# Stably stratified stars containing magnetic fields whose toroidal components
are much larger than the poloidal ones in general relativity – A perturbation
analysis –

Shijun Yoshida1<EMAIL_ADDRESS>1Astronomical Institute, Tohoku
University, Sendai 980-8578, Japan

###### Abstract

We construct the stably stratified magnetized stars within the framework of
general relativity. The effects of magnetic fields on the structure of the
star and spacetime are treated as perturbations of non-magnetized stars. By
assuming ideal magnetohydrodynamics and employing one-parameter equations of
state, we derive basic equations for describing stationary and axisymmetric
stably stratified stars containing magnetic fields whose toroidal components
are much larger than the poloidal ones. A number of the polytropic models are
numerically calculated to investigate basic properties of the effects of
magnetic fields on the stellar structure. According to the stability result
obtained by Braithwaite, which remains a matter of conjecture for general
magnetized stars, certain of the magnetized stars constructed in this study
are possibly stable.

###### pacs:

04.40.Dg, 97.60.Jd

##  I Introduction

It has been well-accepted that soft-gamma repeaters (SGRs) and anomalous x-ray
pulsars (AXPs) are magnetars, highly magnetized neutron stars whose strength
of the surface field is as large as $\sim 10^{14}-10^{15}$ G duncan ; paczynsk
; thompsona ; thompsonb ; woods . The existence of the magnetar has
reactivated studies on equilibrium configurations of magnetized stars.

In order to elucidate basic properties of equilibrium configurations of
magnetized stars, a large number of studies have been performed so far since
the pioneering work of Chandrasekhar and Fermi chandra . A large fraction of
those studies have been done within the framework of Newtonian
magnetohydrodynamics and Newton’s theory of gravity (cf., e.g., Refs.
prendergast ; woltjer ; roxburgh ; trehanb ; monaghana ; monaghanb ; sinha ;
trehana ; ioka ; miketinac ; miketinac2 ; tomimura ; yoshida0 ; yoshida ;
lander ; fujisawaa ; fujisawab ; duez ). Since neutron stars are very compact
in the sense that their compactness $M/R$ is as large as $\sim 0.1-0.2$ with
$M$ and $R$ being their mass and radius in geometrical units, general
relativity is required to describe the gravitational field of neutron stars.
Therefore, general relativistic models of magnetized stars have been
investigated as well. Bocquet et al. bocquet  and Cardall et al. cardall
obtained relativistic neutron star models with purely poloidal magnetic
fields. Using a perturbative technique, Konno et al. konno  calculated similar
models to those obtained in Refs. bocquet ; cardall . Kiuchi and Yoshida
kiuchi  computed magnetized stars with purely toroidal fields (cf., also, Ref.
frieben ). Ioka and Sasaki Ioka2004 , Colaiuda et al. colaiuda , and Ciolfi et
al. ciolfia ; ciolfib  derived relativistic stellar models having both
toroidal and poloidal magnetic fields with perturbative techniques (cf., also,
Ref. ciolfic ). Yoshida et al. yoshidaa  included the effects of the stable
stratification in the magnetized star model obtained in Ref. Ioka2004 . Uryu
et al. uryua ; uryub  obtained magnetized stars with mixed poloidal-toroidal
magnetic fields by solving a full set of Einstein equations,
magnetohydrodynamics equations, and coordinate conditions numerically. By
assuming simpler conformally flat spacetime, Pili et al. pilia ; pilib ;
bucciantini ; pilic  calculated many models of magnetized stars. Although
great progress has been achieved in this field, as mentioned before, further
studies are required because all the magnetized star models are constructed by
some particular magnetic-field configurations which are not necessarily
realistic. In particular, it is not still clear at all whether stable models
exist.

Stability of magnetized stars with a relatively simple magnetic field
structure have been examined with analytical approaches. The pioneering work
was done by Tayler tayler73 , who showed that stars with purely toroidal
magnetic fields are unstable. Wright wright73  subsequently showed that the
same instability mechanism, the pinch-type instability mechanism, operates in
stars with purely poloidal magnetic fields. He also suggested the possibility
that stars having mixed poloidal-toroidal magnetic fields may be stable if the
strength of both components is comparable (cf., also, Refs. markey ; tayler80
; assche ). Flowers and Ruderman flowers  found that another type of
instability occurs in purely poloidal magnetic field configurations. All those
classical stability analyses were based on a method of an energy principle in
the framework of Newtonian dynamics (cf., also, Refs. pitts ; goossens ).
Another approach is a local analysis, with which Acheson acheson1978
investigated the stability of rotating magnetized stars containing purely
toroidal fields in detail within the framework of Newtonian dynamics (cf.,
also, Refs. acheson1979 ; spruit ) and derived detailed stability conditions
for purely toroidal magnetic fields buried inside rotating stars with
dissipation. Bonanno and Urpin analyzed the axisymmetric stability bonanno
and the non-axisymmetric stability bonanno2  of cylindrical equilibrium
configurations possessing mixed poloidal-toroidal fields, while ignoring
compressibility and stratification of the fluid.

Recently the stability problem of the magnetized star has been approached from
another direction, dynamical simulation approaches, and some significant
progress has been made. By following the time evolution of small random
initial magnetic fields around a spherical star in the framework of Newtonian
resistive magnetohydrodynamics, Braithwaite and Spruit Braithwaite2004 ;
Braithwaite2006  obtained stable equilibria of magnetized stars that are
formed as a self-organization phenomenon. The resulting stable magnetic fields
have both poloidal and toroidal components with comparable strength and
support the conjecture for stability conditions of the magnetized star given
by the classical studies mentioned before (cf., also, Ref. duez10 ). By using
the numerical magnetohydrodynamic simulation, Braithwaite Braithwaite2009
studied stability conditions for the magnetized stars and obtained a stability
condition for his models given in terms of the ratio of the poloidal magnetic
energy to the total magnetic energy. The stability condition is given by

$\tilde{a}{E_{\rm EM}\over|W|}<{E_{\rm EM}^{(p)}\over E_{\rm EM}}\lesssim
0.8\,,$ (1)

where $E_{\rm EM}$, $E_{\rm EM}^{(p)}$, and $W$ are the total magnetic energy,
the poloidal magnetic energy, and the gravitational energy, respectively, and
$\tilde{a}$ is a dimensionless factor related to the buoyancy properties of
the star. For neutron stars and main-sequence stars, the dimensionless factor
$\tilde{a}$ is of order $10^{3}$ and $10$, respectively. Lander and Jones
examined the stability of magnetized stars by numerically solving the time
evolution of linear perturbations of the stars in their series of papers
lander11 ; lander11b ; lander12 . For the stars with purely toroidal and
purely poloidal magnetic fields, their results are consistent with those of
the classical stability analysis, i.e., the pinch-type instability occurs near
the symmetry and the magnetic axes for the cases of the purely toroidal and
the purely poloidal magnetic fields, respectively (cf., also, Refs. kiuchia ;
kiuchib ; ciolfid ; lasky ; ciolfie ; zink ). They also assessed the stability
of various magnetized stars with mixed poloidal-toroidal fields and found that
all their models considered suffer from the pinch-type instability even for
the cases in which the poloidal and toroidal components have comparable
strength lander12 . At first glance, it seems that the results by Lander and
Jones are incompatible with those by Braithwaite and his collaborators
Braithwaite2009 ; duez10 . Mitchell et al. mitchell  made numerical
simulations smilar to those of Braithwaite and his collaborators
Braithwaite2004 ; Braithwaite2006  but for the case of the non-stratified
star. They then obtained no stable equilibrium for the non-stratified case and
showed that stable stratification of the fluid will be a key ingredient, which
is taken into account in the analyses of Refs. Braithwaite2009 ; duez10  but
not in the analyses of Ref. lander12 . In other words, the results obtained by
Mitchell et al. mitchell  suggest that stable stratification is required to
avoid instability for some magnetic-field configurations inside the star. Note
that in the simulations by Braithwaite and his collaborators Braithwaite2004 ;
Braithwaite2006 , resistive dissipation will also play a crucial role.
Therefore, effects of the resistive dissipation on dynamical stability of the
magnetized star need to be closely examined.

Despite the fact that a large number of studies on equilibria and stabilities
of magnetized stars have been made so far, as mentioned before, the magnetic
field structure of the neutron star has not yet been elucidated not only
theoretically but also observationally. The formation process of the neutron
star would however provide us with some clues. During the core collapse events
which produce neutron stars, the poloidal magnetic field lines would get
wrapped around the rotation axis because of the differential rotation of the
core (cf., e.g., Ref. kotake ). As a result, the toroidal field would be
significantly amplified. It is therefore likely to expect that the toroidal
component of the magnetic field is much larger than the poloidal one inside
the neutron star at least soon after its birth.

To investigate properties of the magnetized star whose toroidal fields are
much larger than the poloidal ones, Kiuchi and Yoshida kiuchi  constructed the
magnetized stars completely neglecting the poloidal component of the magnetic
field. Although studies on stars with purely toroidal magnetic fields can
elucidate approximate properties of magnetized stars whose toroidal fields are
much larger than the poloidal ones, purely toroidal magnetic fields inside the
star are unstable as mentioned before. To stabilize the toroidal magnetic
field inside the star, the inclusion of the poloidal magnetic field is
necessary. To our knowledge, however, equilibrium states of the magnetized
star whose toroidal fields are much larger than the poloidal ones, which are
plausible neutron star models, have not been constructed so far, except the
case of purely toroidal magnetic fields. As mentioned before, another
important stabilizing agent for magnetic fields inside the star is a stable
stratification of the fluid. In order to construct neutron star models with
more realistic interior magnetic field structure, in this study, we
investigate stably stratified stars having magnetic fields characterized by
the condition of ${E_{\rm EM}^{(p)}/E_{\rm EM}}\ll 1$, i.e., the toroidal
field is much larger than the poloidal one, within the framework of general
relativity.

The strength of the effects of magnetic fields on the stellar structure can be
roughly estimated by an approximate ratio of the magnetic field energy,
$E_{\rm EM}$, to the gravitational energy, $W$, given by

${E_{\rm EM}\over|W|}\approx 10^{-6}\left({B_{0}\over 10^{15}\,{\rm
G}}\right)^{2}\left({R\over 10\,{\rm km}}\right)^{4}\left({M\over
1.4\,M_{\odot}}\right)^{-2}\,,$ (2)

where $B_{0}$, $R$, and $M$ are the strength of the magnetic field, the
radius, and the mass of the star, respectively. This ratio is very small even
if a magnetar characterized by $B_{0}\sim 10^{15}\,{\rm G}$ is considered. In
order to investigate effects of magnetic fields on the neutron star structure,
therefore, perturbation approaches are generally quite efficient in the sense
that they are tractable and give sufficiently accurate results. We therefore
make use of a perturbation approach to study the structure of the magnetized
star in this work.

The present paper is organized as follows. In Section II, we introduce the
general formalism for general relativistic ideal magnetohydrodynamics. Section
III presents the formalism used to construct the stably stratified magnetized
star whose toroidal fields are much larger than the poloidal ones. In Section
IV, we exhibit examples of the stably stratified magnetized stars calculated
numerically. Finally, we give the discussion and summary in Sections V and VI,
respectively. In Appendix, we give a Newtonian analysis of the same magnetized
star as that discussed in this paper. In the following, we choose the
signature $(-,+,+,+)$ for the spacetime metric and, unless otherwise stated,
we adopt geometrical units with $c=G=1$, where $c$ and $G$ are the speed of
light and Newton’s gravitational constant, respectively.

