Quark matter with strong magnetic field and possibility of the third family of compact stars

0.56 MB·

by  Hajime Sotani

Mon.Not.R.Astron.Soc.000,000–000 (0000) Printed24January2017 (MNLATEXstylefilev2.2)
Quark matter with strong magnetic field and possibility of
the third family of compact stars
7
1 Hajime Sotani1 ⋆ and Toshitaka Tatsumi2
0
1Divisionof Theoretical Astronomy, National Astronomical Observatory of Japan, 2-21-1 Osawa, Mitaka, Tokyo 181-8588, Japan
2
2Department of Physics, Kyoto University,Kyoto606-8502, Japan

n
a
J
24January2017
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2

ABSTRACT
]
E We consider the possibility for the existence of the third family of compact objects,
H consideringthe effect ofstrongmagnetic fields inside the hybridstars.As a result, we
demonstratesuchnewsequencesofstableequilibriumconfigurationsforsomehadronic
.
h equations of state. Through the analysis of the adiabatic index inside stars, we find
p the conditions for appearing the third family of compact objects, i.e., for hadronic
- starswithoutquarks,thatthe maximummassshouldbe small,the centraldensityfor
o
the maximum mass should be also small, and the radius for the the maximum mass
r
t should be large. Even for soft hadronic equations of state, the two solar-mass stars
s
a might survive as the third family of compact objects, once quark matter with strong
[ magneticfield,suchas∼O(1019G),istakenintoaccount.Itmightgiveahinttosolve

the so-called hyperon puzzle in nuclear physics.
1
v Key words: equation of state – magnetic fields – stars: neutron –
1
6
3
6
0 1 INTRODUCTION theoretically, there are many attempts in terms of the
1. crustal torsional oscillations (e.g., Steiner& Watts (2009);
Nowadays, properties of matter under the extreme condi-
0 Gearheart et al. (2011); Sotani (2011, 2014)) and/or the
tions such as high-density or high-temperature region have
7 magnetic oscillations (e.g., Sotani, Kokkotas& Stergioulas
beenoneofthemostinterestingsubjects.Recentprogressin
1 (2007, 2008); Sotani, Colaiuda & Kokkotas (2008);
observation of neutron stars has provided us of interesting
: Sotani & Kokkotas (2009); Colaiud & Kokkotas (2011);
v resultsaboutthemaximummass,themagneticfield,surface
Passamonti & Andersson (2012); Gabler et al. (2012,
i temperatureandsoon.Theoretically, however,thereisstill
X 2013)).Throughsuchattempts,itisshownthat,identifying
large ambiguity about theequation of state (EOS) at high-
r the observed quasi periodic oscillations with the crustal
a densityregion. Inparticular, thedensity insidetheneutron torsional oscillations, one can constrain the EOS in the
stars can bewell over thenuclear saturation density,which
crust region (Sotani et al 2012, 2013a,b; Sotani 2016;
leadstothedifficultytodeterminethestructuresofneutron
Sotani, Iida & Oyamatsu 2016, 2017).
stars. Thus, via the observations of neutron stars itself and
thephenomenaassociatedwiththeneutronstars,onecould Ontheotherhand,discoveryofthe2M⊙ neutronstars
obtain theinformation about such a high density region. also can put a strong constraint on the EOS for neutron
The asteroseismology is one of the most power- star matter (Demorest et al. 2010; Antoniadiset al. 2013),
ful techniques to extract the interior information (e.g., where M⊙ denotes the solar mass. That is, the existence
Andersson & Kokkotas (1996); Sotani, Tominaga & Maeda of these massive compact objects rules out several soft
(2001); Sotani, Kohri & Harada (2004); Sotani et al. EOSs,with which theexpected maximum mass of compact
(2011);Donevaet al.(2013)).Thegravitationalwavesmust objectislessthantheobservedmaximummass.Meanwhile,
be suitable observables for adopting the asteroseismol- in nuclear physics, it is becoming a standard conception
ogy, although the direct observations of the gravitational that hyperons would appear at the high density region
waves have not been successful yet. Even excluding the such as ∼ (2 − 3) × n0, where n0 denotes the nuclear
gravitational waves, fortunately, there are evidences of saturation density. But, the introduction of hyperons
observations of neutron star oscillations. That is the quasi makes the EOS soft enough to be ruled out by the ex-
periodic oscillations discovered in the afterglow of the istence of 2M⊙ neutron stars. This is a serious problem
giant flares. To explain the quasi periodic oscillations in nuclear physics, which is the so-called hyperon puzzle.
So far, there are several discussions in the astrophysi-
cal scenario in the context of the hyperon puzzle (e.g.,
⋆ E-mail:[email protected] Takatsuka (2004); Takatsuka,Nishizaki & Tamagaki
2 H. Sotani & T. Tatsumi
(2008); Peres, Oertel & Novak (2013); an unsolved problem. In addition to the standard neu-
van Dalen, Colucci & Sedrakian (2014); Oertel et al tron stars, the existence of strongly magnetized neutron
(2015, 2016); Fortin et al. (2016)). stars, the so-called magnetars, has been suggested theo-
Compact objects can be left after the death of mas- retically(Duncan & Thompson1992;Thompson & Duncan
sive stars. Depending on the mass of progenitors, the re- 1993, 1996) and observationally (Kouveliotou et al. 1998;
sultant compact stars are different. In general, two fami- Hurley et al. 1999; Melatos 1999). However, the configura-
lies of compact stars are well-known. One is white dwarfs, tion and strength of magnetic fields inside the star are still
which are supported by the electron degeneracy pressure. quite uncertain. Using the virial theorem, the maximum
Second is neutron stars, which are supported by the neu- limit of the magnetic field strength can be estimated to
tron degeneracy pressure together with the strong nuclear be ∼ 1018 −1019 G for the canonical neutron star model
repulsion forces in the sort range (Shapiro & Teukolsky (Lai & Shapiro 1991; Cardall, Prakash, & Lattimer 2001).
