Relativistic hybrid stars with super-strong toroidal magnetic fields: An evolutionary track with QCD phase transition

Abstract

We investigate structures of hybrid stars, which feature quark core surrounded by a hadronic matter mantle, with super-strong toroidal magnetic fields in full general relativity. Modeling the equation of state (EOS) with a first order transition by bridging the MIT bag model for the description of quark matter and the nuclear EOS by Shen et al., we numerically construct thousands of the equilibrium configurations to study the effects of the phase transition. It is found that the appearance of the quark phase can affect distributions of the magnetic fields inside the hybrid stars, making the maximum field strength about up to 30 % larger than for the normal neutron stars. Using the equilibrium configurations, we explore the possible evolutionary paths to the formation of hybrid stars due to the spin-down of magnetized rotating neutron stars. We find that the energy release by the phase transition to the hybrid stars is quite large ($\la 10^{52} \rm erg$) even for super strongly magnetized compact stars. Our results suggest that the strong gravitational-wave emission and the sudden spin-up signature could be observable signals of the QCD phase transition, possibly for a source out to Megaparsec distances. Comment: 17 pages, 10 figures. accepted to MNRAS

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arXiv:0910.0327v1 [astro-ph.HE] 2 Oct 2009
Mon. Not. R. Astron. Soc. 000, 1–16 (2008)Printed 2 October 2009 (MN LATEX style file v2.2)
Relativistic hybrid stars with super-strong toroidal
magnetic fields: An evolutionary track with QCD phase
transition
Nobutoshi Yasutake1⋆, Kenta Kiuchi2†, Kei Kotake1,3‡
1Division of Theoretical Astronomy, National Astronomical Observatory of Japan, 2-21-1 Osawa, Mitaka, Tokyo 181-8588, Japan
2Department of Physics, Waseda University, 3-4-1 Okubo, Shinjuku-ku, Tokyo 169-8555, Japan
3Center for Computational Astrophysics, National Astronomical Observatory of Japan, 2-21-1, Osawa, Mitaka, Tokyo, 181-8588, Japan
Typeset 2 October 2009; Accepted
ABSTRACT
We investigate structures of hybrid stars, which feature quark core surrounded by
a hadronic matter mantle, with super-strong toroidal magnetic fields in full general
relativity. Modeling the equation of state (EOS) with a first order transition by bridg-
ing the MIT bag model for the description of quark matter and the nuclear EOS by
Shen et al., we numerically construct thousands of the equilibrium configurations to
study the effects of the phase transition. It is found that the appearance of the quark
phase can affect distributions of the magnetic fields inside the hybrid stars, making the
maximum field strength about up to 30 % larger than for the normal neutron stars.
Using the equilibrium configurations, we explore the possible evolutionary paths to
the formation of hybrid stars due to the spin-down of magnetized rotating neutron
stars. We find that the energy release by the phase transition to the hybrid stars is
quite large (? 1052erg) even for super strongly magnetized compact stars. Our results
suggest that the strong gravitational-wave emission and the sudden spin-up signature
could be observable signals of the QCD phase transition, possibly for a source out to
Megaparsec distances.
Key words:
stars:neutron
dense matter – equation of state – gravitation – stars: rotation –
1 INTRODUCTION
A very hot issue in hadronic and nuclear physics is to
search the phase transition from baryons to their consti-
tutes - deconfined quarks. Heavy ion colliders such as RHIC
(Brookhaven) and LHC (CERN) are now on line to explore
the QCD phase diagram for the high temperature and small
baryon density regimes, for which lattice QCD calculations
predict a smooth crossover to the QCD phase transition (see
Stephanov (2004) for review).
Conversely for the low temperature and high baryon
density regimes, compact stars are expected to provide a
unique window on the phase transition at their extreme den-
sity with super-strong magnetic field. It has been suggested
long ago that quark matter may exist in the interior of com-
pact objects (Itoh 1970; Bodmer 1971; Witten 1984). Hybrid
stars (and strange quark stars) are considered to be such ob-
jects, which feature quark cores surrounded by a hadronic
⋆E-mail: yasutake@th.nao.ac.jp
† kiuchi@gravity.phys.waseda.ac.jp
‡ kei.kotake@nao.ac.jp
matter mantle (or quark cores only) (for reviews, e.g., Weber
(1999); Glendenning (2001)). Even if the relevant conditions
could be reached in a laboratory in the near future (Senger
2004), the conditions prevailing in the compact stars are dif-
ferent from those produced in accelerators, i.e., the matter is
long-lived, charge neutral and in β-equilibrium with respect
to weak interactions. It is therefore important to investigate
the properties of such “exotic” stars, providing hints about
the main features of matter at those extreme conditions.