##  II Basic equations describing dynamics of perfectly conductive fluids

The dynamics of perfect fluids coupled with electromagnetic fields may be
described by the magnetohydrodynamics equations summarized as follows. Baryon
mass conservation equation:

$\nabla_{\mu}\left(\rho u^{\mu}\right)=0\,,$ (3)

where $\rho$ and $u^{\mu}$ are the rest-mass density and the fluid four-
velocity, respectively. Here, $\nabla_{\mu}$ denotes the covariant derivative
associated with the metric $g_{\mu\nu}$, and spacetime indices are denoted by
lower case Greek letters ($\alpha,\beta,\gamma,\cdots$). The Maxwell
equations:

$\displaystyle\nabla_{\alpha}F_{\mu\nu}+\nabla_{\mu}F_{\nu\alpha}+\nabla_{\nu}F_{\alpha\mu}=0\,,$
(4) $\displaystyle\nabla_{\nu}F^{\mu\nu}=4\pi J^{\mu}\,,$ (5)

where $F_{\mu\nu}$ and $J^{\mu}$ are the Faraday tensor and the current four-
vector, respectively. The conservation law of the energy-momentum tensor:

$\nabla_{\nu}T^{\mu\nu}=0\,,$ (6)

where $T^{\mu\nu}$ is the energy-momentum tensor, defined by

$\displaystyle T^{\mu\nu}$ $\displaystyle=$ $\displaystyle\rho
hu^{\mu}u^{\nu}+Pg^{\mu\nu}$ (7) $\displaystyle+{1\over
4\pi}\left[F^{\mu\alpha}{F^{\nu}}_{\alpha}-{1\over
4}g^{\mu\nu}F^{\alpha\beta}F_{\alpha\beta}\right]\,,$

where $h$ and $P$ are the specific enthalpy and the pressure, respectively.
Here, the specific enthalpy may, in terms of the specific internal energy
$\varepsilon$, the pressure $P$, and the rest-mass density $\rho$, be defined
by

$h=1+\varepsilon+{P\over\rho}\,.$ (8)

As for the equations of state, we supply one-parameter equations of state,
given by

$P=P\left(\rho\right)\,,\quad\varepsilon=\varepsilon\left(\rho\right)\,.$ (9)

The electric field $E_{\mu}$ and the magnetic field $B_{\mu}$ observed by an
observer associated with the fluid four-velocity $u^{\mu}$ are defined by

$\displaystyle E_{\mu}$ $\displaystyle=$ $\displaystyle F_{\mu\nu}u^{\nu}\,,$
(10) $\displaystyle B_{\mu}$ $\displaystyle=$ $\displaystyle{1\over
2}\epsilon_{\nu\mu\alpha\beta}u^{\nu}F^{\alpha\beta}\,,$ (11)

where $\epsilon_{\mu\nu\alpha\beta}$ is the Levi-Civita tensor with
$\epsilon_{0123}=\sqrt{-g}$. Here, $g$ denotes the determinant of the metric
$g_{\mu\nu}$. Since the neutron-star matter may be approximately assumed as a
perfect conductor, in this study, we may further impose the condition of
perfect conductivity, given by

$E_{\mu}=F_{\mu\nu}u^{\nu}=0\,.$ (12)

Equation (6) may be divided into two equations, the energy equation and the
momentum equations, respectively, given by

$\displaystyle-u_{\mu}\nabla_{\nu}T^{\mu\nu}$ $\displaystyle=$ $\displaystyle
u^{\mu}\nabla_{\nu}\left\\{\rho\left(1+\varepsilon\right)\right\\}+\rho
h\nabla_{\nu}u^{\nu}$ (13) $\displaystyle=$ $\displaystyle\rho
u^{\nu}\nabla_{\nu}\varepsilon+P\nabla_{\nu}u^{\nu}=0\,,$ $\displaystyle
q_{\mu\alpha}\nabla_{\nu}T^{\alpha\nu}$ $\displaystyle=$ $\displaystyle\rho
hu^{\nu}\nabla_{\nu}u_{\mu}+q_{\mu}^{\nu}\nabla_{\nu}P-F_{\mu\nu}J^{\nu}$ (14)
$\displaystyle=$ $\displaystyle 0\,,$

where $q_{\mu\nu}=g_{\mu\nu}+u_{\mu}u_{\nu}$. Note that the perfect
conductivity condition (12) has been used in the derivation of Equation (13).

##  III Master equations for equilibrium solutions of the magnetized star

In order to obtain equilibrium solutions of the relativistic stars containing
the mixed poloidal-toroidal magnetic fields, in this study, we make the
following assumptions: (i) Equilibrium models are stationary and axisymmetric,
i.e., the spacetime has the time Killing vector $t^{\mu}$ and the rotational
Killing vector $\varphi^{\mu}$, and Lie derivatives of the equilibrium
quantities along the Killing vectors $t^{\mu}$ and $\varphi^{\mu}$ vanish.
(ii) There is no fluid flow. (iii) The magnetic fields are sufficiently weak
in the sense that the magnetic effects on the equilibrium structures may be
treated as perturbations of stars including no electromagnetic field. (iv) The
toroidal component of magnetic fields is much larger than the poloidal one.
Under these assumptions, we may derive the master equations for describing
equilibrium states of the relativistic stars containing the mixed poloidal-
toroidal magnetic fields using the magnetohydrodynamic equations summarized in
the previous section.

In order to give a clear and definite description of the assumptions (iii) and
(iv), we introduce two dimensionless smallness parameters $\varepsilon_{t}$
and $\varepsilon_{p}$ representing the amplitudes of the toroidal and the
poloidal components of magnetic fields, respectively. We may then write that
${}^{(t)}F_{\mu\nu}{}^{(t)}F^{\mu\nu}\,P^{-1}=O\left(\varepsilon_{t}^{2}\right)$
and
${}^{(p)}F_{\mu\nu}{}^{(p)}F^{\mu\nu}\,P^{-1}=O\left(\varepsilon_{p}^{2}\right)$
where ${}^{(t)}F_{\mu\nu}$ and ${}^{(p)}F_{\mu\nu}$ stand for the toroidal and
poloidal components of the Faraday tensor $F_{\mu\nu}$, respectively. Note
that $F_{\mu\nu}$ can be divided into the two parts, ${}^{(t)}F_{\mu\nu}$ and
${}^{(p)}F_{\mu\nu}$, because of the assumptions of the stationary and axially
symmetric magnetic fields and the perfectly conducting fluid without flow. By
the assumption (iii) we have $\varepsilon_{t}\ll 1$ and $\varepsilon_{p}\ll
1$. By the assumption (iv) we further impose that
$\varepsilon_{p}\ll\varepsilon_{t}\ll 1$.

In this study, as mentioned before, the unperturbed state is assumed to be a
static and spherically symmetric star without magnetic fields. Around the
spherically symmetric star, we may impose as perturbations the magnetic
fields, given by

$\displaystyle
F_{\mu\nu}=\varepsilon_{t}{}^{(t)}F_{\mu\nu}+\varepsilon_{p}{}^{(p)}F_{\mu\nu}\,.$
(15)

By this magnetic field, the matter distribution deviates from spherical
symmetry. In this study, we are primarily interested in the lowest-order
effects of the poloidal magnetic field on the structure of the star including
purely toroidal magnetic fields. We therefore consider perturbations of order
$\varepsilon_{t}^{2}$ and $\varepsilon_{t}\varepsilon_{p}$ on the structure of
the spherically symmetric star but neglect perturbations of order higher than
$\varepsilon_{p}^{2}$. Note that because of the assumption (iv), i.e.,
$\varepsilon_{p}\ll\varepsilon_{t}\ll 1$, we have the inequality
$\varepsilon_{t}^{0}\varepsilon_{p}^{0}\gg\varepsilon_{t}^{2}\varepsilon_{p}^{0}\gg\varepsilon_{t}^{1}\varepsilon_{p}^{1}\gg\varepsilon_{t}^{0}\varepsilon_{p}^{2}\,.$

###  III.1 Static and spherically symmetric stars without magnetic fields: the
$\varepsilon_{t}^{0}\varepsilon_{p}^{0}$-order equations

The line element of static and spherically symmetric spacetime may be given by

$\displaystyle ds^{2}$ $\displaystyle=$ ${}^{(0)}g_{\mu\nu}dx^{\mu}dx^{\nu}$
(17) $\displaystyle=$ $\displaystyle-e^{2\nu}dt^{2}+e^{2\lambda}dr^{2}$
$\displaystyle\quad\quad+r^{2}\left(d\theta^{2}+\sin^{2}\theta\,d\varphi^{2}\right)\,,$

where ${}^{(0)}g_{\mu\nu}$ denotes the unperturbed metric, and $\nu$ and
$\lambda$ are functions of $r$ only. The equilibrium state of the unperturbed
star is described by the following equations (cf., e.g., Ref. mtw ):

$\displaystyle{dM_{r}\over dr}=4\pi
r^{2}{\,}^{(0)}\rho\left(1+{\,}^{(0)}\varepsilon\right)\,,$ (18)
$\displaystyle{d{\,}^{(0)}P\over
dr}=-e^{2\lambda}{\,}^{(0)}\rho{\,}^{(0)}h{M_{r}+4\pi{\,}^{(0)}Pr^{3}\over
r^{2}}\,,$ (19) $\displaystyle{d\nu\over
dr}=-{1\over{\,}^{(0)}\rho{\,}^{(0)}h}{d{\,}^{(0)}P\over dr}\,,$ (20)

where $M_{r}$ is defined in terms of the metric function by

$M_{r}={r\over 2}\left(1-e^{-2\lambda}\right)\,,$ (21)

and ${\,}^{(0)}\rho$, ${\,}^{(0)}\varepsilon$, ${\,}^{(0)}P$, and
${\,}^{(0)}h$ are, respectively, the rest-mass density, specific internal
energy, pressure, and specific enthalpy for the unperturbed star.

###  III.2 Magnetic fields around a spherical star: the
$\varepsilon_{t}^{1}\varepsilon_{p}^{0}$ and
$\varepsilon_{t}^{0}\varepsilon_{p}^{1}$ order equations

Due to the assumption of no fluid flow, the assumption (ii), the fluid four-
velocity is given by

$\displaystyle u^{\mu}=\gamma\,t^{\mu}\,,$ (22)

where $\gamma$ is the function determined by the normalization condition
$u^{\mu}u_{\mu}=-1$. The perfect-conductivity condition (12) then becomes

$\displaystyle F_{\mu\nu}t^{\nu}=F_{\mu t}=0\,.$ (23)

As argued by Kiuchi and Yoshida kiuchi , the toroidal component of the
magnetic field may be characterized by the conditions, given by

${}^{(t)}F_{\mu\nu}\varphi^{\nu}=0\,.$ (24)

Thus, we see that the non-zero component of the toroidal magnetic field
${}^{(t)}F_{\mu\nu}$ is ${}^{(t)}F_{r\theta}\left(=-^{(t)}F_{\theta r}\right)$
only. The poloidal component of the magnetic field may be given in term of the
poloidal flux function $\Psi$, which is actually the $\varphi$ component of
the vector potential $A_{\mu}$, i.e., $\Psi=A_{\varphi}$ (cf., e.g., Ref.
bocquet ). Under the present assumptions, therefore, the Faraday tensor may be
given by

$\displaystyle F_{\mu\nu}$ $\displaystyle=$
$\displaystyle\varepsilon_{t}{}^{(t)}F_{\mu\nu}+\varepsilon_{p}{}^{(p)}F_{\mu\nu}$
(34) $\displaystyle=$
$\displaystyle\varepsilon_{t}\,^{(t)}F_{r\theta}\left(\begin{array}[]{cccc}0&0&0&0\\\
0&0&1&0\\\ 0&-1&0&0\\\ 0&0&0&0\end{array}\right)$
$\displaystyle+\varepsilon_{p}\left(\begin{array}[]{cccc}0&0&0&0\\\
0&0&0&\partial_{r}\Psi\\\ 0&0&0&\partial_{\theta}\Psi\\\
0&-\partial_{r}\Psi&-\partial_{\theta}\Psi&0\end{array}\right)\,,$

where ${}^{(t)}F_{r\theta}$ and $\Psi$ are functions of $r$ and $\theta$ only.
Thanks to the introduction of the poloidal flux function $\Psi$, we see that
the Faraday tenser (34) automatically satisfies one of the Maxwell equations
(4). The other Maxwell equation (5) is used to determine the current four-
vector $J^{\mu}$ in ideal magnetohydrodynamic theory. The explicit form of the
current four-vector $J^{\mu}$ is, from equations (5), (17) and (34), given by

$\displaystyle J^{t}=0\,,\quad J^{\varphi}=O\left(\varepsilon_{p}\right)\,,$
(35) $\displaystyle J^{r}=\varepsilon_{t}{1\over 4\pi
e^{\nu+\lambda}r^{2}\sin\theta}\partial_{\theta}\left(e^{\nu-\lambda}\sin\theta^{(t)}F_{r\theta}\right)$
$\displaystyle\quad\quad\quad\quad+O\left(\varepsilon_{t}^{3}\right)\,,$ (36)
$\displaystyle J^{\theta}=\varepsilon_{t}{-1\over 4\pi
e^{\nu+\lambda}r^{2}\sin\theta}\partial_{r}\left(e^{\nu-\lambda}\sin\theta^{(t)}F_{r\theta}\right)$
$\displaystyle\quad\quad\quad\quad+O\left(\varepsilon_{t}^{3}\right)\,.$ (37)

This current four-vector $J^{\mu}$ is used to write the momentum equation (14)
explicitly, which becomes, in the present situation,

$\displaystyle-\partial_{\mu}\ln\gamma+\frac{1}{\rho
h}\partial_{\mu}P-\frac{1}{\rho h}F_{\mu\nu}J^{\nu}=0\,.$ (38)

Because of equation (23), we see that the time component of equation (38) is
automatically satisfied. The toroidal component ($\varphi$ component) and the
poloidal components ($r$ and $\theta$ components) of equation (38),
respectively, lead