1983). The possibility for the existence of the third fam- Note that such estimation is simply derived from the New-
ily of compact stars before the gravitational collapse to tonianvirialtheorembasedontheassumptionthatthefield
black holes has been an interesting and important issue strength insidethestar is constant.On theother hand,the
in nuclear physics and astrophysics (Glendenning 2000; relativistic version of virial theorem is more complicated
Haensel, Potekhin & Yakovlev 2007). One of the necessary (Gourgoulhon & Bonazzola 1994) and the possible maxi-
conditions is related to the gravitational stability of super- mum strength in relativistic case could be enhanced. The
densestars; variation of the magnetic field inside stars should also en-
hanceit.
4 R
Γ> 1+K s , (1)
3(cid:16) R (cid:17) Furthermore, the origin of the magnetic fields inside
neutronstarsisstilluncertain.Thesimplest scenariomight
in terms of the adiabatic index Γ within general relativ-
ity, where R is the stellar radius, R = 2MG/c2 is the be the explanation due to the fossil magnetic field inher-
s
itedfrom progenitorstars.That is,thesmallmagneticfield
Schwarzschildradius,K isoforderunity,andM isthestel-
would be amplified during the gravitational collapse with
lar mass. Since the third family of compact objects should
theconservationofmagneticflux,whichleadstothestrong
be the new stable sequence with the central density ρ be-
c
magnetic field in neutron stars (Chanmugam 1992). The
yond the maximum central density of neutron stars ρ ,
c,max
canonicalmagneticfieldofneutronstarsmightbeexplained
this condition should be fulfilled for superdense stars with
withthissimplescenario,butthecaseofmagnetarsisquite
thecentraldensityρ >ρ .However,thisconditionmay
c c,max
difficult(Tatsumi2000).Asanothergenerationmechanism,
notbesufficientfortheexistenceofthethirdfamily.Asan-
themagnetohydrodynamicdynamohasbeenalsosuggested,
other condition, early qualitative discussion of Gerlach has
where the rapidly rotating protoneutron star with the spin
suggested that the third family is possible if the adiabatic
period smaller than 3 ms may amplify a seed of the mag-
index (or speed of sound) abruptly increases beyond ρ
c,max netic field up to ∼ 1015 G (Duncan& Thompson 1992;
(Gerlach 1968).
Thompson & Duncan 1993), although this scenario seems
Ifthereisathirdfamily,itshouldbehybridstars.Inthe
tobeunacceptablefromtheobservationsofsupernovarem-
seminalpaper,Baym(1977)hasdiscussed thepossibility of
nants related to the magnetar candidates (Vink& Kuiper
hybrid stars as the third family of compact stars, where he
2006). Moreover, as an origin of a strong magnetic field in-
concludedthathybridstarswithρ >ρ aremostlikely
c c,max
unstable, because Γ → 4/3 or the sound velocity c2 → 1/3 side neutron stars, one of the authors (T.T.) has suggested
s
the possibility of ferromagnetism of quark liquid inside the
as a limiting behavior of quark matter at high-density due
stars (Tatsumi 2000).
to theasymptotic freedom.
In our previous paper, we have discussed the ef- In this paper, we consider the strong magnetic
fect of the strong magnetic field on EOS of quark mat- field inside hybrid stars, which are composed primary
ter, and shown that Γ → 2 within the bag model EOS of free quarks. Up to now, it has been also dis-
(Sotani & Tatsumi 2015). Thus, we have pointed out that cussed how the strong the magnetic field affects behav-
themaximummassmaywellcleartherecentobservationof ior of quark matter in high density region, e.g., strange
2M⊙1.Inthispaper,weconsiderthepossibilityofthethird quark star models in strong magnetic fields within a
family of compact stars in the same context. The EOS of confining model (Sinha,Huang & Sedrakian 2013), self-
quark matter can then satisfy Eq. (1). Taking Shen’s EOS consistent configuration of magnetic field in hybrid stars
(Shen et al. 1998) and modified Shen’s EOSs (Shen et al. (Franzon, Dexheimer& Schramm 2016), and effect of tem-
2011; Ishizukaet al. 2008) as typical ones for neutron star perature on two-flavor superconducting quark matter with
matter,wewill discusswhetherthethirdfamily couldexist themagneticfieldeffect(Mandal & Jaikumar2016).Mean-
or not, and such conditions if the third family exists. We while,asanextremecase,wefocusontheeffectofthelowest
remark that the Shen EOS is one of the EOSs adopted in Landau level on the EOS of quark matter in this paper. In
many numerical simulations so far. practice, it was shown that quarks can settle on only the
The magnetic field inside neutron stars has been lowest Landau level, if the magnetic field strength is much
strongerthanacriticalfieldstrength.Asaresult,thequark
EOS becomes so stiff, and quark matter largely occupies
1 The stiffening effect of the magnetic field on the EOS has inside the resultant hybrid star (Sotani & Tatsumi 2015).
been also studied in the context of the possibility of the super- We especially examine the possibility for existence of the
Chandrasekhar white dwarfs (Kundu&Mukhopadhyay 2012; third family of compact objects by considering such hybrid
Das&Mukhopadhyay 2012,2013). stars. Unless otherwise noted, we adopt the geometric unit
The 3rd family of compact objects 3
of c=G=1, where c and G denote the speed of light and
Table1.Expectedmaximummassesofneutronstarsconstructed
thegravitational constant, respectively. withthehadronicEOSswithouttheeffectofmagneticfields,and
thecorrespondingstellarradiiandthecentral densities.
2 HYBRID STARS WITH STRONG EOS Mmax/M⊙ R(km) εc (g/cm3)
MAGNETIC FIELD
HShen 2.21 12.6 1.82×1015
We assume the hadron-quark deconfinement transition in HShenΛ 1.90 12.5 1.86×1015
HShenY 1.64 13.5 1.32×1015
thecore region. Themechanism or properties of thedecon-
HShenYπ 1.66 12.6 1.62×1015
finement transition has large ambiguity at high-density re-
gion, while many studies have suggested it. To construct
hybrid stars, the hadronic EOS have to be connected to
oversimplified(seeIshizukaet al.(2008)fordetailaboutthe
the quark EOS in some wise. So far, in order to dis-
introductionofpion).Evenso,asoneofthehadronicEOSs,
cuss the dependence of the stellar models on the EOSs,
wealsoadoptsuchanEOS.Hereafter,werefertotheHShen
sometimes the EOS, i.e., p(ε) with p and ε being pres-
EOS modified in Ishizukaet al. (2008) as “HShen Y EOS”
sure and the energy density, respectively, are simply con-
nected in the p−ε plane (e.g., Rhoades & Ruffini (1974); without the pion contribution and “HShen Yπ EOS” with
thepion contribution,respectively.