The formation of the quark cores in compact stars
is expected to take place by a first-order phase transition
(Glendenning 1992, 2001). Albeit still very uncertain (e.g.,
Horvath (2007)), such a transition would proceed by the
conversion of initially metastable hadronic matter in the
core into the new deconfined quark phase. The metastable
phase could be formed as the central density of neutron stars
exceeds a critical value, due to mass accretion, spin-down
or cooling. Possible astrophysical cites are in the protoneu-
tron stars during the collapse of of supernova cores (near
the epoch of bounce (Takahara & Sato 1985; Yasutake et al.
2007) or at the late postbounce phase (Gentile et al. 1993;

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2
N. Yasutake, K. Kiuchi, and K. Kotake
Nakazato et al. 2008; Sagert et al. 2008)) and in old neu-
tron stars accreting from their companions (Benvenuto et al.
1994; Chau 1997; Lin et al. 2006). In whichever cases, the
sudden nucleation of the exotic phase in the hadronic
star will be accompanied by a core-quake and huge en-
ergy release of the gravitational binding energy. Such en-
ergy release has been proposed to explain the central en-
gines of the gamma-ray bursts (Bombaci & Datta 2000;
Berezhiani et al. 2003). Possible observables of these tran-
sient phenomenon should be glitches, magnetar flares, and
superbursts (Alford et al. 2007). In addition, the detec-
tion of gravitational waves (Ioka 2001; Yasutake et al. 2007;
Lin et al. 2006; Abdikamalov et al. 2008) and neutrinos
(Nakazato et al. 2008; Sagert et al. 2008) generated at the
moment of the phase transition, should supply us implica-
tions to unveil the mechanism of the phase transition.
Here it should be noted that most of these cal-
culations/estimations concerning
inside compact stars are limited to a non-rotating case,
in which the Tolman-Oppenheimer-Volkoff equation is
solved to obtain their structures (Haensel et al. 1986;
Zdunik et al. 1987; Muto & Tatsumi 1990; Drago et al.
2004; Yasutake & Kashiwa
Gourgoulhon et al.(1999);
Zdunik et al. (2007), inwhich relativistic
configurations of rotating strange stars, beyond the so-
called slow rotation approximations (see references in
Glendenning (2001)), are constructed for estimating the en-
ergy release. In Gourgoulhon et al. (1999); Yasutake et al.
(2005), hadron matter is assumed to be converted fully to
quark matter, leading to the formation of strange quark
stars like in Alcock et al. (1986); Benvenuto & Horvath
(1989); Olesen & Madsen (1991); Lugones et al. (1994).
However, it is recently pointed out that the strange quark
stars could be ruled out by their too fast spin-down rates via
gravitational radiations from r-modes instability (Madsen
2000). The quasi-periodic oscillations (QPOs) of strange
quark stars could not be reconciled with the observations
(Watts & Reddy 2007). Moreover, Mallick et al. (2009)
has recently claimed that magnetars cannot convert to
purely quark stars, but only to hybrid stars. These suggest
us to pay attention to hybrid stars rather than stars
made of purely deconfined quarks. In a very recent work
by Zdunik et al. (2008), the phase transition of rotating
hybrid stars is discussed, however, the equation of state
(EOS) is made very idealistic, assumed to model a strong
phase transition, seemingly less sophisticated than the
EOSs in the recent literature cited above. In addition to
rotation, neutron stars observed in nature are magnetized
with the typical magnetic field strength ∼ 1011–1013G
(Lyne & Graham-Smith 2005). The field strength is often
much larger than the canonical value as ∼ 1015G for
a special class of the neutron stars such as magnetars
(Lattimer & Prakash 2007;
In a series of our recent papers (Kiuchi & Kotake 2008;
Kiuchi & Yoshida 2008; Kiuchi et al. 2008d), we have stud-
ied equilibrium configurations of relativistic magnetized
compact stars to apply for the understanding of their
formations and evolutions, but with hadronic EOSs.