$\displaystyle\varepsilon_{t}\varepsilon_{p}\partial_{\theta}\left(e^{\nu-\lambda}\sin\theta^{(t)}F_{r\theta}\right)\,\partial_{r}\Psi$
$\displaystyle-\varepsilon_{t}\varepsilon_{p}\partial_{r}\left(e^{\nu-\lambda}\sin\theta^{(t)}F_{r\theta}\right)\,\partial_{\theta}\Psi$
$\displaystyle+O\left(\varepsilon_{t}^{3}\varepsilon_{p}\right)=0\,,$ (39)
$\displaystyle-\partial_{C}\ln\gamma+\frac{1}{\rho h}\partial_{C}P$
$\displaystyle+\varepsilon_{t}^{2}{{}^{(t)}F_{r\theta}\over
4\pi^{(0)}\rho^{(0)}he^{\nu+\lambda}r^{2}\sin\theta}\partial_{C}\left(e^{\nu-\lambda}\sin\theta^{(t)}F_{r\theta}\right)$
$\displaystyle+O\left(\varepsilon_{p}^{2}\right)=0\,,$ (40)

where the index $C$ is used to denote poloidal indices and runs from 1($r$) to
2($\theta$). Note that equations (39) and (40) are the
$\varepsilon_{t}\varepsilon_{p}$-order accurate expression of the momentum
equation (38). The integrability conditions for equations (39) and (40)
require that

$\displaystyle\Psi$ $\displaystyle=$
$\displaystyle\Psi\left[e^{\nu-\lambda}\sin\theta^{(t)}F_{r\theta}\right]\,,$
(41) $\displaystyle e^{\nu-\lambda}\sin\theta^{(t)}F_{r\theta}$
$\displaystyle=$ $\displaystyle
K\left[{}^{(0)}\rho^{(0)}he^{2\nu}r^{2}\sin^{2}\theta\right]\,,$ (42)

where $K$ is an arbitrary function of
${}^{(0)}\rho^{(0)}he^{2\nu}r^{2}\sin^{2}\theta$. Equations (41) and (42) are
only the conditions that the magnetic fields have to satisfy. Therefore, the
magnetic field distribution can be specified by the two arbitrary functions
$\Psi=\Psi\left[w\right]$ and $K=K\left[w\right]$ with $w$ being
$w=^{(0)}\rho^{(0)}he^{2\nu}r^{2}\sin^{2}\theta$ as far as the corresponding
magnetic field satisfies the physically reasonable boundary conditions.

###  III.3 Deformation of the star and spacetime due to the magnetic field:
the $\varepsilon_{t}^{2}\varepsilon_{p}^{0}$ and
$\varepsilon_{t}^{1}\varepsilon_{p}^{1}$ order equations

In this subsection, we derive the master equations for the deformation of the
star and spacetime due to the magnetic field discussed in the previous
subsection. Since the fluid four-velocity is proportional to the time Killing
vector $t^{\mu}$, as given in equation (22), the baryon mass conservation
equation (3) and the energy equation (13) are automatically satisfied. We do
not therefore need to consider them further. The only fluid equation that we
have to consider is equation (40). Because of equations (9) and (42), the
momentum equation (40) has the first integral, given by

$\displaystyle-\ln\gamma+\int\frac{dP}{\rho
h}+\varepsilon_{t}^{2}\frac{1}{4\pi}\int\frac{K(w)}{w}\frac{dK}{dw}dw$
$\displaystyle\quad=C+O\left(\varepsilon_{p}^{2}\right),$ (43)

where $C$ is a constant of integration. This equation is sometimes called the
equation of hydrostatic equilibrium. From equation (43), it is seen that the
poloidal magnetic field does not affect the fluid distribution within the
$\varepsilon_{t}\varepsilon_{p}$ order accuracy. As shown later, the poloidal
magnetic field does affect the spacetime geometry, which is determined by the
Einstein equations.

The functions $\gamma$ and $\displaystyle\int\frac{dP}{\rho h}$ and the
constant of integration $C$ may be expanded as follows:

$\displaystyle\gamma$ $\displaystyle=$ $\displaystyle
e^{-\nu}+\varepsilon_{t}^{2}{}^{(2)}\gamma(r,\theta)+O\left(\varepsilon_{p}^{2}\right)\,,$
(44) $\displaystyle\int\frac{dP}{\rho h}$ $\displaystyle=$
$\displaystyle\int\frac{d^{(0)}P}{{}^{(0)}\rho{}^{(0)}h}+\varepsilon_{t}^{2}\frac{{}^{(2)}P(r,\theta)}{{}^{(0)}\rho{}^{(0)}h}+O\left(\varepsilon_{p}^{2}\right)\,,$
(45) $\displaystyle C$ $\displaystyle=$
${}^{(0)}C+\varepsilon_{t}^{2}{}^{(2)}C+O\left(\varepsilon_{p}^{2}\right)\,,$
(46)

where ${}^{(2)}\gamma$, ${}^{(2)}P$, and ${}^{(2)}C$ are perturbations of
order $\varepsilon_{t}^{2}$. Substituting equations (44)–(46) into equation
(43), we obtain

$\displaystyle\int\frac{d^{(0)}P}{{}^{(0)}\rho{}^{(0)}h}+\nu=^{(0)}C\,,$ (47)
$\displaystyle\frac{{}^{(2)}P(r,\theta)}{{}^{(0)}\rho{}^{(0)}h}-e^{\nu}{}^{(2)}\gamma(r,\theta)$
$\displaystyle\quad+\frac{1}{4\pi}\int\frac{K(w)}{w}\frac{dK}{dw}dw=^{(2)}C\,.$
(48)

Note that the derivative of equation (47) with respect to $r$ yields equation
(20).

The stress-energy tensor given in equation (7) is divided into the fluid part
${}^{(F)}T^{\mu}_{\nu}$ and the electromagnetic part ${}^{(EM)}T^{\mu}_{\nu}$,
defined, respectively, by

${}^{(F)}{T^{\mu}}_{\nu}$ $\displaystyle=$ $\displaystyle\rho
hu^{\mu}u_{\nu}+P{\delta^{\mu}}_{\nu}\,,$ (49) ${}^{(EM)}{T^{\mu}}_{\nu}$
$\displaystyle=$ $\displaystyle{1\over
4\pi}\left(F^{\mu\alpha}F_{\nu\alpha}-{1\over
4}{\delta^{\mu}}_{\nu}\,F^{\alpha\beta}F_{\alpha\beta}\right)\,.$ (50)

For the Faraday tensor given in equation (34), the electromagnetic part of the
stress-energy tensor is given by

${}^{(EM)}{T^{\mu}}_{\nu}$
$\displaystyle=\varepsilon_{t}^{2}\,{e^{-2\lambda}\over 8\pi
r^{2}}\left({}^{(t)}F_{r\theta}\right)^{2}\left(\begin{array}[]{cccc}-1&0&0&0\\\
0&1&0&0\\\ 0&0&1&0\\\ 0&0&0&-1\end{array}\right)$ (55)
$\displaystyle+\varepsilon_{t}\varepsilon_{p}{e^{-2\lambda}\,{}^{(t)}F_{r\theta}\over
4\pi r^{2}}\left(\begin{array}[]{cccc}0&0&0&0\\\
0&0&0&-\partial_{\theta}\Psi\\\ 0&0&0&\ \ \partial_{r}\Psi\\\
0&\displaystyle-r^{-2}e^{2\lambda}{\partial_{\theta}\Psi\over\sin^{2}\theta}&\displaystyle{\partial_{r}\Psi\over\sin^{2}\theta}&0\end{array}\right)$
(60) $\displaystyle+O\left(\varepsilon_{p}^{2}\right)\,.$ (61)

From the expression (61), we may confirm that whereas the circularity
conditions for the spacetime, given by

$\displaystyle
t^{\alpha}{T_{\alpha}}^{[\beta}t^{\gamma}\varphi^{\delta]}=0\,,\quad\varphi^{\alpha}{T_{\alpha}}^{[\beta}t^{\gamma}\varphi^{\delta]}=0\,,$
(62)

(cf., e.g., Ref. wald ) are fulfilled up to the $\varepsilon_{t}^{2}$ order,
they are violated at the $\varepsilon_{t}\varepsilon_{p}$ order. This implies
that up to the $\varepsilon_{t}^{2}$ order, the spacetime around the
magnetized stars considered may be described by a simpler form of the metric
used for stationary and axisymmetric rotating stars without magnetic fields
(cf., e.g., Ref. thorne ). Note that the solutions within accuracy up to
$\varepsilon_{t}^{2}$ order correspond to a perturbation version of the star
containing purely toroidal magnetic fields constructed by Kiuchi and Yoshida
kiuchi .

To calculate particular models of the magnetized star, we need to specify
completely the arbitrary functions $K$ and $\Psi$, given in equations (41) and
(42). In the present study, the two arbitrary functions are assumed to be
given by

$\displaystyle K=b\,w=b\,^{(0)}\rho^{(0)}he^{2\nu}r^{2}\sin^{2}\theta\,,$ (63)
$\displaystyle\Psi=a\,w=a\,^{(0)}\rho^{(0)}he^{2\nu}r^{2}\sin^{2}\theta\,,$
(64)

where $b$ and $a$ are constants. Note that this choice of the function $K$ is
the same as that of the $k=1$ case considered in Ref. kiuchi . For these
arbitrary functions, the regularity of the magnetic field on the symmetry axis
is satisfied. In this study, we assume that there is no magnetic field outside
the star and that there is no surface current. Thus, the magnetic field has to
vanish on the surface of the star. For the arbitrary functions given in
equations (63) and (64), the magnetic field $B^{\mu}$ becomes

$\displaystyle B^{\mu}$ $\displaystyle=$
$\displaystyle\varepsilon_{t}b\left(0,0,0,e^{\nu}{}^{(0)}\rho^{(0)}h\right)$
(65) $\displaystyle+$
$\displaystyle\varepsilon_{p}a\,e^{2\nu-\lambda}\left(0,2\,{}^{(0)}\rho^{(0)}h\cos\theta,\right.$
$\displaystyle-$ $\displaystyle\left.{\sin\theta\over r}\left\\{r{d\over
dr}\left({}^{(0)}\rho^{(0)}h\right)+2(^{(0)}\rho^{(0)}h)\left(r{d\nu\over
dr}+1\right)\right\\},0\right)$ $\displaystyle+$ $\displaystyle
O\left(\varepsilon_{t}^{3}\right)\,.$

This magnetic field vanishes if the two conditions ${}^{(0)}\rho^{(0)}h=0$ and
$\displaystyle{d\over dr}\left({}^{(0)}\rho^{(0)}h\right)=0$ are fulfilled. On
the surface of the star, therefore, we require the conditions, given by

${}^{(0)}\rho^{(0)}h=0\,,\quad{d\over
dr}\left({}^{(0)}\rho^{(0)}h\right)=0\,,$ (66)

which are, as a matter of fact, conditions for the equation of state.

The explicit expression for non-zero components of ${}^{(EM)}{T^{\mu}}_{\nu}$
is summarized as follows:

${}^{(EM)}{T^{t}}_{t}$ $\displaystyle=$
${}^{(EM)}{T^{\varphi}}_{\varphi}=-^{(EM)}{T^{r}}_{r}=-^{(EM)}{T^{\theta}}_{\theta}$
(67)
$\displaystyle=-\varepsilon_{t}^{2}\frac{b^{2}}{8\pi}r^{2}e^{2\nu}\left({}^{(0)}\rho^{(0)}h\right)^{2}\sin^{2}\theta$
$\displaystyle+O\left(\varepsilon_{p}^{2}\right)\,,$
${}^{(EM)}{T^{r}}_{\varphi}$ $\displaystyle=$ $\displaystyle
r^{2}\sin^{2}\theta\,e^{-2\lambda}{\,}^{(EM)}{T^{\varphi}}_{r}$ (68)
$\displaystyle=$
$\displaystyle-\varepsilon_{t}\varepsilon_{p}\frac{ab}{2\pi}r^{2}e^{3\nu-\lambda}\left({}^{(0)}\rho^{(0)}h\right)^{2}\sin^{2}\theta\cos\theta$
$\displaystyle+O\left(\varepsilon_{t}^{3}\right)\,,$
${}^{(EM)}{T^{\theta}}_{\varphi}$ $\displaystyle=$
$\displaystyle\sin^{2}\theta{\,}^{(EM)}{T^{\varphi}}_{\theta}$ (69)
$\displaystyle=$
$\displaystyle\varepsilon_{t}\varepsilon_{p}\frac{ab}{4\pi}re^{3\nu-\lambda}\left({}^{(0)}\rho^{(0)}h\right)$
$\displaystyle\times$ $\displaystyle\left\\{r{d\over
dr}\left({}^{(0)}\rho^{(0)}h\right)+2(^{(0)}\rho^{(0)}h)\left(r{d\nu\over
dr}+1\right)\right\\}$ $\displaystyle\times$
$\displaystyle\sin^{3}\theta+O\left(\varepsilon_{t}^{3}\right)\,.$