Sotani, Tominaga & Maeda (2001); Alford, Han & Prakash
SincethreeEOSswithhyperonsconsideredhereareim-
(2013); Bedaque & Steiner (2015)). In the same way as in
provedHShenEOSbyaddingthecontributionsofhyperons,
such previous works, in this paper, we simply connect the
those become completely equivalent to the HShen EOS for
hadronicEOSwiththequarkEOSatthetransitiondensity
thedensity region lower than thecritical density where the
ε to avoid thecomplexity;
c
hyperon and/or pion can appear. In Fig. 1, we show the
p(ε)= pH(ε) ε<εc , (2) neutronstarmass foreachEOSasafunction ofthecentral
(cid:26)pQ(ε) ε>εc density in the left panel and that as a function of stellar
radius in the right panel. In the figure, the solid and bro-
where p denotes the pressure in the hadronic (quark)
H(Q) kenlines correspond to thestable and unstableequilibrium
phase and ε is defined by p (ε ) = p (ε ). We re-
c H c Q c models, respectively, and the marks denote the maximum
mark that the baryon number density becomes dis-
massexpectedforeachEOS.Thepropertiesofneutronstar
continuous at the interface between the hadronic and
modelwithmaximummassforeachEOSareshowninTable
quark phases (see Table 2), i.e., this simple connec-
1. From this figure, one can observe that the stellar masses
tion is thermodynamically inconsistent. If the deconfine-
constructedwithEOSswithhyperonscannotbeover2M⊙,
ment transition is of the first order, we must take
which is the maximum mass observed so far, even though
into account the mixed phase by way of the Maxwell
theoriginalHShenEOScanconstructthestellarmodelwith
construction or more elaborated Gibbs construction
the mass larger than 2M⊙. Hereafter, we mainly focus on
(Glendenning 1992; Heiselberg, Pethick & Staubo 1993;
the HShen and HShen Y EOSs to discuss the possibility of
Voskresensky,Yasuhira& Tatsumi 2002; Maruyama et al.
thirdfamilyofcompactobjects.Thisisbecause,theHShen
2007). Fortunately, we can check that the bulk properties
Y EOS is an example for the case where the third family
of hybrid stars have little dependence on the details of
can exist, while the HShen EOS is for the case where the
the phase transition, at least in the maximum mass region
third family cannot exist, as shown later.
(Sotani & Tatsumi 2015).
For the quark matter EOS we use here the bag model
In this paper, we adopt four hadronic EOSs as shown
EOS for simplicity (Farhi& Jaffe 1984; Sotani & Tatsumi
in Table 1, i.e., one EOS composed of only nucleons and 2015)2. The original form of the bag model EOS can be
three EOSs including the hyperons, in order to construct
written as,
hybrid stars. Those are based on the relativistic mean field
1
approach.Shen et al.(1998)haveoriginallyderivedtheEOS p= (ε−4B), (3)
table composed of only nucleons in the wide ranges of den- 3
sity,temperature,andprotonfraction,wheretheyalsoused where B denotes thebag constant and theenergy density ε
theThomas-Fermiapproximationtodescribetheheavynu- can bewritten with thebaryon numberdensity n as
b
clei in the low density region. Afterward, they modified
3
their EOS in the same framework as the original one, but ε= π2/3¯hcn4/3+B (4)
4 b
the contribution of Λ hyperons was also taken into account
for SU(3) flavor symmetricmassless quark matter.
(Shen et al. 2011). Hereafter, we refer to the original and
As mentioned before, in order to construct the third
modified EOSs as “HShen EOS” and “HShen Λ EOS”, re-
family of compact objects, at least, one has to take into
spectively.
account the additional physics to support the gravity of
On the other hand, Ishizuka et al. (2008) have consid-
ered the contributions of Λ, Σ, and Ξ hyperons with and
without pions into the HShen EOS, where they adopted
UΛ(N) = −30 MeV, UΣ(N) = +30 MeV, and UΞ(N) = −15 2noIntpiesrtauprboaptuilvaereEffOecStfboyrtqhuearbkagmcaotntesrt,anmtinBi.mEaOllySibnacslueddionngtthhee
MeV, as standard values of hyperon potentials at the satu-
Nambu-Jona-Lasiniomodelmaybealsoavailable(Menezes etal.
ration density. We remark that they consider the pion con- 2009), where the effective bag constant corresponding to B can
tributions to EOS, assuming that the pion interaction is bereduced fromEOS.Basically,thebothshouldshareasimilar
neglected. Thus, the effect of introduction of pions may be featureathighdensity,wherechiralsymmetryisrestored.
4 H. Sotani & T. Tatsumi
2.5
2.5
HShen
HShen ! HShen
2.0 2.0 HShen !
M! 1.5 ! 1.5
/ HShen Y ! HShen Y"
M 1.0 !/ 1.0
HShen Y" HShen Y
0.5
0.5
0
1014 1015 0.011 12 13 14 15 16
! (g/cm3) R (km)
c
Figure1.WiththeHShen,HShenΛ,HShenY,andHShenYπEOSs,theneutronstarmassisshownasafunctionofthecentraldensity
intheleftpanel,whilethemass-radiusrelationsareshownintherightpanel.Inthebothpanels,themarksdenotethemaximummass
expected witheachEOS.Thesolidandbrokenlinescorrespondtothestableandunstablestellarmodels,respectively.