Above situations motivate us to investigate the struc-
tures of relativistic hybrid stars with magnetic fields, and
by using them, to estimate the energy release at the phase
the phasetransition
2009).Exceptions
Yasutake et al.
are
(2005);
equilibrium
for
Woods & Thompson2006).
transition. This study is posed as an extension to the
study by Yasutake et al. (2005), in which the phase tran-
sition to the rotating strange stars was investigated. Due
to the unavailability of the method to construct a fully
general relativistic star with arbitrarily magnetic structures
(namely both with toroidal and poloidal fields), we here
consider the equilibrium with purely toroidal fields as in
Kiuchi & Yoshida (2008). It is noted that the outcomes of
the recent stellar evolution calculations (Heger et al. 2005)
and the MHD(magnetohydrodynamics) simulations of core-
collapse supernovae (Kotake et al. (2004); Takiwaki et al.
(2004); Obergaulinger et al. (2006); Dessart et al. (2007);
Sawai et al. (2008); Kiuchi et al. (2008c); Takiwaki et al.
(2009)), suggesting much dominance of the toroidal fields,
are not in contradiction with the assumption. Following the
scenario proposed in Yasutake et al. (2005), we consider the
possible evolutionary tracks of a rapidly rotating and mag-
netized neutron star to a slowing rotating hybrid star due to
the spin-down via gravitational radiation and/or magnetic
braking. During the evolutions, the baryon mass and the
magnetic field strength are taken to be constant for simplic-
ity. The energy release can be estimated from the difference
in the mass-energies between the hadronic star and the hy-
brid star along each sequence. By constructing thousands of
equilibrium configurations, we hope to clarify the possible
maximum energy release at the moment of the transition
and discuss their astrophysical implications.
The paper is organized as follows. The method for con-
structing the EOS with the phase transition and the nu-
merical scheme for the stellar equilibrium configurations are
briefly summarized in Sec. 2. Sec. 3 is devoted to showing
numerical results. Summary and discussion follow in Sec. 4.
In this paper, we use geometrical units with G = c = 1.
2 EQUATION OF STATE AND NUMERICAL
METHOD
2.1Equations of State with a first-order phase
transition
As mentioned, we assume that the deconfinement of the
quarks takes place at the first order phase transition. In this
case, a mixed phase can form and it is typically described us-
ing two separate EOSs, one for the hadronic and the other
for the quark phase. Here the bulk Gibbs construction is
used to bridge the two phases. In the following, we briefly
summarize the adopted EOSs for each phase and explain
the features of the resulting EOS with the first order phase
transition.
Since lattice QCD is yet to make solid predictions for
large density regimes in compact objects, the quark mat-
ter EOS is currently computed using phenomenological de-
scriptions such as the MIT bag or the Nambu-Jona-Lasinio
(NJL) models. For the quark phase, we choose the EOS
based on the very simple but widely applied MIT bag model
(see Weber (1999); Glendenning (2001) for reviews). Using
the model, one can express the energy density, ǫ, and the
pressure, P of strange quark matter as functions of baryon
number density, n in the following form,
ǫ=
?
f
ǫf+ B, (1)

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Relativistic hybrid stars with super-strong magnetic fields
3
ǫf
=
3m4
8π2[xf(2x2
n2d(ǫ/n)
d n
f
f+ 1)
?
1 + x2
f− arsinh xf], (2)
P=
, (3)
where mf is the quark mass of the flavor of f taken to be
mu = md = 5 MeV and ms = 150 MeV, xf = k(f)
is a normalized Fermi wave number of k(f)
bag constant. Baryon number density of strange matter is
n =
ber density of f quarks. We use a simple MIT bag model
of self-bound strange quark matter (Chodos et al. 1974),
neglecting the quark interactions except for the confine-
ment effects described by the bag constant (see, e.g., ref-
erences in Bonanno et al. (2007) for sophistication of the
model). We set that the bag constants are 200 MeV fm−3
and 250 MeV fm−3in this paper. These values seem con-
sistent with the implications in the recent lattice QCD re-
sults (Ivanov et al. 2005). The quark EOS with lower bag
constant than 200 MeV fm−3may be too soft to be com-
patible with the observation of the massive neutron star like
pulsars Ter 5 I and J (> 1.68M⊙ with 95 % confidence)
(Ransom et al. 2005; Alford et al. 2007). For the EOS of
the hadron phase, we adopt the nuclear EOS developed
by Shen et al. (1998), which are often employed in recent
MHD simulations relevant for magnetars (see Kotake et al.