From this stress-energy tensor for the electromagnetic field, we may expect
that the line element of the spacetime around the magnetized star is given by

$\displaystyle ds^{2}$ $\displaystyle=$
$\displaystyle-e^{2\nu}\left[1+2\epsilon_{t}^{2}\left\\{h_{0}(r)+h_{2}(r)P_{2}(\cos\theta)\right\\}\right]dt^{2}$
(70) $\displaystyle+e^{2\lambda}\left[1+{2\epsilon_{t}^{2}e^{2\lambda}\over
r}\left\\{m_{0}(r)+m_{2}(r)P_{2}(\cos\theta)\right\\}\right]dr^{2}$
$\displaystyle+r^{2}\left[1+2\epsilon_{t}^{2}\left\\{v_{2}(r)-h_{2}(r)\right\\}P_{2}(\cos\theta)\right]$
$\displaystyle\quad\quad\times\left(d\theta^{2}+\sin^{2}\theta
d\varphi^{2}\right)$
$\displaystyle-2\epsilon_{t}\epsilon_{p}d_{2}(r)\sin\theta{\partial\over\partial\theta}P_{2}(\cos\theta)\,drd\varphi$
$\displaystyle+O\left(\varepsilon_{p}^{2}\right)\,,$

where $P_{l}$ is the Legendre polynomial of degree $l$. The normalization
factor of the fluid four-velocity $\gamma$ defined by equation (22) (cf.,
also, equation (44)) is then given by

$\displaystyle\gamma$ $\displaystyle=$ $\displaystyle
e^{-\nu}+\varepsilon_{t}^{2}\,{}^{(2)}\gamma(r,\theta)+O\left(\varepsilon_{p}^{2}\right)$
(72) $\displaystyle=$ $\displaystyle
e^{-\nu}\left[1-\epsilon_{t}^{2}\left\\{h_{0}(r)+h_{2}(r)P_{2}(\cos\theta)\right\\}\right]$
$\displaystyle+O\left(\epsilon_{p}^{2}\right)\,.$

The $\epsilon_{t}^{2}$-order pressure perturbation given in equation (48) is
written by

$\displaystyle\frac{{}^{(2)}P(r,\theta)}{{}^{(0)}\rho{}^{(0)}h}$
$\displaystyle=$ $\displaystyle\frac{\delta
P_{0}(r)}{{}^{(0)}\rho{}^{(0)}h}+\frac{\delta
P_{2}(r)}{{}^{(0)}\rho{}^{(0)}h}P_{2}(\cos\theta)$ $\displaystyle=$
${}^{(2)}C+e^{\nu(r)}{}^{(2)}\gamma(r,\theta)-\frac{1}{4\pi}\int\frac{K(w)}{w}\frac{dK}{dw}dw$
$\displaystyle=$
${}^{(2)}C-h_{0}(r)-\frac{1}{6\pi}b^{2}{\,}^{(0)}\rho^{(0)}h\,e^{2\nu}r^{2}$
$\displaystyle-$
$\displaystyle\left\\{h_{2}(r)-\frac{1}{6\pi}b^{2}{\,}^{(0)}\rho^{(0)}h\,e^{2\nu}r^{2}\right\\}P_{2}(\cos\theta)\,,$

where $\delta P_{0}$ and $\delta P_{2}$ are coefficients in the Legendre
expansion of ${}^{(2)}P(r,\theta)$. Thus, we have

$\displaystyle\frac{\delta P_{0}(r)}{{}^{(0)}\rho{}^{(0)}h}$ $\displaystyle=$
$\displaystyle-
h_{0}(r)-\frac{1}{6\pi}b^{2}{\,}^{(0)}\rho^{(0)}h\,e^{2\nu}r^{2}$ (74)
$\displaystyle+^{(2)}C\,,$ $\displaystyle\frac{\delta
P_{2}(r)}{{}^{(0)}\rho{}^{(0)}h}$ $\displaystyle=$ $\displaystyle-
h_{2}(r)+\frac{1}{6\pi}b^{2}{\,}^{(0)}\rho^{(0)}h\,e^{2\nu}r^{2}\,.$ (75)

From the relations obtained so far and the perturbed Einstein equations,
following standard procedures (cf., e.g., Refs. Ioka2004 ; yoshidaa ), we
obtain the master equations for the deformation of the magnetized star with
mixed poloidal-toroidal fields as follows.

The $\varepsilon_{t}^{2}$ order equations:

$\displaystyle{dm_{0}\over dr}$ $\displaystyle=$ $\displaystyle
r^{2}\left({}^{(0)}\rho{}^{(0)}h\right)$ (76)
$\displaystyle\times\left\\{4\pi\left({d(^{(0)}\rho+{\,}^{(0)}\rho{\,}^{(0)}\epsilon)\over
d{\,}^{(0)}P}\right)\frac{\delta P_{0}(r)}{{}^{(0)}\rho{}^{(0)}h}\right.$
$\displaystyle\left.\quad\quad+\frac{1}{3}\,b^{2}e^{2\nu}r^{2}\left({}^{(0)}\rho{}^{(0)}h\right)\right\\}\,,$
$\displaystyle{d\over dr}\left(\frac{\delta
P_{0}(r)}{{}^{(0)}\rho{}^{(0)}h}\right)$ $\displaystyle=$
$\displaystyle-\frac{e^{4\lambda}}{r^{2}}\left(1+8\pi\,r^{2}{\,}^{(0)}P\right)m_{0}$
(77) $\displaystyle-4\pi
e^{2\lambda}r\left({}^{(0)}\rho{}^{(0)}h\right)\left(\frac{\delta
P_{0}(r)}{{}^{(0)}\rho{}^{(0)}h}\right)$
$\displaystyle-b^{2}e^{2\nu}\left\\{\frac{r^{2}}{6\pi}{d\over
dr}\left({}^{(0)}\rho{}^{(0)}h\right)\right.$
$\displaystyle\left.+\frac{r^{3}}{3}e^{2\lambda}\left({}^{(0)}\rho{}^{(0)}h\right)\left({}^{(0)}\rho{}^{(0)}h+4{\,}^{(0)}P\right)\right.$
$\displaystyle\left.+\frac{r}{6\pi}\left(1+e^{2\lambda}\right)\left({}^{(0)}\rho{}^{(0)}h\right)\right\\}\,,$
$\displaystyle h_{0}$ $\displaystyle=$ $\displaystyle-\frac{\delta
P_{0}(r)}{{}^{(0)}\rho{}^{(0)}h}-\frac{1}{6\pi}\,{b^{2}e^{2\nu}r^{2}}\left({}^{(0)}\rho{}^{(0)}h\right)$
(78) $\displaystyle+^{(2)}C\,,$ $\displaystyle{dh_{2}\over dr}$
$\displaystyle=$ $\displaystyle-\frac{2e^{2\lambda}}{\displaystyle
r^{2}{d\nu\over dr}}\,v_{2}$ $\displaystyle-\left[2{d\nu\over
dr}-\frac{e^{2\lambda}}{\displaystyle r^{3}{d\nu\over dr}}\left\\{4\pi
r^{3}\left({}^{(0)}\rho^{(0)}h\right)-2M\right\\}\right]h_{2}$
$\displaystyle-\frac{b^{2}\,e^{2\nu}}{\displaystyle 3{d\nu\over
dr}}\,r^{2}\left(2\left(r{d\nu\over
dr}\right)^{2}+e^{2\lambda}\right)\left({}^{(0)}\rho^{(0)}h\right)^{2}\,,$
$\displaystyle{dv_{2}\over dr}$ $\displaystyle=$ $\displaystyle-2{d\nu\over
dr}\,h_{2}-\frac{2}{3}\,b^{2}\,e^{2\nu}r^{3}\left(1+r{d\nu\over
dr}\right)\left({}^{(0)}\rho^{(0)}h\right)^{2}\,,$ $\displaystyle\frac{\delta
P_{2}(r)}{{}^{(0)}\rho{}^{(0)}h}$ $\displaystyle=$ $\displaystyle-
h_{2}+\frac{1}{6\pi}\,b^{2}\,e^{2\nu}r^{2}\left({}^{(0)}\rho^{(0)}h\right)\,,$
(81) $\displaystyle m_{2}$ $\displaystyle=$
$\displaystyle-e^{-2\lambda}\,r\,h_{2}-\frac{2}{3}\,b^{2}\,e^{2\left(\nu-\lambda\right)}r^{5}\left({}^{(0)}\rho^{(0)}h\right)^{2}\,.$
(82)

The $\varepsilon_{t}\varepsilon_{p}$ order equations:

$\displaystyle
d_{2}=-\frac{2}{3}\,a\,b\,e^{\lambda+3\nu}r^{4}\left({}^{(0)}\rho^{(0)}h\right)^{2}\,.$
(83)

Regular solutions of the master equations (76), (77), (LABEL:dh2_eq), and
(LABEL:dv2_eq) near the center of the star may be written as

$\displaystyle m_{0}$ $\displaystyle=$ $\displaystyle
r^{3}\left(m_{00}+m_{02}r^{2}\cdots\right)\,,$ (84) $\displaystyle\frac{\delta
P_{0}(r)}{{}^{(0)}\rho{}^{(0)}h}$ $\displaystyle=$ $\displaystyle
h_{00}+h_{02}r^{2}+\cdots\,,$ (85) $\displaystyle h_{2}$ $\displaystyle=$
$\displaystyle r^{2}\left(h_{20}+h_{22}r^{2}+\cdots\right)\,,$ (86)
$\displaystyle v_{2}$ $\displaystyle=$ $\displaystyle
r^{4}\left(v_{20}+v_{22}r^{2}+\cdots\right)\,,\ $ (87)

where $m_{00}$, $m_{02}$, $h_{00}$, $h_{02}$, $h_{20}$, $h_{22}$, $v_{20}$,
and $v_{22}$ are expansion coefficients. In this expansion solution, we may
obtain a unique regular solution if values of $h_{00}$ and $h_{20}$ are given.
In the present situation, a value of $h_{20}$ is determined by the boundary
condition at infinity, which will be argued in the next paragraph. In order to
determine a value of $h_{00}$, on the other hand, an extra condition is
required. To determine the extra condition, in this study, we consider the
following two distinct situations: (1) the baryon rest-mass density at the
center of the star keeps constant when magnetic fields are imposed. (2) the
total baryon rest-mass of the star keeps constant when magnetic fields are
imposed. The situation (1) is realized by the condition of $h_{00}=0$ (cf.,
equation (100)). As for the situation (2), we will argue in the next
subsection.

The solutions of the $\varepsilon_{t}^{2}$-order vacuum Einstein equations
suitable for the exterior spacetime of the isolated star are analytically
given by

$\displaystyle m_{0}=m_{0}(R)={\rm const.}\,,\quad h_{0}=-{m_{0}(R)\over
r-2M}\,,$ (88) $\displaystyle h_{2}=DQ_{2}^{2}(y)\,,\quad v_{2}$
$\displaystyle=$ $\displaystyle-{2D\over\sqrt{y^{2}-1}}Q_{2}^{1}(y)\,,$ (89)

where $\displaystyle y={r\over M}-1$ and, $Q_{l}^{m}$ and $D$ are the
associated Legendre function of the second kind and a constant, respectively
(cf., e.g., Refs. Ioka2004 ; yoshidaa ; thorne ). At the surface of the star,
the external solutions, given by equations (88) and (89), are matched to the
internal solutions obtained by integrating equations (76), (77),
(LABEL:dv2_eq), and (LABEL:dh2_eq) from the center of the star outwards with
the boundary conditions given in equations (84)–(87). The physically
acceptable solutions for the whole spacetime may then be obtained.

###  III.4 Global quantities characterizing the magnetized star

In order to investigate properties of equilibrium solutions of the magnetized
star, global quantities are used in the following discussion. For equilibrium
states of the magnetized star, the total baryon rest-mass $\widetilde{M}^{*}$,
the internal thermal energy $\widetilde{E}_{\rm int}$, and the electromagnetic
energy $\widetilde{E}_{\rm EM}$ may be defined as

$\displaystyle\widetilde{M}^{*}=\int\rho\gamma\sqrt{-g}\,d^{3}x\,,$ (90)
$\displaystyle\widetilde{E}_{\rm
int}=\int\rho\varepsilon\gamma\sqrt{-g}\,d^{3}x\,,$ (91)
$\displaystyle\widetilde{E}_{\rm EM}={1\over 8\pi}\int
B^{\mu}B_{\mu}\gamma\sqrt{-g}\,d^{3}x\ \,,$ (92)

(cf., e.g., Ref. kiuchi ).