compact objects. In this paper, as such an additional levelofquarkmatter,andthatquarkmatterbecomesflavor
physics, we consider the strong magnetic fields inside stars symmetric, the EOS for quark matter settling only in the
(Sotani & Tatsumi 2015). With respect to the EOS, theef- lowest Landau level can be expressed within the MIT bag
fect of the magnetic fields should be more important in model, as
quark matter rather than in hadronic matter, because (i)
p=ε−2B, (5)
the quark masses are much smaller than the baryon mass
and (ii) all quarks have net electric charge while the neu- where theenergy density ε can bewritten as
trons are dominant in the hadronic phase. With such rea-
sons, for simplicity, we consider the effect of the magnetic ε= 5π2¯h2c2n2+B, (6)
fieldsonlyonquarkmatter,asinSotani & Tatsumi(2015), 2eB b
where we omit the effect of magnetic field on the hadronic (Sotani& Tatsumi 2015). Hereafter, we refer to this quark
EOS. In the following we assume the density dependent EOSas“LandauEOS”.Therelationbetweenpandεgiven
magnetic field inside stars, without recourse to its details by Eq. (5) is satisfied independently of the magnetic field
(Bandyopadhyay,Chakrabarty & Pal 1997), i.e., the weak strength, when the strength would be larger than the crit-
magneticfieldinthelowdensityregionandthestrongmag- ical strength, B . We remark that the Landau EOS is the
c
netic field in the high density region. An actual calculation limiting case of a stiff EOS, because the adiabatic sound
hasshownthatthebulkpropertiesofhadronicEOSislittle speed, c = (dp/dε)1/2, becomes the speed of light c. We,
s
affected up to B = O(1017G) (Casali., Castro & Menezes hereafter, use the notation ε for the transition density ε
L c
2014). Thus, although for a realistic case one might see the to specify the quark EOS. We also remark that the Lan-
effect ofmagnetic field evenin thehadronicEOS with such dau EOS is independent, even if one simply considers the
a strong magnetic field as in this paper, as a first step we effects of magnetic field on EOS such as p→p+B2/2 and
omit such effect and focus on the effect of magnetic field ε→ε+B2/2 (e,g,. Casali., Castro & Menezes (2014))4.
onlyinquarkmatterwiththereasonsmentionedtheabove. In Table 2, for given the values of ε , we show the
L
Ingeneral, manyLandaulevels areoccupied byquarks corresponding values of the bag constant, critical magnetic
if the magnetic fields are not so strong. But, one may have fieldstrength,baryonnumberdensitywiththecriticalmag-
toconsidertheeffectoftheLandaulevels,ifthestrengthof netic field in the quark phase, and baryon number density
magnetic fields is strong enough for quarks to occupy only in the hadronic phase for the HShen and HShen Y EOSs.
some lower levels. In particular, the extreme case is when Wealso remark that thereisstill left alarge uncertaintyin
quarkssettle only in thelowest Landau level, which can re- thevalueofthebagconstant.TheMITgrouphasoriginally
alizeifthestrengthofmagneticfieldsbecomes∼O(1019 G) obtained B = 57.5 MeV/fm3 (DeGrand et al. 1975). Sub-
(Sotani & Tatsumi2015)3.Assumingthattheuniformmag- sequently, the various values of B are extracted by several
neticfieldlocallypointstowardaspecificdirection,thatthe groupsvia fittingtolight hadron spectra, which reaches up
quark mass does not contribute significantly to the energy to ∼ 351.7 MeV/fm3 (Carlson, Hansson & Peterson 1983).
Recently, the possibility of B =O(500) MeV/fm3 has been
suggested by evaluating the energy-momentum tensor in
3 Compered to the observations of field strength of magnetars QCD or using the data of the deconfinement temperature
such as ∼ 1015 G, the order of ∼ 1019 G may be much larger. within the lattice QCD calculations (Buballa 2005). So, in
However, it could be possible as in the case of white dwarf, i.e., thispaperwe considerin thewide rangeof B, i.e., B <∼ 500
the observed surfacestrength ofmagnetic field ofwhite dwarfis
atmostintherangeof106−108G(Koester&Chanmugam1990;
Kepleretal. 2013), while the central field strength in the order
of1012 Gcanbetheoreticallyconstructed (Ostriker&Hartwick 4 In more realistic case, one might see anisotropy in the mag-
1968).Asanotherpossibility,suchastrongmagneticfieldincore neticpressure,dependingonthemagneticfieldconfiguration. In
region may be screened out by the ambient high-conductivity addition, one might take into account the magnetization of the
plasma(HariDass&Soni2012). system(e.g.,Das&Mukhopadhyay (2012)).
The 3rd family of compact objects 5
Table2. AtthegiventhevaluesoftransitiondensityεL,thecorrespondingvaluesofthebagconstantB,criticalmagneticfieldstrength
Bc, baryon number density with the critical magnetic field in the quark phase n(bQ), and baryon number density in the hadron phase
n(H) fortheHShenandHSnenYEOSs.
b
EOS εL (g/cm3) B (MeV/fm3) Bc (G) n(bQ) (fm−3) n(bH) (fm−3)
HShen 1.40×1015 284.5 5.10×1019 0.893 0.671
1.83×1015 352.5 5.92×1019 1.116 0.825
2.40×1015 442.2 6.86×1019 1.391 1.012
HShenY 1.0×1015 247.1 4.04×1019 0.629 0.526
1.4×1015 339.9 4.81×1019 0.818 0.710
2.0×1015 470.2 5.82×1019 1.088 0.969
MeV/fm3,todiscusstheexistenceofthirdfamilyofcompact 2.4
objects. HShen 1.40
2.3
Finally we directly connect thehadronicEOS with the
Landau EOS at the transition density ε , as shown in Fig.
L ! 2.2
2, i.e., we set εc = εL in Eq. (2). In particular, in Fig. 2, M
we show the HShen (left panel) and HShen Y EOSs (right M/ 2.1
panel) as an example, which are connected to the EOS de- 1.83
scribing quark matter as in Eq. (5). In this figure, thesolid 2.0
2.40
line corresponds to the HShen or HShen Y EOS, while the
dotted lines are EOS for quark matter with different val- 1.9
9 10 11 12 13 14
uesof εL bychanging thebag constant B. The thirdfamily R (km)
of compact objects is a sequence of the stable equilibrium
configurations of compact objects, which are more compact
than neutron stars, and the unstable equilibrium models Figure3.Mass-radiusrelationsconstructedwiththeHShenEOS
of neutron stars should exist between the sequence of the connectedtotheLandauEOSatεL=1.40,1.83,and2.40×1015
thirdfamily ofcompact stars andthatof theusualneutron g/cm3.Inthefigure,thesolidanddottedlinescorrespondtothe
stable and unstable equilibrium configurations, respectively. For
stars. Thus, in definition, the transition density ε should
L
reference,themaximummassofneutronstarsexpectedwiththe
be larger than the central density with which the neutron
HShenEOSdenotes bythefilledcircleinthefigure.
starmassconstructedwiththehadronicEOSbecomesmaxi-
mum.NotethattheLandauEOSexhibitsastrongstiffening
after the phase transition.
stable stellar model can not reach ε . The reason why the
L
third family of compact objects cannot appear for adopt-
ing the HShen EOS, may be the fact that the stellar mass
3 THE THIRD FAMILY OF COMPACT constructed with the HShen EOS is so large that even the
OBJECTS stiffness dueto theLandau EOS cannot support.