(2006) for a review). The Shen EOS is based on the rela-
tivistic mean field theory with a local density approxima-
tion, which has been constructed to reproduce the exper-
imental data of masses and radii of stable and unstable
nuclei (see references in Shen et al. (1998)). At the max-
imum densities higher than two times of saturation den-
sity, muons may appear (Wiringa et al. 1988; Akmal et al.
1998). However, we neglect it, since the muon contribution
to pressure at the higher density has been pointed to be very
small (Douchin & Haencel 2001).
In modeling the phase transition to quark matter, there
is a main physical uncertainty, the critical density for the
onset of the mixed phase. Under the MIT bag model, the
transition is determined by the value of the bag constant.
We use the bulk Gibbs construction to bridge the two phases
using the technique developed by Glendenning (1992). In the
transition, two conserved quantities are the baryon number
density (nB) and the electric charge density (: YenB with Ye
being the electron fraction),
F/mf
F, and B is the
1
3(nu + nd+ ns), where nf = k(f) 3
F
/π2is the num-
nB
=χnB,Q+ (1 − χ)nB,H, (4)
YenB
=χYC,Q nB,Q+ (1 − χ) YC,H nB,H. (5)
Here χ is the volume fraction of matter in the quark phase.
The subscripts H and Q label the number density and the
charge fraction, YC, in the hadronic and in the quark phase,
respectively. Since matter in cold compact stars is in chemi-
cal equilibrium under β-decay with vanishing neutrino chem-
ical potentials, the following equations have to be satisfied,
µe+ µp
=µn, (6)
µn
=µu+ 2µd, (7)
µp
=2µu+ µd, (8)
µd
=µs, (9)
together with the mechanical equilibrium, the equality of
the pressure in the two phases,
PQ= PH.(10)
Here for simplicity, the finite size effects on the phase
transition (Endo et al. 2006; Maruyama et al. 2006) are ig-
nored. Five unknown variables to describe the mixed phase
(χ,nB,Q,nB,H,YC,Q,YC,H) can be solved by the five sets of
the equations of (3,4,6,7,9).
Figures 1 and 2 show the constructed EOS with the
first-order phase transition. For comparison, we also plot
the hadronic “Shen EOS” mentioned above and pure quark
EOS(”quarkB200” and ”quarkB250”). The left and right
panels are the pressure and the energy per baryon as a
function of the baryon density, respectively. The labels of
“mixB200(quarkB200)” and “mixB250(quarkB250)” indi-
cate the two different bag constants, B=200 MeV fm−3and
250 MeV fm−3. The EOS consists of the three phases of the
“Hadron”, “Mixed”, and the “Quark”. The critical densities
with n1 (open circles) and n2 (closed circles) marking the
boundaries between the hadron, mixed, and quark phase,
are also given in Table 1. From the left panel and Table 1,
it can be seen that the critical density for the mixed phase
becomes higher for the larger bag constant. This is because
for the large bag constant, the density should be larger to
achieve the equilibrium between the mixed phase and the
pure hadron phase, remembering that the pressure of the
quark matter becomes smaller with increasing the bag con-
stant (from Eqs. (1,2)). The higher n1 for the large bag con-
stant also leads to the higher n2, the transition to the quark
phase (Table 1). From the right panel of Figure 1 and 2, it
is seen that the energy with the quark phase (”mixB200”
and ”mixB250”) are clearly lower than the case only with
the hadronic phase (”shen”), and also that the energy is
generally lower for the smaller bag constant above n1.
The actual conversion process from nuclear matter to
quark matter has been of a topic of hot debate, such as deto-
nation or deflagration (Drago et al. 2007). The range of the
conversion timescale is estimated to be very wide, roughly
0.1 − 100 sec, depending upon the bag constant, tempera-
ture, mass of neutron stars, and so forth (Olesen & Madsen
1991). Since our main aim of this paper is the estimation of
the liberated energy by the conversion from neutron stars
to hybrid stars, we do not consider the detailed combustion
processes in this study.