For the unperturbed spherical star, the gravitational mass $M$, the total
baryon rest-mass $M^{*}$, and the internal thermal energy $E_{\rm int}$ may be
given by

$\displaystyle M=M_{r}(R)\,,$ (93) $\displaystyle
M^{*}=4\pi\int_{0}^{R}{\,}^{(0)}\rho e^{\lambda}r^{2}dr\,,$ (94)
$\displaystyle E_{\rm int}=4\pi\int_{0}^{R}{\,}^{(0)}\rho{\,}^{(0)}\varepsilon
e^{\lambda}r^{2}dr\,,$ (95)

where $R$ denotes the circumferential radius of the star determined by the
condition ${\,}^{(0)}P(R)=0$. The gravitational potential energy $W$ for the
unperturbed star may be defined by

$\displaystyle|W|=M^{*}+E_{\rm int}-M\,.$ (96)

The $O(\epsilon_{t}^{2})$ magnetic effects on the gravitational mass
$\widetilde{M}$, the total baryon rest-mass $\widetilde{M}^{*}$, and the
internal thermal energy $\widetilde{E}_{\rm int}$ may, respectively, be given
by

$\displaystyle\Delta M=\epsilon_{t}^{2}\,m_{0}(R)\,,$ (97)
$\displaystyle\Delta M^{*}=4\pi\epsilon_{t}^{2}\int_{0}^{R}{\,}^{(0)}\rho
e^{\lambda}r^{2}$ $\displaystyle\quad\quad\times\left({d\ln{\,}^{(0)}\rho\over
d{\,}^{(0)}P}\delta P_{0}+{e^{2\lambda}m_{0}\over r}\right)dr\,,$ (98)
$\displaystyle\Delta E_{\rm
int}=4\pi\epsilon_{t}^{2}\int_{0}^{R}{\,}^{(0)}\rho{\,}^{(0)}\varepsilon
e^{\lambda}r^{2}$
$\displaystyle\quad\quad\times\left({d\ln\left({\,}^{(0)}\rho{\,}^{(0)}\varepsilon\right)\over
d{\,}^{(0)}P}\delta P_{0}+{e^{2\lambda}m_{0}\over r}\right)dr\,.$ (99)

As mentioned in the previous subsection, we study the sequences of equilibrium
states of the magnetized star characterized by the fixed total baryon rest-
mass. Thus, the condition of $\Delta M^{*}=0$ is used to determine values of
$h_{00}$ in equation (85), which are related to $O(\epsilon_{t}^{2})$ changes
in the central density of the star $\Delta\rho_{c}$, given by

$\displaystyle\Delta\rho_{c}=\epsilon_{t}^{2}\,\left.{d{}^{(0)}\rho\over
d{}^{(0)}P}\right|_{r=0}{}^{(0)}\rho(0){}^{(0)}h(0)\,h_{00}\,.$ (100)

The electromagnetic energy $E_{\rm EM}$ is decomposed as

$\displaystyle E_{\rm EM}=E_{\rm EM}^{(p)}+E_{\rm EM}^{(t)}\,,$ (101)

where $E^{(p)}_{\rm EM}$ and $E_{\rm EM}^{(t)}$ are the poloidal and toroidal
magnetic-field energies, respectively, given by

$\displaystyle E_{\rm EM}^{(p)}=\epsilon_{p}^{2}\,{1\over
3}\,a^{2}\int_{0}^{R}r^{2}e^{4\nu-\lambda}$
$\displaystyle\times\left[\left\\{r{d\over
dr}\left({\,}^{(0)}\rho{\,}^{(0)}h\right)+2\left({\,}^{(0)}\rho{\,}^{(0)}h\right)\left(r{d\nu\over
dr}+1\right)\right\\}^{2}\right.$
$\displaystyle\quad\quad\quad\quad\left.+2e^{2\lambda}\left({\,}^{(0)}\rho{\,}^{(0)}h\right)^{2}\right]\,dr+O\left(\varepsilon_{p}^{4}\right)\,,$
(102) $\displaystyle E_{\rm EM}^{(t)}=\epsilon_{t}^{2}\,{1\over
3}\,b^{2}\int_{0}^{R}e^{\lambda+2\nu}r^{4}\left({\,}^{(0)}\rho{\,}^{(0)}h\right)^{2}\,dr$
$\displaystyle\quad\quad\quad\quad+O\left(\varepsilon_{t}^{4}\right)\,.$ (103)

Multipole moments of the star may characterize the equilibrium star globally.
The constant of integration $D$ appearing in the exterior solution given in
equation (89) is related to the mass quadrupole moment $\Delta Q$, given by

$\Delta Q=\epsilon_{t}^{2}\,{8\over 5}\,M^{3}D\,,$ (104)

(cf., e.g., Refs. ioka ; thorne ).

Deformation of the surface of the star due to the magnetic stress also
characterizes equilibrium solutions of the magnetized star. The surface of the
star is defined by the algebraic equation
$P(r)={\,}^{(0)}P(r)+\epsilon_{t}^{2}\delta P_{0}(r)+\epsilon_{t}^{2}\delta
P_{2}(r)P_{2}(\theta)+O\left(\varepsilon_{p}^{2}\right)=0$. Thus, the
$O(\epsilon_{t}^{2})$ radial displacement of the fluid elements on the stellar
surface, $\Delta r$, is given by

$\displaystyle\Delta r$ $\displaystyle=$ $\displaystyle\left(\Delta
r\right)_{0}+\left(\Delta r\right)_{2}P_{2}$ (105) $\displaystyle=$
$\displaystyle-\epsilon_{t}^{2}\,\left(\delta P_{0}(R)+\delta
P_{2}(R)P_{2}\right){dr\over d{\,}^{(0)}P}(R)\,.$

$(\Delta r)_{0}$ may be interpreted as an average change in the radius of the
star induced by the magnetic effects. The degree of the quadrupole surface
deformation due to the magnetic stress is well described by the ellipticity
$e^{*}$, given by

$\displaystyle e^{*}=-\epsilon_{t}^{2}\,{3\over 2}\left[{\left(\Delta
r\right)_{2}\over R}+v_{2}(R)-h_{2}(R)\right]\,,$ (106)

where $e^{*}$ is defined as a relative difference between the equatorial and
polar circumference radii of the star (cf., e.g., Refs. Ioka2004 ; thorne ).
Thus, $e^{*}<0$ ($e^{*}>0$) means that the star is prolate (oblate).

An important quantity of magnetized objects is the total magnetic helicity
${\cal H}$, which is a conserved quantity in ideal magnetohydrodynamics
defined by

$\displaystyle{\cal H}=\int H^{0}\sqrt{-g}\,d^{3}x\,,$ (107)

where $H^{0}$ is the time component of the magnetic helicity four-current
$H^{\mu}$, defined by

$\displaystyle H^{\mu}=-{1\over
2}\epsilon^{\mu\nu\alpha\beta}A_{\nu}F_{\alpha\beta}\,.$ (108)

We may confirm that the magnetic helicity ${\cal H}$ is a conserved quantity
in ideal magnetohydrodynamics as follows. Taking the the divergence of
equation (108) yields

$\displaystyle\nabla_{\mu}H^{\mu}=-{1\over 2}*F^{\mu\nu}F_{\mu\nu}\,,$ (109)

where $*F_{\mu\nu}$ is the Hodge dual of the Faraday tensor $F_{\mu\nu}$. We
then have $\nabla_{\mu}H^{\mu}=0$ if the perfect conductivity condition
$F_{\mu\nu}u^{\nu}=0$ is satisfied. For the present model, the total magnetic
helicity is explicitly written as

$\displaystyle{\cal H}$ $\displaystyle=$ $\displaystyle{16\pi\over
3}\,\epsilon_{t}\epsilon_{p}ab\int_{0}^{R}e^{\lambda+3\nu}r^{4}\left({\,}^{(0)}\rho{\,}^{(0)}h\right)^{2}\,dr\,,$
(110)

where we use the vector potential $A_{\mu}$, given by

$\displaystyle
A_{\mu}=\left(0,\epsilon_{t}b\,e^{\lambda+\nu}r^{2}{\,}^{(0)}\rho{\,}^{(0)}h\cos\theta,0,\epsilon_{p}\psi\right)\,.$
(111)

The dimensionless magnetic helicity, defined by ${\cal H}_{M}={\cal H}/M^{2}$,
is used when its numerical value is shown. The magnetic helicity is a measure
of the net twist of a magnetic-field configuration. Thus, the magnetic
helicity vanishes for purely poloidal fields and for purely toroidal fields.
Some experiments and numerical computations show an interesting fact that the
total magnetic helicity is likely to be conserved even when the resistivity
cannot be ignored Braithwaite2004 ; Hsu2002 . If this fact is retained for the
neutron star formation process, the total magnetic helicity has to be
approximately conserved during the formation process.

##  IV Numerical Results

In this section, we give some numerical examples of the star including mixed
poloidal-toroidal magnetic fields to examine the magnetic effects on the
stellar structure. For the one-parameter equations of state (9), we use the
polytrope and the gamma-law equation of state, respectively, given by

$\displaystyle P$ $\displaystyle=$ $\displaystyle\kappa\,\rho^{1+{1\over
n}}\,,$ (112) $\displaystyle\varepsilon$ $\displaystyle=$
$\displaystyle{1\over\Gamma-1}{P\over\rho}\,,$ (113)

where $\kappa$, $n$, $\Gamma$ are constants. The constant $n$ is called the
polytrope index. The constant $\Gamma$ stands for the adiabatic index, defined
by

$\displaystyle\Gamma=\left({\partial\ln P\over\partial\ln\rho}\right)_{\rm
ad}\,,$ (114)

where “ad” means that the derivative is evaluated along an adiabatic process
curve. A general relativistic version of the Schwarzschild discriminant $A$
for the background star may be defined by

$\displaystyle
A={1\over{}^{(0)}\rho{}^{(0)}h}{d\left({}^{(0)}\rho+{}^{(0)}\rho{}^{(0)}\varepsilon\right)\over
dr}-{1\over\Gamma\,{}^{(0)}P}{d{}^{(0)}P\over dr}\,.$ (115)

Following Ipser and Lindblom ipser , we employ a definition of the general
relativistic Brunt-Väisälä frequency $N$, given by

$\displaystyle N^{2}=-gA\,,$ (116)

where $g$ is an effective gravitational acceleration, defined by

$\displaystyle
g=-e^{2\nu-2\lambda}{1\over{}^{(0)}\rho{}^{(0)}h}{d{}^{(0)}P\over dr}\,.$
(117)

The sign of the discriminant $A$ therefore determines whether or not a stellar
medium is stable against convection. A stellar medium, where
$d{}^{(0)}\rho/dr\leq 0$ and $d{}^{(0)}P/dr\leq 0$ are fulfilled, is
convectively unstable if $A>0$. If we assume that $\Gamma$ is given by

$\displaystyle{1\over\Gamma}={n\over n+1}+\tilde{\delta}\,,$ (118)

where $\tilde{\delta}$ is a constant, then the Schwarzschild discriminant can
be written by

$\displaystyle A=-{\tilde{\delta}\over{}^{(0)}h}\,{d\ln{}^{(0)}P\over dr}\,.$
(119)

For a star characterized by the equations of state given in equations (112)
and (113), thus, a stable stratification of the fluid density is realized if
the following condition is fulfilled,

$\displaystyle\tilde{\delta}<0\,,\quad{\rm or}\quad\Gamma>{n+1\over n}\,.$
(120)

For the isentropic case, we have $\Gamma=\left(n+1\right)/n$ (or
$\tilde{\delta}=0$) and the stellar medium is marginally stable against
convection.

As argued in the last section, the conditions (66) have to be satisfied at the
surface of star in order for the magnetic field to vanish there. These
conditions are automatically fulfilled if $n>1$. In this study, we consider
the case of $n=1.05$ only as an example of equations of state for “neutron
star-like” models. Note that the $n=1$ polytrope is frequently used to study
neutron stars. For the models with $n=1$, however, the conditions (66) cannot
be satisfied at the surface of the star as long as equation (64) is assumed.

As for the adiabatic index, we choose two cases, $\Gamma=(n+1)/n\approx
1.95238$ and $\Gamma=2.05$. The former is a non-stratified case and the latter
a stably stratified case (cf. equation (120)). As discussed in, e.g., Refs.
yoshidaa ; reisenegger2008 , magnetic buoyancy instability can be weakened in
a stably stratified stellar medium. As shown numerically by Mitchell et al.
mitchell , the stable stratification will play a crucial role in order for
large-scale magnetic fields inside the star to survive for a sufficiently long
time (cf., also, Ref. reisenegger2008 ). In fact, the neutron star core is
expected to be strongly stably stratified due to a smooth composition gradient
reisenegger . Note that equations of state similar to those of the present
study are employed in Ref. yoshidaa .

Figure 1: Gravitational mass $M$ and baryon mass $M^{*}$, given as functions
of the central density ${}^{(0)}\rho(0)$. All the quantities are given in
units of $\kappa=1$.