Next, we consider the hybrid stars with the improved
3.1 Hybrid stars as the third family of compact
HShen EOS including the hyperons, i.e., the HShen Λ,
stars
HShen Y, and HShen Yπ EOSs. However, since the results
First, we considerhybridstars with theHShenEOS,which with the HShenYπ EOS are very similar to those with the
is composed of only nucleons. In Fig. 3, we show the HShen Y EOS, we remove the results with the HShen Yπ
mass-radius relation for the cases of ε = 1.40, 1.83, and EOS from the following figures to avoid a complication. In
L
2.40×1015 g/cm3, where the solid and dotted lines corre- Fig. 4, we especially show the mass-radius relation for the
spondtothestableandunstableconfigurations.Forthecase casesofε =1.0,1.4,and2.0×1015 g/cm3 byadoptingthe
L
ofε =1.40×1015 g/cm3,duetothestiffnessoftheLandau HShenYEOSasthehadronicEOS.Inthisfigure,thesolid
L
EOS, the maximum mass of hybrid star can become larger anddottedlinescorrespondtothestableandunstableequi-
thanthatofneutronstarconstructedwiththeHShenEOS. librium configurations of thehybridstars. From this figure,
However, since ε in this case is less than the central den- one can obviously observe the third family of compact ob-
L
sity with which theneutron starmass constructed with the jects for ε =1.4 and 2.0×1015 g/cm3. Unfortunately, the
L
HShen EOS becomes maximum, one can not see the third case of ε =2.0×1015 g/cm3 has to be ruled out, because
L
family of compact objects. For the other cases of εL =1.83 theexpectedmaximummassislessthan2M⊙,butthecase
and2.40×1015 g/cm3,themaximummassesarecompletely ofεL =1.4×1015 g/cm3 canstillsurvivethe2M⊙ problem.
equivalenttothatpredictedfromtheHShenEOS,whereone Additionally, from this figure, we find that the maximum
cannotobservetheadditionalequilibriumsequence.Thatis, massofthethirdfamilyofcompactobjectsdecreasesasthe
the third family of compact objects cannot be constructed value of ε increases. And, the third family of compact ob-
L
by adopting the HShen EOS. We remark that the quark jects eventually becomes unstable when the value of ε is
L
phasedoesnotappearinsidethestellarmodelsforε =1.83 over a critical value. By definition, the radius of the third
L
and 2.40×1015 g/cm3, because the central density for the family of compact object should be smaller than that of a
6 H. Sotani & T. Tatsumi
1.83
3m) 1036 1.40 2.40 3m) 1036 1.0
g/c g/c 2.0
p (er HShen p (er 1035 HShen Y 1.4
1035 1034
1015 1015
! (g/cm3) ! (g/cm3)
Figure 2. EOS for constructing hybrid stars. In this figure, as an example, we show the HShen EOS (left panel) and HShen Y EOS
(right panel) as hadronic EOS, which are connected to the EOS for quark matter. The solid line corresponds to the HShen or HShen
YEOS, whilethe dotted linescorrespondto theEOSforquarkmatter withdifferent valueof thetransitiondensity εL denoted inthe
unitof 1015 g/cm3.For reference, the vertical linesdenote the central density withwhichthe mass of neutron starbecomes maximum
foreachhadronicEOS,asshowninTable1.
2.4 2.5
HShen Y
2.2 HShen
2.3
2.0 !
! 1.4 1.0 M 2.1
M 1.8 2.0 /x J1614-2230
/ a
M m 1.9
1.6 M HShen !
1.4 1.7
HShen Y
xxxxxxxx
1.2 1.5
8 9 10 11 12 13 14 15 0.5 1.0 1.5 2.0 2.5
R (km)
! (1015 g/cm3)
L
Figure 4. Mass-radius relations constructed with the HShen Y
EOSconnectedtotheLandauEOSatεL=1.0,1.4,and2.0×1015 Fixxgure 5. The maxxximum mass of sxxtar is shown as xxa function
g/cm3.Inthefigure,thesolidanddottedlinescorrespondtothe of the transition density εL. The thick parts correspond to the
stableandunstableequilibriumconfigurations,respectively.One thirdfamilyofcompact objects. Forreference, theobservational
can observe the third familyof compact objects for the cases of constraintonthemassofthemillisecondpulsarJ1614-2230isalso
εL=1.4and2.0×1015 g/cm3.Forreference,themaximummass shown,whichisM =(1.97±0.04)M⊙ (Demorestetal.2010).
ofneutronstarsexpectedwiththeHShenYEOSdenotesbythe
filledcircleinthefigure.
tronstars,theparameterspaceofε andthehadronicEOS
L
forrealizing thethirdfamily ofcompact objects seem tobe
usualneutronstar,whilethemassofthethirdfamilymight quite limited. In Fig. 5, for reference, we also add the ob-
be comparable to that of a usual neutron star. Thus, one servational evidence of massive neutron stars, whose mass
possibility to be distinguish whether an object is the third is estimated to be M = (1.97±0.04)M⊙ (Demorest et al.
family or usual neutron star is the observations of two ob- 2010). Dueto theexistence of such a massive neutron star,
jects with different radii but comparable masses, i.e., the the third family of compact objects constructed with only
so-called twin stars. Additionally, the careful observations theHShenYEOScansurvive,althoughtheallowedparam-
of cooling history of compact objects tell us whether quark eterrangeofε isnotsolarge. Meanwhile, thethirdfamily
L
matterexistsinsidethestar,becausethecoolinghistoryde- constructedwith theHShenΛis marginal. Weremarkthat
pends on the existence of quark matter. This might be a thethird family with theHShenYπ EOS can beproduced,
hintto considera possibility of thethird family of compact but the maximum mass of any compact object in the third
objects. family cannot reach 2M⊙.