Proto-neutron stars left after supernova explosion are
very hot (∼ 50 MeV) initially, however, cool down be-
low ∼ 1MeV in some tens of seconds (Burrows & Lattimer
1986). The newly formed neutron stars stabilize at prac-
tically zero temperature. As mentioned, in cold neutron
stars, the β equilibrium in the weak interactions can be
well validated, with neutrinos and antineutrinos freely es-
caping from the star. Combining the zero-temperature, zero-
neutrino fraction, and the beta-equilibrium conditions with
the charge neutrality condition, the thermodynamic vari-
ables depending on the three parameters (e.g. the pressure as
P(ρ,Ye,T) with Ye being the electron fraction), can only be
determined by a single variable (barotropic), which we take
to be the energy density, namely P(ǫ) (Shapiro & Teukolsky
1983) in the following.
In the following, we introduce the baryon mass density,
ρ0, for convenience, defined to be the number density, n,
multiplied a baryon mass, m0 = 1.6605 × 10−24g.

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4
N. Yasutake, K. Kiuchi, and K. Kotake
Table 1. Critical densities making the transition to
the mixed, n1, and the quark phase, n2, respectively.
EOSn1 [/fm3]n2 [/fm3]
mixB200
mixB250
2.30E-01
2.63E-01
1.29
1.51
2.2 Constructing method of equilibrium stellar
configurations
Employing the constructed EOS with the phase transition,
we construct the equilibrium stellar configurations. As men-
tioned in introduction, we pay attention to the general rela-
tivistic and toroidally magnetized stellar configurations. The
basic equations and the numerical methods for the purpose
are already given in Kiuchi & Yoshida (2008). Hence, we
only give a brief summary for later convenience.
Assumptions to obtain the equilibrium models are sum-
marized as follows ; (1) Equilibrium models are stationary
and axisymmetric. (2) The matter source is approximated
by a perfect fluid with infinite conductivity. (3) There is no
meridional flow of the matter. (4) The EOS for the matter
is barotropic, which is satisfied as mentioned. (5) Magnetic
axis and rotation axis are aligned. Because the circularity
condition (Wald 1984) holds under these assumptions, the
metric can be written in the form
Cook et al. 1992),
(Komatsu et al. 1989;
ds2= −eγ+ρdt2+ e2α(dr2+ r2dθ2)
+eγ−ρr2sin2θ(dϕ − ωdt)2, (11)
where the metric potentials, γ, ρ, α, and ω, are functions
of r and θ only. We see that the non-zero component of
Faraday tensor Fµν in this coordinate is F12. Integrability
of the equation of motion of the matter requires,
eγ−2αsinθF12 = K(u);u ≡ ρ0he2γr2sin2θ, (12)
where K is an arbitrary function of ρ0he2γr2sin2θ. The vari-
ables ρ0 and h represent the baryon density and relativistic
specific enthalpy, respectively. Integrating the equation of
motion of the matter, we arrive at the equation of hydro-
static equilibrium,
?P(ǫ)
0
dP
ǫ + P+ρ + γ
2
+1
2ln(1 − v2)
+1
4π
?
K(u)
u
dK
dudu = C,(13)
where v = (Ω−ω)r sinθe−ρwith Ω being the angular veloc-
ity of the matter and C is an integration constant. Here, we
assume the rigid rotation. The first term of equation (13)
depends only on EOSs. Hence, we prepare the values of the
integral with EOS tables, precisely. This preparation enable
us to calculate equation (13) precisely, though our EOSs are
quite different from polytrope models. To compute specific
models of the magnetized stars, we need specify the func-
tion form of K, which determines the distribution of the
magnetic fields. We take the following simple form,
K(u) = buk, (14)
where b and k are constants. Regularity of toroidal magnetic
field on the magnetic axis requires that k ? 1. If k ? 1,
the magnetic fields vanish at the surface of the star. In this
study, we consider the k = 1 case because in the general rel-
ativistic MHD simulation, Kiuchi et al. (2008c) have found
that magnetic distribution with k ?= 1 is unstable against
axisymmetric perturbations.
To solve the master equations numerically, we employ
the Cook-Shapiro-Teukolsky scheme (Cook et al. 1992) ex-
tended by Kiuchi & Yoshida (2008), which does not care
about the function form of EOS. Hence, it is straight for-
ward to update our numerical code for incorporating the
EOS with the phase transition.