For the background stars considered in this study, we plot in Figure 1 the
gravitational mass $M$ and the baryon rest-mass $M^{*}$ as functions of the
central density ${}^{(0)}\rho(0)$. Throughout this paper, units of $\kappa=1$
are used when numerical results are shown. The maximum gravitational mass is
$M\approx 0.17028$ and $0.17307$ for the stars with $\Gamma=(n+1)/n$ and
$2.05$, respectively. When the magnetized stars associated with the condition
of $\Delta\rho_{c}=0$ are considered, the effects of the magnetic field on the
stellar structure are examined for the background stars given in Figure 1.
When the the magnetized stars associated with the condition of $\Delta
M^{*}=0$ are considered, we focus on the particular background stars with
$M/R=0.1$ and $0.2$. The compactness of $M/R=0.2$ is typical for neutron
stars. In Table 1, some global and physical quantities for these background
stars are tabulated. Note that all the background stars given in Table 1 are
dynamically stable against radial collapse because values of their
gravitational mass are less than those of the maximum one.

Table 1:  Global and physical quantities for the background stars in units of $\kappa=1$.  $\Gamma$ | $M/R$ | ${}^{(0)}\rho(0)$ | $M$ | $M^{*}$ | $E_{\rm{int}}/\left|W\right|$
---|---|---|---|---|---
1.95238 | 0.100000 | 0.0625033 | 0.116229 | 0.122150 | 0.429344
| 0.200000 | 0.252634 | 0.170052 | 0.185529 | 0.601263
2.05000 | 0.100000 | 0.0620501 | 0.116627 | 0.122976 | 0.389322
| 0.200000 | 0.244169 | 0.172482 | 0.190336 | 0.543018

Figure 2: Equi–$F_{r\theta}$ contours (Left) and equi–$\Psi$ contours (Right)
on the meridional cross section for the model with $\Gamma=2.05$ and
$M/R=0.2$. Here, $z$ and $\varpi$ are defined by $z=r\cos\theta$ and
$\varpi=r\sin\theta$, respectively. The thick quarter circle shows the surface
of the star, on which $F_{r\theta}=0$ and $\Psi=0$ are required by the
boundary condition.

The distribution of magnetic fields is completely determined by the two
arbitrary functions, $\Psi(w)$ and $K(w)$, with $w$ being the function of the
background quantities. For the two arbitrary functions, we have assumed
equations (63) and (64) in this paper. In Figure 2, we give the profiles of
magnetic fields: the toroidal magnetic field $F_{r\theta}$ and the poloidal
flux function $\Psi$ for the background star with $\Gamma=2.05$ and $M/R=0.2$.
The right of Figure 2 shows how lines of the magnetic force on the meridional
cross section behave because an equi–$\Psi$ line corresponds to a line of the
magnetic force. Note that there is no magnetic field outside the star in the
model constructed in this study as mentioned before.

For investigating the effects of magnetic fields on the structure of the star,
it is helpful to introduce the quantities that represent typical strength of
the magnetic field of the star. The norms of the toroidal and poloidal
magnetic fields are, respectively, given by

$\displaystyle\left|{}^{(t)}B\right|^{2}$ $\displaystyle=$
$\displaystyle\varepsilon_{t}^{2}b^{2}r^{2}\sin^{2}\theta\,e^{2\nu}\left({}^{(0)}\rho^{(0)}h\right)^{2}\,,$
(121) $\displaystyle\left|{}^{(p)}B\right|^{2}$ $\displaystyle=$
$\displaystyle\varepsilon_{p}^{2}a^{2}\,4e^{4\nu}\left({}^{(0)}\rho^{(0)}h\cos\theta\right)^{2}$
(122) $\displaystyle+e^{4\nu-2\lambda}\sin^{2}\theta$
$\displaystyle\times\left\\{r{d\over
dr}\left({}^{(0)}\rho^{(0)}h\right)\right.$
$\displaystyle\left.\quad+2\left({}^{(0)}\rho^{(0)}h\right)\left(r{d\nu\over
dr}+1\right)\right\\}^{2}\,.$

The maximum value of the norm of the toroidal magnetic field is then given by

$\displaystyle\left|{}^{(t)}B_{\rm
max}\right|^{2}=\varepsilon_{t}^{2}b^{2}\,{\rm
max}\left[r^{2}\,e^{2\nu}\left({}^{(0)}\rho^{(0)}h\right)^{2}\right]\,,$ (123)

where ${\rm max}\left[f(r)\right]$ means the maximum value of the function
$f(r)$. The maximum value of the norm of the poloidal magnetic field is mostly
attained at the center of the star. Thus, the norm of the poloidal magnetic
field at the center of the star, given by

$\displaystyle\left|{}^{(p)}B_{c}\right|^{2}=\varepsilon_{p}^{2}a^{2}\,4e^{4\nu(0)}\left({}^{(0)}\rho(0)^{(0)}h(0)\right)^{2}\,,$
(124)

may be used as a representative value of the strength of the poloidal magnetic
field. By using these values of the norms, we may obtain the two dimensionless
quantities representing magnetic-field strength, given by

${}^{(t)}{\cal R}_{M}={\left|{}^{(t)}B_{\rm max}\right|^{2}R^{4}\over
4M^{2}}\,,^{(p)}{\cal R}_{M}={\left|{}^{(p)}B_{c}\right|^{2}R^{4}\over
4M^{2}}\,,$ (125)

which are as large as the ratios of the toroidal and the poloidal magnetic
energies to the gravitational energy, respectively. The perturbations due to
the magnetic field given in equation (34) basically depend on the
dimensionless smallness parameters $\varepsilon_{t}$ and $\varepsilon_{p}$,
which can be arbitrarily set depending on the desired magnetic field-strength
as long as $1\gg\varepsilon_{t}\gg\varepsilon_{p}$. This arbitrariness in
perturbation quantities is then removed by using the two dimensionless
quantities ${}^{(t)}{\cal R}_{M}$ and ${}^{(p)}{\cal R}_{M}$ when numerical
results are shown in this paper.

Figure 3: Normalized nondimensional changes in the gravitational mass $\Delta
M/M{}^{(t)}{\cal R}_{M}$ and the baryon rest-mass $\Delta
M^{*}/M^{*}{}^{(t)}{\cal R}_{M}$, given as functions of the central density of
the background star ${}^{(0)}\rho(0)$. The solid circle indicates the result
obtained within the framework of Newton’s dynamics and theory of gravity (cf.,
Appendix). Figure 4: Normalized nondimensional changes in the internal thermal
energy $\Delta E_{\rm int}/E_{\rm int}{}^{(t)}{\cal R}_{M}$, given as
functions of the central density of the background star ${}^{(0)}\rho(0)$. The
solid circle indicates the result obtained within the framework of Newton’s
dynamics and theory of gravity (cf., Appendix). Figure 5: Normalized
nondimensional changes in the mean radius $(\Delta r)_{0}/R{}^{(t)}{\cal
R}_{M}$, given as functions of the central density of the background star
${}^{(0)}\rho(0)$. The solid circle indicates the result obtained within the
framework of Newton’s dynamics and theory of gravity (cf., Appendix). Figure
6: Normalized nondimensional changes in the mass quadrupole moment $\Delta
Q/MR^{2}{}^{(t)}{\cal R}_{M}$, given as functions of the central density of
the background star ${}^{(0)}\rho(0)$. The solid circle indicates the result
obtained within the framework of Newton’s dynamics and theory of gravity (cf.,
Appendix). Figure 7: Normalized changes in the ellipticity
$e^{*}/{}^{(t)}{\cal R}_{M}$, given as functions of the central density of the
background star ${}^{(0)}\rho(0)$. The solid circle indicates the result
obtained within the framework of Newton’s dynamics and theory of gravity (cf.,
Appendix). Figure 8: Normalized nondimensional changes in the toroidal
magnetic energy $E_{\rm EM}^{(t)}/\left|W\right|{}^{(t)}{\cal R}_{M}$, given
as functions of the central density of the background star ${}^{(0)}\rho(0)$.
The solid circle indicates the result obtained within the framework of
Newton’s dynamics and theory of gravity (cf., Appendix). Figure 9: Normalized
nondimensional changes in the poloidal magnetic energy $E_{\rm
EM}^{(p)}/\left|W\right|{}^{(p)}{\cal R}_{M}$, given as functions of the
central density of the background star ${}^{(0)}\rho(0)$. The solid circle
indicates the result obtained within the framework of Newton’s dynamics and
theory of gravity (cf., Appendix). Figure 10: Normalized changes in the
nondimensional magnetic helicity ${\cal H}_{M}/\sqrt{{}^{(t)}{\cal
R}_{M}{}^{(p)}{\cal R}_{M}}$, given as functions of the central density of the
background star ${}^{(0)}\rho(0)$. The solid circle indicates the result
obtained within the framework of Newton’s dynamics and theory of gravity (cf.,
Appendix).

First, we examine properties of the magnetized stars obtained under the
condition of $\Delta\rho_{c}=0$, i.e., their central densities are kept
constant when the magnetic fields are imposed. In Figure 3, normalized
nondimensional changes in the gravitational mass $\Delta M\left/M{}^{(t)}{\cal
R}_{M}\right.$ and the baryon rest-mass $\Delta M^{*}\left/M^{*}{}^{(t)}{\cal
R}_{M}\right.$ are plotted as functions of the central density of the
background star ${}^{(0)}\rho(0)$. In Figures 4 through 10, we plot, as
functions of the central density of the background star ${}^{(0)}\rho(0)$,
normalized nondimensional changes in the internal thermal energy $\Delta
E_{\rm int}\left/E_{\rm int}{}^{(t)}{\cal R}_{M}\right.$, the mean radius
$\left(\Delta r\right)_{0}\left/R{}^{(t)}{\cal R}_{M}\right.$, the mass
quadrupole moment $\Delta Q\left/MR^{2}{}^{(t)}{\cal R}_{M}\right.$, the
ellipticity $e^{*}\left/{}^{(t)}{\cal R}_{M}\right.$, the toroidal magnetic
energy $E_{\rm EM}^{(t)}\left/|W|{}^{(t)}{\cal R}_{M}\right.$, and the
poloidal magnetic energy $E_{\rm EM}^{(p)}\left/|W|{}^{(p)}{\cal
R}_{M}\right.$, the magnetic helicity ${\cal H}_{M}\left/\sqrt{{}^{(t)}{\cal
R}_{M}{}^{(p)}{\cal R}_{M}}\right.$, respectively. In Figures 3 through 10,
the solid circles on the vertical axis indicate the results obtained by the
calculation based on Newtonian magnetohydrodynamics and Newton’s theory of
gravity (cf., Appendix). In these figures, we see that the present general
relativistic results in the Newtonian limit (the limit of ${}^{(0)}\rho(0)\to
0$) are nicely agreement with those obtained by the Newtonian calculations.
This fact serves as a useful consistency check of our numerical code.

Properties of the magnetized stars with $\Delta\rho_{c}=0$ observed in Figures
3 through 10 are summarized as follows: The results for the models with
$\Gamma=2.05$ are little different from those for the models with
$\Gamma=(n+1)/n\approx 1.95238$ (cf. Figures 3 through 10). This implies that
small stratification has little effect on the equilibrium structure of the
magnetized stars. The imposition of the toroidal magnetic fields results in a
decrease in the total baryon rest-mass, i.e., $\Delta M^{*}<0$. Due to the
imposition of the toroidal magnetic fields, values of the gravitational mass
decrease, i.e., $\Delta M<0$, for the background stars with
${}^{(0)}\rho(0)\lessapprox 0.255$ while they increase, i.e., $\Delta M>0$,
for the background stars with ${}^{(0)}\rho(0)\gtrapprox 0.255$ (cf. Figure
3). The mean radius of the star $\left(\Delta r\right)_{0}$ increases when the
toroidal magnetic field is imposed (cf. Figure 5). The values of the mass
quadrupole moment $\Delta Q$ and the ellipticity $e^{*}$ are negative, which
reflects the fact that the star is prolate (cf. Figures 6 and 7). The prolate
deformation is typical for stars containing dominant toroidal magnetic fields
(cf., e.g., Refs kiuchi ; frieben ).

