoourFigs.6and7).However
0 0
MNRAS000,1–12(2016)
Low-mass neutron stars 9
103 103
OI180/220 OI180/220
OI230/220 OI230/220
OI230/350 OI230/350
OI280/220 OI280/220
OI280/350 102 OI280/350
OI360/220 OI360/220
¯I102 1.0ρ0 OI360/350 ¯Q 1.0ρ0 OI360/350
FPS FPS
SLy4 SLy4
Miyatsu 101 1.5ρ0 Miyatsu
1.5ρ0 Shen 2.0ρ0 Shen
BSk19 BSk19
2.0ρ0 BSk20 BSk20
BSk21 BSk21
101 100
100 100
10−1 1.0ρ0 10−1 1.0ρ0
¯I 1.5ρ0 ¯Q 1.5ρ0
δ10−2 2.0ρ0 δ 10−2 2.0ρ0
10−3 10−3
50 100 150 200 50 100 150 200
η[MeV] η[MeV]
Figure 6.Left:fitofI¯≡I/M3.Right:FitofthereducedquadrupolemomentQ¯≡Q(rot)∗/(M3χ2).Thehorizontal dashedlineinthe
left(right)panelmarksthevalueofI¯(Q¯)foraNSwithM/M⊙=0.89andtheShenEOS.
109 109
OI180/220 OI180/220
108 OI230/220 108 OI230/220
OI230/350 OI230/350
107 OI280/220 107 OI280/220
OI280/350 OI280/350
106 OI360/220 106 OI360/220
¯(rot)λ105 1.0ρ0 OFPIS360/350 ¯(tid)λ105 1.0ρ0 OFPIS360/350
SLy4 SLy4
104 Miyatsu 104 Miyatsu
1.5ρ0 SBhSekn19 1.5ρ0 SBhSekn19
103 103
2.0ρ0 BSk20 2.0ρ0 BSk20
BSk21 BSk21
102 102
100 100
¯(rot)δλ1100−−21 211...050ρρρ000 ¯(tid)δλ1100−−21 211...050ρρρ000
10−3 10−3
50 100 150 200 50 100 150 200
η[MeV] η[MeV]
Figure 7. Left: Fit of the rotational Love number λ¯(rot). Right: Fit of the tidal Love number λ¯tid. The horizontal dashed linein the
left(right)panelmarksthevalueofλ¯(rot) (λ¯(tid))foraNSwithM/M⊙=0.89andtheShenEOS.
the characteristic gravitational-wave amplitude depends on timedelay.AccordingtotheirMonteCarlosimulations,the
a (generally unknown) geometrical factor involving the ori- mass of a NS with spin frequency f = 500 Hz and ellip-
entationoftheNSandtheantennapatternofthedetectors ticity 10−6 at 1 kpc is typically measurable with an ac-
(seee.g.Bonazzola & Gourgoulhon1996;Dhurandharet al. curacy of 20% using the Einstein Telescope. Higher-order
2011).Ithasrecentlybeenproposedthatgravitational-wave terms in the Shapiro time delay will depend on the higher
measurementsofastochasticgravitational-wavebackground multipolemomentsI andQ,andtheymayallowustomea-
fromrotatingNSscouldbeusedtoconstraintheaverageNS sure these moments and constrain η. It may also be possi-
ellipticity,and(ifconstraintsonthemassescanbeobtained) ble to combine the empirical relations of the present work
these measurements could also constrain η. This possibility with similar fitting relations that have been developed in
seemsmostpromisingforthird-generationdetectorssuchas the context of gravitational-wave asteroseismology (see e.g.
theEinstein Telescope (Talukderet al. 2014). Doneva& Kokkotas2015).
Ono et al. (2015) proposed to estimate the mass of Last but not least, as mentioned in the introduc-
an isolated rapidly rotating NS by exploiting the mass- tion, Advanced LIGO observations of binary systems in-
dependent logarithmic phase shift caused by the Shapiro volving NSs could yield measurements of masses and
MNRAS000,1–12(2016)
10 Hector O. Silva et al
100 100
OI180/220 OI180/220
2.0ρ0 OI230/220 2.0ρ0 OI230/220
OI230/350 OI230/350
10−1
OI280/220 OI280/220
OI280/350 OI280/350
∗eQ 10−1 1.5ρ0 OOFPIIS336600//232500 (rot)k2 10−2 1.5ρ0 OOFPIIS336600//232500
SLy4 1.0ρ0 SLy4
1.0ρ0 Miyatsu Miyatsu
Shen 10−3 Shen
BSk19 BSk19
BSk20 BSk20
BSk21 BSk21
10−2 10−4
100 100
∗δeQ 1100−−21 211...050ρρρ000 (rot)δk2 1100−−21 211...050ρρρ000
10−3 10−3
50 100 150 200 50 100 150 200
η[MeV] η[MeV]
Figure 8. Left: Fit of the quadrupole ellipticity e∗. Right: fit of the ℓ = 2 rotational apsidal constant k(rot). Both quantities behave
Q 2
similarly,becomingnearlyindependentofηforρc=2.0ρ0.Forlargevalueofη,weseethatk2rot≈0.7irrespectiveofthecentraldensity.
Thehorizontaldashedlineintheleft(right)panel marksthevalueofe∗Q (k2(rot))foraNSwithM/M⊙=0.89andtheShenEOS.
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