After obtaining solutions, it is useful to compute global
physical quantities characterizing the equilibrium configura-
tions to clearly understand the properties of the sequences
of the equilibrium models. In this paper, we compute the
following quantities: the gravitational mass M, the baryon
rest mass M0, the total angular momentum J, the total ro-
tational energy T, the total magnetic energy H, the mag-
netic flux Φ, the gravitational energy W and the mean de-
formation rate ¯ e, whose definitions are explicitly given in
Kiuchi & Yoshida (2008). More explicitly, the mean defor-
mation rate ¯ e is defined as
¯ e ≡Izz− Ixx
Izz
, (15)
where Ixx
2π?
matter. Circumferential radius Rcir is defined as Rcir ≡
e(γ−ρ)/2re with re being the coordinate radius at the stellar
equatorial surface.
We checked the convergence of the presented results by
doubling the mesh numbers from the standard set of radial
and angular direction mesh points of 400×260. By checking
the relativistic virial identities (Bonazzola & Gourgoulhon
1994) for all the models, we confirm that the typical values
are orders of magnitude 10−3, and become 10−2at worse
(10−4at best). These values, which is a measure of the
numerical convergence, are almost same for the polytropic
EOS case (Kiuchi & Yoshida 2008). In general, the conver-
gence is known to become much worse for realistic EOSs, be-
cause their density and pressure profile are not smooth due
to phase transitions. In this respect, our numerical scheme
works well. In Sec. 3.2, we will discuss energy releases by
the QCD phase transition and find that the values of our
interest are ∼ 10−2M⊙, for which the numerical accuracy
above is certificated mostly. By doubling the mesh points,
i.e., 800(r) × 520(θ), we checked that the order of the mag-
nitude of the released energy does not change.
In constructing one equilibrium sequence, we have three
parameters to choose, namely the central density ρc, the
strength of the magnetic field parameter b, and the axis ra-
tio rp/re. Changing these parameters, we seek solutions in
as wide parameter range as possible to study the proper-
ties of the equilibrium sequences. We need to calculate more
than 50 models changing ρc, b, to follow one evolutionary
sequence for a fixed baryon mass and magnetic flux. In ad-
dition, we change the initial angular momentum of each se-
quence by 10 models to model the rapidly rotating to the
non-rotating case and also 3 different EOSs of mixB200,
mixB250, and Shen are employed. This means that we have
=π?
ǫr4sinθ(1 + cos2θ)drdθ and Izz
ǫr4sin3θdrdθ with ǫ being the energy density of the
=

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Relativistic hybrid stars with super-strong magnetic fields
5
to construct at least 1500 models to explore properties of
the magnetized compact objects with quark cores systemati-
cally. In doing so, we use the Rosenbrock and Gram-Schmidt
method, which is helpful to obtain the convergence of the
solutions efficiently.
3 NUMERICAL RESULTS
First of all, we discuss how the EOS with the phase transi-
tion affects the equilibrium configurations in subsection 3.1,
where we pay attention to non-rotating models. Since the
magnetars and the high field neutron stars observed so far
are all slow rotators, such non-rotating but highly magne-
tized static models could well be approximated to such stars.
Moreover the static models merit that one can see purely
magnetic effects on the equilibrium properties because there
is no centrifugal force and all the stellar deformation is at-
tributed to the magnetic stress. Then in subsection 3.2, we
move on to discuss the the releasable energy of the phase
transition from the rotating and magnetized hadronic stars
to the hybrid stars.
3.1 Effect of the phase transition on the
equilibrium configurations
Now we discuss the equilibrium configurations of the
non-rotatingandstrongly
with/without the quark cores. Figures 3 and 4 are one ex-
ample for the hybrid star with mixB200 EOS and for the
neutron star with the hadronic “Shen” EOS, showing the
distributions of the baryon density (left panel) and the mag-
netic field (right panel) in the meridional planes, respec-
tively. These two models have the same baryon mass of
1.86M⊙, and the same magnetic flux of 2.00 × 1030G cm2,
which mean that they are really highly magnetized models
with the central magnetic fields of ∼ 1018G.