In Fig. 5, we show the maximum mass of hybrid stars Comparing the results obtained up to now with Fig. 1
as a function of ε for each hadronic EOS. In particular, andTable1,wemaybeabletoextracttheconditionstore-
L
the thick lines correspond to the third family of compact alizethethirdfamilyofcompactstarswhosemaximummass
objects,wherethevalueofεL islargerthanthecentralden- becomesover2M⊙:fortheneutronstarmodelsconstructed
sity with which the mass of the neutron star constructed with the hadronic EOSs, the maximum mass and the cor-
with the hadronic EOS becomes maximum. From this fig- responding central density should be small, and the cor-
ure,onecanseethathybridstarswiththeimprovedHShen responding stellar radius should be large. Thus, we remark
EOSswithhyperonscanrealizethethirdfamilyofcompact thattheremightbeapossibilitythatthesoftEOSswithhy-
objects.However,consideringtheexistenceofthe2M⊙ neu- peronsand/ortheothercomponentswhichhavebeenruled
The 3rd family of compact objects 7
5.0 2.7
) HShen Y
3 4.5 2.5
m
/c 4.0 ! 2.3
g M
15 3.5 /x 2.1 HShen Y
0 3.0 a
(1x 2.5 HShen ! Mm 1.9 HShen !
ma 2.0 1.7
!
1.5 1.5
0.5 1.0 1.5 2.0 2.5 9 10 11 12
R (km)
! (1015 g/cm3) max
L
Figure 7.Themaximum massof staris shownas afunctionof
Figure 6. The central density with which the mass of the star thestellarradiusforthehybridstarwiththemaximummass.The
becomesmaximumisshownasafunctionofthejunctiondensity thickpartscorrespondtothethirdfamilyofcompact objects.
εL. The thick parts correspond to the third family of compact
objects.
3.5
outafterthediscoveryofthe2M⊙neutronstar,couldrevive 3.3
owing to theshift to thethird family of compact objects. HShen "
3.1
We discuss some properties of the third family of com- x
a
pact objects with the additional figures. In Fig. 6, we show m
! 2.9
thecentraldensitywithwhichthehybridstarbecomesmax- HShen Y
imum mass, ε , as a function of ε for the HShen Λ and
max L 2.7
HShenYEOSs,wherethethicklinescorrespondtothethird
family of compact objects. From this figure, we find that 2.5
1.5 1.7 1.9 2.1 2.3 2.5 2.7
the value of ε does not monotonically increase with ε .
max L ! /!
In Fig. 7, we show the maximum mass of hybrid stars as a max !
function of the stellar radius when the mass of hybrid star
becomesmaximum,R ,aschangingthevalueofε forthe
max L Figure 8.Adiabatic index at the central density for the hybrid
HShen Λ and HShen Y EOSs. One can see that the stellar
starwithmaximummassisshownasafunctionofthemaximum
radiusRmaxandthemaximummassofhybridstargenerally mass of star. The thick parts correspond to the third family of
decrease as the value of εL increases up to a critical value. compactobjects.
In Fig. 8, we show the adiabatic index at the stellar center
for the maximum mass hybrid star, Γ , as a function of
max
the maximum mass of hybrid star by changing thevalue of
for theLandau EOS.Then,wecan observethatΓ becomes
ε , where the adiabatic index Γ is defined as
L
very large at the transition density, since ε is relatively
L
p+ε dp close to 2B. Meanwhile, Γ approaches the asymptotic value
Γ= . (7)
p (cid:16)dε(cid:17) of2 in thehigh-densityregion. Asan example, inFig. 9 we
showtheadiabaticindexasafunctionoftheenergydensity
As the value of ε increases and reaches the critical value
L
especially with the HShen and HShen Y EOSs, where the
with which the third family of compact objects becomes
solid and broken lines correspond to the adiabatic indexes
unstable equilibrium, the value of Γ also becomes max-
max
for compact objects without and with the phase transition
imum, although such stellar models are ruled out from the
into quark matter. In the figure, the numbers denote the
2M⊙ neutron star observation.
transition density ε in the unit of 1015 g/cm3. From this
L
figure, one can observe that the adiabatic index for com-
pactobjectswiththephasetransitionincreasesabruptlyat
3.2 Criteria for the third family of compact stars ε = ε , and that the difference between the adiabatic in-
L
dexes without and with the phase transition decreases as
Now, we would like to give a conjecture about the condi-
ε increases. On theother hand,as shown before, thecom-
tions when the third family of compact object exists. As L
pact object cannot recover the stability, when the value of
mentioned before, at least the central density of the com-
ε exceeds a critical value. From these results, we consider
pact object should be larger than that for the neutron star L
that the difference between the adiabatic indexes without
withthemaximummass.Inaddition,inordertorecoverthe
and with thephase transition should be larger than a criti-
stability,EOSfortheinnerregionwherethephasetransition
calvalue,ifthethirdfamilyofcompactobjectsexist.So,in
arises from the hadronic into quark matter, must be signif-
Fig.10,weplotthedifferenceδΓasafunctionofε forthe
icantly stiff. To check this point, we focus on the adiabatic L
HShen, HShen Λ, and HShen Y EOSs. Here, δΓ is defined
index calculated with Eq. (7), i.e.,
as
2(ε−B)
Γ= ε−2B , (8) δΓ=ΓHS−ΓNS, (9)
8 H. Sotani & T. Tatsumi
8 the phase transition must occurs at relatively low density
and accordingly the gravitational instability occurs before
7
reaching the maximum mass constructed with the normal
6 EOS, which is the EOS without the phase transition. This
! 5 HShen " HShen Y is a result owing to the appearance of the mixed phase
" (Glendenning& Kettner 2000; Schaffner-Bielich et al.