Table 2:  Global and physical quantities. $(\Gamma,M/R)$ | $\displaystyle{\Delta\rho_{c}\over{}^{(0)}\rho(0){}^{(t)}{\cal R}_{M}}$ | $\displaystyle{\Delta M\over M{}^{(t)}{\cal R}_{M}}$ | $\displaystyle{\Delta E_{\rm int}\over E_{\rm int}{}^{(t)}{\cal R}_{M}}$ | $\displaystyle{(\Delta r)_{0}\over R{}^{(t)}{\cal R}_{M}}$ | $\displaystyle{\Delta Q\over MR^{2}{}^{(t)}{\cal R}_{M}}$ | $\displaystyle{e^{*}\over{}^{(t)}{\cal R}_{M}}$ | $\displaystyle{E_{\rm EM}^{(t)}\over|W|{}^{(t)}{\cal R}_{M}}$ | $\displaystyle{E_{\rm EM}^{(p)}\over|W|{}^{(p)}{\cal R}_{M}}$ | $\displaystyle{{\cal H}_{M}\over\sqrt{{}^{(t)}{\cal R}_{M}{}^{(p)}{\cal R}_{M}}}$
---|---|---|---|---|---|---|---|---|---
(1.95238,0.1) | 0.2424 | $1.565\times 10^{-2}$ | $-0.2413$ | 0.3048 | $-0.1105$ | $-0.2294$ | 0.2097 | $9.747\times 10^{-2}$ | 1.512
(1.95238,0.2) | 0.8077 | $2.338\times 10^{-2}$ | 0.4077 | $-1.607\times 10^{-2}$ | $-3.946\times 10^{-2}$ | $-0.1213$ | 0.1626 | $7.349\times 10^{-2}$ | 1.596
(2.05,0.1) | 0.2457 | $1.646\times 10^{-2}$ | $-0.2397$ | 0.3056 | $-0.1116$ | $-0.2316$ | 0.2110 | $9.904\times 10^{-2}$ | 1.527
(2.05,0.2) | 0.5677 | $2.257\times 10^{-2}$ | 0.2043 | $8.193\times 10^{-2}$ | $-4.134\times 10^{-2}$ | $-0.1271$ | 0.1672 | $7.792\times 10^{-2}$ | 1.654

Next, we examine properties of the magnetized stars obtained under the
condition of $\Delta M^{*}=0$, i.e., their total baryon rest-masses are kept
constant when the magnetic fields are imposed. Table 2 lists global and
physical quantities characterizing the magnetized stars with $\Delta M^{*}=0$;
the changes in the central density $\Delta\rho_{c}$, the gravitational mass
$\Delta M$, the internal thermal energy $\Delta E_{\rm int}$, the mean radius
$\left(\Delta r\right)_{0}$, the mass quadrupole moment $\Delta Q$, the
ellipticity $e^{*}$, the toroidal magnetic energy $E_{\rm EM}^{(t)}$, the
poloidal magnetic energy $E_{\rm EM}^{(p)}$, and the magnetic helicity ${\cal
H}$. In this table, all the quantities are normalized to be nondimensional, as
given in the first row.

Properties of the magnetized star with $\Delta M^{*}=0$ observed in Table 2
are summarized as follows: The imposition of the toroidal magnetic fields
results in an increase in the central density, i.e., $\Delta\rho_{c}>0$. The
values of the mass quadrupole moment $\Delta Q$ and the ellipticity $e^{*}$
are negative, which reflects the fact that the star is prolate. These
properties concerning $\Delta\rho_{c}$ and $\Delta Q$ are attributed to the
magnetic hoop stress around the symmetry axis due to the toroidal magnetic
field, which tends to make the star prolate like a rubber belt fastening
around the waist of a star. The gravitational mass increase due to the
imposition of the toroidal magnetic fields, i.e., $\Delta M>0$.

Since the deformation of the star considered in this study is caused by
toroidal magnetic fields only, even though poloidal magnetic fields make the
deformation of the spacetime, the results obtained in this study can be
compared with those obtained by Kiuchi and Yoshida kiuchi , who derived
general relativistic stars having purely toroidal magnetic fields with a non-
perturbative approach. Although weakly magnetized stars cannot be calculated
with Kiuchi and Yoshida’s method because of their non-perturbative approach,
it is found that the present results are consistent with those obtained by
Kiuchi and Yoshida kiuchi  (Compare, e.g., Table 2 with Figure 6 of Ref.
kiuchi ).

##  V Discussion

In this study, as mentioned before, the general relativistic magnetized stars
are constructed under the condition of $\varepsilon_{p}\ll\varepsilon_{t}\ll
1$, and the effects of magnetic fields are investigated within accuracy
$O\left(\varepsilon_{t}\varepsilon_{p}\right)$. The terms higher than
$O\left(\varepsilon_{p}^{2}\right)$ in the equations are then discarded. The
deformation of the star and spacetime occurs in the
$\varepsilon_{t}^{2}$-order, which is attributed to the magnetic effects due
to the toroidal field. This deformation due to the toroidal magnetic field is
the same as that of the weakly magnetized star with purely toroidal fields
within accuracy $O\left(\varepsilon_{t}^{2}\right)$. Thus, the present results
include those for the weakly magnetized star with purely toroidal fields. To
our knowledge, such general relativistic magnetized stars having purely
toroidal fields have been constructed with a perturbative approach for the
first time. The $\varepsilon_{t}\varepsilon_{p}$-order effects appear in the
deformation of the spacetime only. Therefore, the
$\varepsilon_{t}\varepsilon_{p}$-order effects are general relativistic ones
and disappear in Newton’s dynamics and theory of gravity. Within the framework
of Newtonian magnetohydrodynamics, in other words, the poloidal magnetic field
does not affect the deformation of the star within accuracy
$O\left(\varepsilon_{t}\varepsilon_{p}\right)$ (cf. Appendix). From a general
relativistic point of view, an interesting fact is that the
$\varepsilon_{t}\varepsilon_{p}$-order effects violate the circularity
conditions (cf. equation (62)). As a result, the $r\varphi$-component of the
metric $g_{r\varphi}$ appears inside the star (cf. equation (83)).

In this paper, we have shown that stationary and axisymmetric solutions of the
magnetized star with mixed poloidal-toroidal fields may indeed be constructed
within accuracy $O\left(\varepsilon_{t}\varepsilon_{p}\right)$. However, such
stationary and axisymmetric solutions cannot be constructed if the
$\varepsilon_{p}^{2}$-order effects on the structure of the star are included.
The reason for this is the following: The $\varepsilon_{p}^{2}$-order
equations are the same as those for the weakly magnetized star with purely
poloidal fields within accuracy $O\left(\varepsilon_{p}^{2}\right)$. For the
weakly magnetized star with purely poloidal fields, the poloidal flux function
$\Psi$ has to satisfy the general relativistic version of the so-called Grad-
Shafranov equation (cf., e.g., Ref. konno ). In the present approximation,
however, the flux function $\Psi$ has to be given by the arbitrary function of
the background quantity $w=^{(0)}\rho^{(0)}he^{2\nu}r^{2}\sin^{2}\theta$ (cf.
equations (39) and (40)). The flux function $\Psi$ given by the arbitrary
function of $w$ does not, in general, fulfill the Grad-Shafranov equation.
Therefore, the $\varepsilon_{p}^{2}$-order equations cannot be solved
consistently with the lower-order equations. This implies that the weakly
magnetized stars constructed in this study cannot be stationary and
axisymmetric when the condition of $\varepsilon_{p}\ll\varepsilon_{t}$ is
violated.

After obtaining equilibrium models of stars, check of their stability is an
important issue because unstable solutions lose their physical meaning in the
sense that they are not realized in nature. Since magnetized stars with purely
toroidal fields are unstable, the present magnetized star models are indeed
unstable when we set $\varepsilon_{p}=0$, which corresponds to the case of
purely toroidal fields. As mentioned in Introduction, both a stable
stratification of the fluid and poloidal magnetic fields act as stabilizing
agents of the toroidal magnetic fields inside the star. The stably stratified
stars with $1\gg\varepsilon_{t}\gg\varepsilon_{p}\neq 0$ constructed in this
study are therefore possibly stable. As mentioned before, unfortunately,
reliable and useful procedures for the diagnosis of the stability for the
magnetized star have not yet been established. (For the moment, numerical
simulations will be the most reliable way to check the stability, but they are
tough work.) Although we are not sure that it is adaptive for the present
magnetized star models, Braithwaite’s stability condition, given in equation
(1), is available to assess their stability. If a magnetized star
characterized by $\Gamma=2.05$, $R\approx 10$ km, $M\approx 1.4\,M_{\odot}$,
and ${}^{(t)}B_{\rm max}\approx 10^{15}$ G is considered, we have
${}^{(t)}{\cal R}_{M}\approx 5\times 10^{-7}$ and $E^{(t)}_{\rm EM}/|W|\approx
8\times 10^{-8}$. We then obtain Braithwaite’s stability condition for the
model, given by

$8\times 10^{-5}\lesssim{E_{\rm EM}^{(p)}\over E_{\rm EM}^{(t)}}\lesssim
0.8\,,$ (126)

where $E_{\rm EM}^{(p)}/E_{\rm EM}\approx E_{\rm EM}^{(p)}/E_{\rm EM}^{(t)}$
is used, and $\tilde{a}\approx 10^{3}$ is assumed because the star is a stably
stratified neutron star model. For the model considered, we have $E_{\rm
EM}^{(p)}/E_{\rm EM}^{(t)}\approx 0.5\,{\cal R}_{M}^{(p)}/{\cal R}_{M}^{(t)}$.
Braithwaite’s stability condition for the model then becomes

$1.6\times 10^{-4}\lesssim{{\cal R}_{M}^{(p)}\over{\cal R}_{M}^{(t)}}\lesssim
1.6\,.$ (127)

Under the condition of $\varepsilon_{p}\ll\varepsilon_{t}\ll 1$, which is the
basic assumption in this study, we can appropriately choose vales of ${{\cal
R}_{M}^{(p)}/{\cal R}_{M}^{(t)}}$ so as to satisfy the inequality given in
equation (127), Braithwaite’s stability condition for the model. Therefore,
the present magnetized star models satisfying the inequality (127) are stable
if Braithwaite’s stability condition is properly adaptive for them. In order
to examine stability properly, however, we have to make stability analyses by
using dynamical simulations or solving linear eigenvalue problems, which
exceed the scope of this work and remain as future challenges.

##  VI Summary

We have constructed the stably stratified magnetized stars within the
framework of general relativity. The effects of magnetic fields on the
structure of the star and spacetime are treated as perturbations of non-
magnetized stars. By assuming ideal magnetohydrodynamics and employing one-
parameter equations of state, we derive basic equations for describing
stationary and axisymmetric stably stratified stars containing magnetic fields
whose toroidal components are much larger than the poloidal ones. A number of
the polytropic models are numerically calculated to investigate basic
properties of the effects of magnetic fields on the stellar structure.
According to the stability result obtained by Braithwaite, which remains a
matter of conjecture for general magnetized stars, certain of the magnetized
stars constructed in this study are possibly stable.

## Acknowledgments

This work was supported by the Grant-in-Aid for Scientific Research (C) No.
18K03606.

##  Appendix A Newtonian analysis

In this appendix, we present the Newtonian version of the magnetized star
considered in this study. The results of the Newtonian analysis can be used to
compare to those of the general relativistic analysis in the Newtonian limit.
We may then check consistency between them. Similar analyses to those given in
this appendix are found in, e.g., Refs. sinha ; asai .

Within the framework of Newtonian magnetohydrodynamics, the dynamics of
perfectly conductive fluids may be described by the following equations:

$\partial_{t}\,\rho+\nabla_{a}\left(\rho\,v^{a}\right)=0\,,$ (128)
$\nabla_{a}B^{a}=0\,,$ (129)
$\partial_{t}\,B^{a}=\nabla_{b}\left(v^{a}B^{b}-v^{b}B^{a}\right)\,,$ (130)
$\displaystyle\left(\partial_{t}+v^{b}\nabla_{b}\right)v_{a}$ $\displaystyle=$
$\displaystyle-{1\over\rho}\nabla_{a}p-\nabla_{a}\Phi$ (131) $\displaystyle+$
$\displaystyle{1\over
4\pi\rho}\left(B^{b}\nabla_{b}B_{a}-B^{b}\nabla_{a}B_{b}\right)\,,$
$\nabla^{b}\nabla_{b}\Phi=4\pi G\rho\,,$ (132)

where $\rho$, $v^{a}$, $B^{a}$, $p$, and $\Phi$ are the mass density, the
fluid velocity, the magnetic field, the pressure, and the gravitational
potential, respectively. Here, $\nabla_{a}$ denotes the covariant derivative
associated with the metric $g_{ab}$, and spatial indices are denoted by lower
case Roman letters ($a,b,c,\cdots$).