Comparing left panels of Figures 3 and 4, more con-
centration of the matter in the center is clearly seen for the
hybrid star (Figure 3). The concentration is also clearly seen
from Figure 5. Inside the inner 6 km of the core of the hybrid
star, the mixed phase appears, leading to the enhancement
in the compression of the matter due to the softening of the
EOS (e.g., Figures 1 and 2).
From right panel of Figure 4, the strong toroidal field
lines are seen to behave like a rubber belt, wrapping around
the waist of the neutron star. It is found that the magnetic
fields frozen-in to the matter, are also compressed by the
presence of the quark phase for the hybrid star (right panel
of Figure 3). In fact, the pinching of the field lines in the
panel corresponds to the surface of the quark core, ∼ 6km
in radius. These qualitative features are also true for the
other equilibrium models.
In addition, a general trend in the equilibrium configu-
rations for the hybrid stars is that they are more compact
than the neutron star, due to the softness of the EOS. Given
the same stellar baryon mass (M0), the gravitational mass is
smaller up to ∼ 0.01M⊙ than for the neutron stars, reflect-
ing their smaller energy of the quark matter (right panel of
Figure 1).
Given the fixed magnetic flux, we can construct equi-
librium sequences by changing the central density. To char-
magnetized compactstars
acterize the features of the hybrid stars, we pay attention to
the model with the maximum mass (dM/dρ0,c = 0) along
each sequence, which we call as the maximum mass sequence
for convenience. In table 2, the important physical quanti-
ties are summarized for the maximum mass sequences with
different magnetic fluxes. It is here noted that the maximum
value of the magnetic flux in the table (Φ = 2.5×1030G cm2)
corresponds to the non-convergence limit, beyond which any
solutions cannot converge with the present numerical scheme
(Kiuchi & Yoshida 2008). Albeit with such limitations, the
field strength is already enough high to affect the configura-
tions and we can well study the magnetic effects on them.
Since the compression of the matter is more enhanced
for the smaller bag constant, the hybrid stars with the
smaller bag constant become more compact (see Rcir Ta-
ble 2). The maximum magnetic fields are found to become
up to about 30 % larger for the hybrid stars than for the
neutron stars (compare Bmax for mixB200 with for Shen).
It is also found that the hybrid stars with the smaller bag
constant become more prolate (smaller values of ¯ e) and also
their maximum masses become smaller up to ∼ 10% than
those for the larger bag constant models. All these features
are helpful to understand the properties of the evolution
tracks of the hybrid stars, which we discuss from the next
section.
3.2 An evolutionary track to a hybrid star
Based on the equilibrium configurations mentioned above,
we now move on to discuss an evolutionary path to the for-
mation of a hybrid star due to the spin-down of magnetized
and rotating neutron stars.
The evolution scenario we have in mind is illustrated
in Figure 6. Let us consider evolution of a single pro-
toneutron star, left after core-collapse supernova explosion,
which rotates with the mass-shedding limit ((A) in Fig-
ure 6). The maximum magnetic fields deep inside the core
are taken to be ∼ 1018G, which could be sustained due
to α − Ω dynamos in such a rapidly rotating neutron star
(Thompson & Duncan 1993, 1996). It should be noted that
the possibility of such ultra-magnetic fields has not been
rejected so far, because what we can learn from the obser-
vations of magnetars by their periods and spin-down rates
is only their surface fields (∼ 1015G). During their evolu-
tion, the baryon mass and the magnetic field strength are
assumed to be constant for simplicity. Such models may
model the evolution of the isolated compact stars, losing
angular momentum via the gravitational radiation and/or
magnetic breaking (from (A) to (B)). The phase transition
to the hybrid stars is expected to take place during the evo-
lution (shown as (B) to (C) in Figure 6). At the moment, the
baryon rest masses and the angular momenta are assumed to
be conserved, which may be justified because the timescale
of the conversion, albeit uncertain of 0.1-100 s, are too short
compared to typical evolutionary timescale of compact stars
(more than 103years) (Olesen & Madsen 1991). After the
transition, the newly born hybrid star evolves, again losing
the angular momentum, to settle down to the magnetized
and no-rotating hybrid star finally (see (C) to (D)).
Now using the left panel of Figure 7, we proceed to dis-
cuss quantitatively the evolution tracks mentioned above.
The red solid the baryon mass ’M0 = 1.86M⊙’ rotating