4 2002). After that, EOS becomes stiff again at higher
densitiesduetotheadditionaleffectsinthenewphase.The
3 HShen
transition from the mixed phase to the pure quark phase
2 maygiverisetoanincrease oftheadiabatic index,which is
1.0 1.5 2.0 2.5
similar to our model for the hadron-quark phase transition
! (1015 g/cm3)
L to generate the third family (Glendenning& Kettner
2000). However, the maximum mass of the third fam-
ily discussed in Glendenning & Kettner (2000) and in
Figure 10.The difference of the adiabatic indexes between the
Schaffner-Bielich et al.(2002)cannotbelargeandprobably
stellar models without and with the phase transition into quark
matter at the transition density εL, are shown as a function of less than the corresponding maximum mass constructed
εL.Thethickpartoneachlinecorrespondstothethirdfamilyof with normal EOS; as already stated the adiabatic index
compactobjects. of the pure quark phase cannot become too large in
the case of the hadron-quark phase transition. In other
words, if we discard the appearance of the mixed phase
where ΓHS and ΓNS denote the adiabatic indexes at ε=εL in the models of Glendenning & Kettner (2000) and of
for the compact stars with and without quark matter. In Schaffner-Bielich et al. (2002), we have no third family and
this figure, the thick part on each line corresponds to the we only have the mass-radius relation with maximum mass
third family of compact object. From this figure, we find less than that with normal EOS. These features are quite
thatthelargejumpofΓatthephasetransitionisnecessary different from our results. We have seen the large increase
so that the third family exists. In fact, δΓ for the HShen of the adiabatic index at the transition point due to the
EOSislessthan∼3forthecentraldensitylargerthanthat effect of magnetic field, which generates the third family of
for the neutron star with the maximum mass. This may be compactstars.Wemustpointoutatthesametimethatthe
a reason why the third family can not exist for the HShen adiabatic index approaches to two at high-density region,
EOS.Namely,theconditionforexistenceofthethirdfamily which supports thelarge maximum mass of hybridstars.
might be that δΓ should be larger than a critical value for Wehavenottakenintoaccountthedetailsofthephase
the stellar models constructed with a central density larger transition such as the mixed phase. One may consider a
than that for the neutron star with the maximum mass, discontinuityoftheenergydensity∆εatthetransitionpoint
where the exact value of such a critical value of δΓ might in Eq. (2),
depend on the hadronicEOS (see AppendixA).
Itshouldbeinterestingtocompareourresultswiththe p(ε)= pH(ε) ε<εc , (11)
ordinary bag model EOS expressed by Eq. (3). The transi- (cid:26)pQ(ε) ε>εc+∆ε
tion densitybecomes veryhigh in thiscase, and thereis no
transition for the large bag constant, for example B = 340 withεc determinedbypH(εc)=pQ(εc+∆ε),whichmayre-
MeV/fm3 withε =1.4×1015 g/cm3 fortheHShenYEOS. sembletheMaxwellconstructioninthecaseofthefirstorder
L
phasetransition(Alford, Han & Prakash2013).TheEOSis
The corresponding adiabatic index can bewritten as
thensoftenednearthetransitiondensitybytheintroduction
4 ε−B of ∆ε, which gives rise to a gravitational instability around
Γ= . (10)
3ε−4B theonset ofthethirdfamily in Fig. 4.However,suchinsta-
Thus,wecanfindbotheffectsthetransitiondensityandδΓ bility is soon recovered by the large δΓ. The appearance of
disfavortheexistenceofthethirdfamilyintheordinarybag themixedphasealsosoftentheEOSnearthetransitionden-
model, besides thestability condition [Eq. (1)]. sity todiminish thestable branchof thethird family, if the
mixed state is taken into account. Anyway, the bulk prop-
ertiessuch as themaximum mass arelittle changed evenin
3.3 Phase transition and the third family that case (Sotani& Tatsumi 2015).
Finally, we would like to make some comments about
the relation between the phase transition and the possi-
ble existence of the third family. In K¨ampfer (1981a,b);
4 CONCLUSION AND DISCUSSION
Glendenning& Kettner (2000); Schaffner-Bielich et al.
(2002), they have also discussed the possibility of the ap- Inthispaper,wehaveconsideredthepossibilityforexistence
pearanceofthethirdfamilyfollowing thephasetransitions. ofthethirdfamilyofcompactobjects,whicharestableequi-
One of the points discussed there is that the softening of librium configurations more compact than neutron stars. It
EOS first occurs at the transition point and then the stiff- is well known that white dwarfs are supported by the elec-
ening follows at high-density. That is, the adiabatic index trondegeneracy pressure,andneutronstarsbytheneutron
abruptly decreases and cannot satisfies the gravitational degeneracy pressure together with the strong nuclear force
stability condition [Eq. (1)] just after the phase transition, in the short range. However, it is not sure whether the sta-
whichleadsstarstobegravitationallyunstable.Inthiscase, ble objects more compact than neutron stars can exist or
The 3rd family of compact objects 9
10 10
9 HShen 9 1.0 HShen Y
1.4
8 8
2.0
7 7
6 6
! 5 1.40 1.83 ! 5
2.40
4 4
3 3
2 2
1 1
0 0
0.5 1.0 1.5 2.0 2.5 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0
! (1015 g/cm3) ! (1015 g/cm3)
Figure 9. With the HShen and HShen Y EOSs, the adiabatic indexes are shown as a function of the energy density, where the solid
and broken lines correspond to the results for the stellar models without and with the phase transition into quark matter. The labels
in the figure denote the transition density εL in the unit of 1015 g/cm3. From this figure, one can obviously observe that the jump of
adiabaticindexwithHShenEOSatthetransitiondensityismuchsmallerthanthecasewithHShenYEOS.Forreference,thevertical
linesdenotethecentraldensitywithwhichthemassofneutronstarconstructed witheachhadronicEOSbecomes maximum.
not. Toconstruct such compact objects, i.e., thethird fam- the maximum mass of neutron stars, one might expect to
ily of compact objects, one has to take into account the observethesecondbouncewhenthecentraldensityreaches
additionalphysicstosupportthegravitygenerated bysuch the transition density where quark matter appears, which
objects.Asadditionalphysics,wehaveconsideredthemag- is similar to the usual bounce in supernovae arising when
neticfieldinsidehybridstars,whosestrengthandgeometry the central density reaches the nuclear saturation density.
are not fixed observationally. If the magnetic fields inside As another possibility, one might distinguish compact ob-
the star would be so strong that quarks settle only in the jectsinthethirdfamilyfromusualneutronstarsbyobserv-
lowest Landau level, the equation of state for quark mat- ingthethermalevolution,becausethecoolinghistorycould
ter becomes quite stiff (Sotani & Tatsumi 2015). To realize strongly depend on theinterior compositions.
suchastiffEOS,themagneticfieldstrengthshouldbecome
In this paper, we have used the simple EOS for quark
∼ O(1019G), where the quark core of hybrid star becomes
matter, based on the MIT bag model. For more realistic
∼(70−90)% ofstellar radius5.Withsuchanextremecase,
description, other EOS, e.g., based on the perturbative
we have examined the possibility whether the third family
QCD or the NJL model might be desirable; especially the
of compact objects can exist.
latter is appropriate to include chiral dynamics, which is
Then, we have shown that thethird family of compact
a basic concept of underlying QCD. Actually the inho-
objectscanexist,eventhoughtherelevantparameterspace
mogeneous chiral phase has been actively studied in the
is not so large and EOS for hadronic matter might be re-
presence of the magnetic field (Bubbala & Carignano
stricted,consideringtheobservationalconstraintonthestel-
2015; Tatsumi, Nishiyama & Karasawa 2015;
lar mass such as 2M⊙. If such stars exist, they are hybrid Nishiyama, Karasawa & Tatsumi 2015;
stars composed of primarily quarks. By way of the numeri-
Yoshiike,Nishiyama & Tatsumi 2015). It should be in-
cal examinations, we have found that, for constructing the
teresting to take into account such aspect in the next step.
third family of compact objects, the hadronic EOSs could
We have adopted the simple procedure to connect the
be favored, with which the maximum mass of neutron star
quark EOS with the hadronic EOS. Thus, we should make
should besmall with asmall centraldensityandalarge ra-
an additional analysis, where we might have to consider
dius.Thatis,consideringthepossibility fortheexistenceof
the physical connection between the EOSs for quark and
third family of compact objects, thehadronic EOSsinclud-
hadronic matters. Moreover, it might be more realistic
ing hyperons may not be ruled out, even if the maximum
to take into account the effects of magnetic field even in
mass of neutron star constructed with such soft hadronic
hadronic matter. Such additional analyses would be done
EOSsis predictedtobeless than2M⊙.Weadditionally re- somewhere in future.
mark that, if the third family exists, the compact objects
whose radii are smaller than neutron stars with the same WehaveseenthatEOSisstronglystiffenedafterthede-
masscanexisteveninthenarrowmassrange,i.e.,theexis- confinementtransitionduetothestrongmagneticfield,and
tence of twin stars. Furthermore, if compact objects in the theadiabaticindexbecomeslargerthanthecriticalvalueof
thirdfamilyisformedinsupernovaewithlargermassesthan the gravitational stability, 4(1+KRs/R)/3. However, only
the large adiabatic index in EOS is not sufficient for the
possible existence of the third family. The transition den-
sity must be not so large, and at the same time the adi-
5 Weconsiderthestrongmagneticfieldinsidethequarkcorefor
abatic index following the phase transition must increase
satisfyingtheconditionthatquarkssettleononlythelowestLan-
enough to large: a combination of both conditions can pro-
dau level. If such a strong magnetic field approaches the stellar
surfacealthough itmaynotberealistic,onecouldexpect toob- ducethethirdfamily.Theseobservationsmaypartiallysup-
serveakindofextremelyactiveelectromagneticphenomenafrom port the conjecture suggested in Gerlach (1968). Addition-
thestellarsurface. ally,wehaveneglected themagneticpressureforconstruct-
10 H. Sotani & T. Tatsumi
ingthestellarmodels. Atleast,thepoloidalmagneticfields 3.3
can increase themaximum massof thestars, which may be HShen #
an advantage for realizing the third family of compact ob-
jects. Such effects would be also considered somewhere in 3.2 HShen Y"
n
future. Furthermore, it should be noted that the magnetic mi
pressuremight dependonthegeometry ofmagneticfields6. !
!
Huanget al.(2010)suggestedthepossibilitythatthetrans- 3.1
versepressuretendstovanish duetothefreezing of matter xno x3rd family xHShen Y
inthelowest Landaulevelifthemagneticfieldissostrong.
Meanwhile, Ferrer et al. (2010) suggested the possibility of 3.0
12.4 12.6 12.8 13.0 13.2 13.4 13.6
apressureanisotropyduetothestrongmagneticfield,while R (km)
PotekhinandYakovlevpointedoutthatthepressureinthe
magnetizedobjectsisindependentofthemagneticdirection
(Potekhin & Yakovlev2012).Ifsuchananisotropicpressure Figuxre A1.TheminixmumvalueofδΓxareplottedasafunction
really appears under the strong magnetic field, one might oftheradiusoftheneutronstarwiththemaximummass.
have to consider such effects on the neutron star structure.
Anyhow, the anisotropic pressure should not be so serious
mass. If such a set of the radius and δΓ is plotted in the
yetforthemagneticfieldofO(1019 G)(Huang et al.2010).
region below the line given by Eq. (A1), the third family
Wearegrateful toC. Ishizukafor preparingsome EOS
mightnotexistwiththeadoptedhadronicEOS,becauseδΓ
data and to A. Ohnishi for discussing the EOSs adopted
would basically decrease as the transition density increases
in this paper. This work was supported in part by Grants-
(seeFig.10)andδΓdeterminedhereshouldbealmostmax-
in-Aid for Scientific Research on Innovative Areas through
imum value of δΓ. For example, with HShen EOS, such ra-
No.24105008andNo.15H00843providedbyMEXT,andby
dius and δΓ become R = 12.56 km and δΓ = 2.39, which
Grant-in-AidforYoungScientists(B)throughNo.26800133
is in the region below the line given by Eq. (A1). Anyway,
provided by JSPS.
to derive the critical line, we should check the correlation
between δΓ andsome properties in hadronicEOS,using
min
more samples.
APPENDIX A: DEPENDENCE OF δΓ ON THE
HADRONIC EOS
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