Following the assumptions given in Sec. III, we assume that there is no fluid
flow, i.e.,

$v^{a}=0\,,$ (133)

and that the magnetized stars are stationary and axisymmetric. Therefore,
physical quantities associated with the magnetized star are independent of the
time coordinate $t$ and the azimuthal angle about the symmetry axis $\varphi$.
Under the assumption of stationarity and axisymmetry, the magnetic fields
$B^{a}$ may, in terms of two functions $B$ and $A_{\varphi}$ independent of
$t$ and $\varphi$, be written by

$\quad B^{a}=B\varphi^{a}+\epsilon^{ab\varphi}\partial_{b}A_{\varphi}\,,$
(134)

where $\varphi^{a}$ denotes the rotational Killing vector, $\epsilon^{abc}$ is
the contravariant spatial Levi-Civita tensor, and $A_{\varphi}$ is the
$\varphi$-component of the vector potential $A_{a}$ or the poloidal flux
function. Due to the assumptions given in equations (133) and (134), equations
(128)–(130) are satisfied automatically, and the $\varphi$-component of
equation (131) becomes

$\displaystyle{1\over
4\pi\rho}\left(B^{b}\nabla_{b}B_{\varphi}-B^{b}\nabla_{\varphi}B_{b}\right)$
$\displaystyle=$ $\displaystyle{1\over
4\pi\rho}\epsilon^{bc\varphi}\partial_{c}A_{\varphi}\partial_{b}\left(B\,\varphi_{\varphi}\right)$
(135) $\displaystyle=$ $\displaystyle 0\,.$

Therefore, the function $B$ has to be given in terms of an arbitrary function
$K(A_{\varphi})$ by

$B={K(A_{\varphi})\over\varphi_{\varphi}}\,.$ (136)

By using equation (136), we may rewrite the poloidal components of the Lorentz
force term in equation (131) as follows:

$\displaystyle{1\over
4\pi\rho}\left(B^{b}\nabla_{b}B_{C}-B^{b}\nabla_{C}B_{b}\right)=-{K\over
4\pi\rho\,\varphi_{\varphi}}\partial_{C}K$
$\displaystyle\quad\quad\quad\quad-{1\over
4\pi\rho\sqrt{g}}\left(\partial_{C}A_{\varphi}\right)\left(\partial_{2}B_{1}-\partial_{1}B_{2}\right)\,,$
(137)

where the index $C$ denote the poloidal indices, i.e., $C=1,2$, and $g$ means
the determinant of the metric $g_{ab}$. If we make the same approximation as
that used in Sec. III, the first and the second terms in the right-hand side
of equation (137) are $O\left(\varepsilon_{t}^{2}\right)$ and
$O\left(\varepsilon_{p}^{2}\right)$, respectively. Under the assumption of
$\varepsilon_{p}\ll\varepsilon_{t}\ll 1$, equation (137) becomes

${1\over
4\pi\rho}\left(B^{b}\nabla_{b}B_{C}-B^{b}\nabla_{C}B_{b}\right)=-{K\over
4\pi\rho\,\varphi_{\varphi}}\partial_{C}K+O\left(\varepsilon_{t}\varepsilon_{p}\right)\,,$
(138)

within accuracy $O\left(\varepsilon_{t}\varepsilon_{p}\right)$. In other
words, similarly to the general relativistic case, poloidal magnetic fields do
not affect the deformation of the star within accuracy
$O\left(\varepsilon_{t}\varepsilon_{p}\right)$. The Euler equation then
becomes

${1\over\rho}\nabla_{C}p+\nabla_{C}\Phi+{1\over
8\pi\rho\,\varphi_{\varphi}}\partial_{C}K^{2}+O\left(\varepsilon_{t}\varepsilon_{p}\right)=0\,.$
(139)

This equation may be integrable if the following conditions for the three
functions $p$, $K$, and $A_{\varphi}$, are assumed:

$p=p(\rho)\,,\quad K=K(\rho\varphi_{\varphi})\,,\quad
A_{\varphi}=A_{\varphi}(\rho\varphi_{\varphi})\,.$ (140)

After giving the actual forms of the three functions $p$, $K$, and
$A_{\varphi}$, we may then obtain the weakly magnetized star models with mixed
poloidal and toroidal fields.

In what follows, the spherical polar coordinates $(r,\theta,\varphi)$ are used
in order to derive the master equations for the weakly magnetized stars with
mixed poloidal-toroidal fields. The metric is then given by

$ds^{2}=dr^{2}+r^{2}d\theta^{2}+r^{2}\sin^{2}\theta d\varphi^{2}\,.$ (141)

The rotational Killing vector $\varphi_{a}$ is given by

$\varphi_{a}=(0,0,r^{2}\sin^{2}\theta)\,.$ (142)

Following the assumptions given in Sec. III, we set the two arbitrary
functions $K$ and $A_{\varphi}$ as follows:

$\displaystyle K$ $\displaystyle=$
$\displaystyle{B_{c}\over\sqrt{2}\,\rho_{c}\alpha}\,\rho\,r^{2}\sin^{2}\theta\,,$
(143) $\displaystyle A_{\varphi}$ $\displaystyle=$ $\displaystyle
b\rho\,r^{2}\sin^{2}\theta\,,$ (144)

where $B_{c}$, $\rho_{c}$, and $\alpha$ are constants that are related to the
magnetic field strength, density, and radius of the star, respectively, and
$b$ is a constant. The absolute value of the magnetic field $B^{a}$ is then
given by

$\sqrt{B^{a}B_{a}}={B_{c}\over\sqrt{2}}\,\hat{\rho}\,\xi\sin\theta+O\left(\varepsilon_{p}^{2}\right)\,,$
(145)

where the two dimension-less quantities $\hat{\rho}=\rho/\rho_{c}$ and
$\xi=r/\alpha$ are introduced. By using equation (143), we may rewrite
equation (139) as

$\displaystyle\nabla_{a}p=-\rho\nabla_{a}\Psi\,,$ (146)
$\displaystyle\Psi=\Phi+{1\over
3}\,\Omega_{A}^{2}r^{2}\,\hat{\rho}\left\\{1-P_{2}\left(\cos\theta\right)\right\\}-\Phi_{0}\,,$
(147)

where $\Omega_{A}$ is the Alfvèn frequency, defined by
$\displaystyle\Omega_{A}=\sqrt{{B_{c}^{2}\over 4\pi\rho_{c}\alpha^{2}}}$, and
$\Phi_{0}$ is a constant. Since $\Omega_{A}=O\left(\varepsilon_{t}\right)$,
within accuracy $O\left(\varepsilon_{t}\varepsilon_{p}\right)$, the function
$\Psi$ fulfills the equation, given by

$\displaystyle\nabla^{a}\nabla_{a}\Psi$ $\displaystyle=$ $\displaystyle 4\pi
G\rho$ (148) $\displaystyle+$ $\displaystyle{1\over
3}\,\Omega_{A}^{2}\left\\{r^{2}{d^{2}\hat{\rho}_{0}\over
dr^{2}}+6r{d\hat{\rho}_{0}\over dr}+6\hat{\rho}_{0}\right.$
$\displaystyle\left.-\left(r^{2}{d^{2}\hat{\rho}_{0}\over
dr^{2}}+6r{d\hat{\rho}_{0}\over
dr}\right)P_{2}\left(\cos\theta\right)\right\\}$
$\displaystyle+O\left(\varepsilon_{p}^{2}\right)\,,$

where $\hat{\rho}_{0}$ is the dimensionless density of the non-magnetized star
normalized by its central value. The function $\Psi$ may be expanded in terms
of the parameter $\Omega_{A}$, and then written by

$\displaystyle\Psi\left(r,\theta\right)$ $\displaystyle=$
$\displaystyle\Psi_{0}\left(r\right)$ (149) $\displaystyle-$ $\displaystyle
2\alpha^{2}\Omega_{A}^{2}\left[\psi_{0}\left(r\right)+\psi_{2}\left(r\right)P_{2}\left(\cos\theta\right)\right]$
$\displaystyle+O\left(\varepsilon_{p}^{2}\right)\,,$

where $\Psi_{0}$ means the function $\Psi$ for the non-magnetized star. Since
we have $p=p(\rho)$ and equation (146), the density $\rho$ is a function of
$\Psi$. Therefore, the density $\rho$ may be also expanded in terms of the
parameter $\Omega_{A}$, and then written by

$\displaystyle\rho\left(r,\theta\right)$ $\displaystyle=$
$\displaystyle\rho_{0}\left(r\right)$ (150) $\displaystyle-$ $\displaystyle
2\alpha^{2}\Omega_{A}^{2}{d\rho_{0}\over
d\Psi_{0}}\left[\psi_{0}\left(r\right)+\psi_{2}\left(r\right)P_{2}\left(\cos\theta\right)\right]$
$\displaystyle+O\left(\varepsilon_{p}^{2}\right)\,,$

where $\rho_{0}$ means the density $\rho$ for the non-magnetized star.
Instituting equations (149) and (150) into equation (148), we obtain

$\displaystyle\alpha^{2}\nabla^{a}\nabla_{a}\Psi_{0}\left(r\right)=4\pi
G\alpha^{2}\rho_{0}\left(r\right)\,,$ (151)
$\displaystyle{1\over\xi^{2}}{d\over d\xi}\left(\xi^{2}\,{d\psi_{0}\over
d\xi}\right)$ $\displaystyle=$ $\displaystyle k(\xi)\psi_{0}$ (152)
$\displaystyle-$ $\displaystyle{1\over 6r^{2}}{d\over dr}\left(r^{2}{d\over
dr}\left(r^{2}\hat{\rho}_{0}\right)\right)\,,$
$\displaystyle{1\over\xi^{2}}{d\over d\xi}\left(\xi^{2}\,{d\psi_{2}\over
d\xi}\right)$ $\displaystyle=$
$\displaystyle\left(k\left(\xi\right)+{6\over\xi^{2}}\right)\psi_{2}$ (153)
$\displaystyle+$ $\displaystyle{1\over 6}\left(r^{2}{d^{2}\hat{\rho}_{0}\over
dr^{2}}+6r{d\hat{\rho}_{0}\over dr}\right)\,,$

where

$k\left(\xi\right)=4\pi G\alpha^{2}{d\rho_{0}\over d\Psi_{0}}\,.$ (154)

In order to solve the three ordinary differential equations (151)–(153),
boundary conditions at the center and surface of the star are necessary. We
require that physical quantities are regular near the center of the star and
that values of the central density are independent of the magnetic field-
strength. At the center of the star, therefore, we have

$\displaystyle{d\Psi_{0}\over
d\xi}\left(0\right)=0\,,\quad\psi_{0}\left(0\right)=0\,,\quad{d\psi_{0}\over
d\xi}\left(0\right)=0\,,$
$\displaystyle\psi_{2}\left(0\right)=0\,,\quad{d\psi_{2}\over
d\xi}\left(0\right)=0\,.$ (155)

To determine the boundary condition at the surface of the star, we need the
equation of the surface of the star, given by

$r=R\left(1+\delta\zeta\right)+O\left(\varepsilon_{p}^{2}\right)\,,$ (156)

where $R$ is the radius for the non-magnetized star. The dimensionless
displacement $\delta\zeta$ is given by

$\displaystyle\delta\zeta={2\alpha^{2}\Omega_{A}^{2}\over\displaystyle
R{d\Psi_{0}\over
dr}(R)}\left[\psi_{0}\left(\xi_{1}\right)+\psi_{2}\left(\xi_{1}\right)P_{2}\left(\cos\theta\right)\right]\,,$
(157)

where $\xi_{1}=R/\alpha$. The displacement $\delta\zeta$ is used to evaluate
the ellipticity of the surface of the star $e^{*}$, defined by

$\displaystyle e^{*}$ $\displaystyle=$
$\displaystyle{R\left(1+\delta\zeta\left(\pi/2\right)\right)-R\left(1+\delta\zeta\left(0\right)\right)\over
R}$ (158) $\displaystyle=$ $\displaystyle-{3\over
2}2\alpha^{2}\Omega_{A}^{2}{1\over\displaystyle R{d\Psi_{0}\over
dr}(R)}\psi_{2}\left(\xi_{1}\right)\,.$

The gravitational potentials inside and outside the star within accuracy
$O\left(\varepsilon_{t}^{2}\right)$ are, respectively, given by

$\displaystyle\Phi$ $\displaystyle=$
$\displaystyle\Psi_{0}\left(r\right)+c_{0}$ (159) $\displaystyle-$
$\displaystyle
2\alpha^{2}\Omega_{A}^{2}\left[c_{1,0}+\psi_{0}\left(\xi\right)+\psi_{2}\left(\xi\right)P_{2}\left(\cos\theta\right)\right]$
$\displaystyle-$ $\displaystyle{1\over
3}\,\Omega_{A}^{2}r^{2}\,\hat{\rho}\left\\{1-P_{2}\left(\cos\theta\right)\right\\}\,,$

and

$\Phi=-{\kappa_{0}\over\xi}-2\alpha^{2}\Omega_{A}^{2}\left[{\kappa_{1,0}\over\xi}+{\kappa_{1,2}\over\xi^{3}}P_{2}\left(\cos\theta\right)\right]\,,$
(160)

where $c_{0}$, $c_{1,0}$, $\kappa_{0}$, $\kappa_{1,0}$ and $\kappa_{1,2}$ are
constants. Since the gravitational potential and its derivative have to be
continuous at the surface of the star, we obtain the following relations:

$\kappa_{0}=\xi_{1}^{2}\partial_{\xi}\Psi_{0}\left(R\right)\,,\
c_{0}=-\Psi_{0}\left(R\right)-\xi_{1}\partial_{\xi}\Psi_{0}\left(R\right)\,,$
(161)
$-c_{1,0}=-{\kappa_{1,0}\over\xi_{1}}+\psi_{0}\left(\xi_{1}\right)\,,\quad{\kappa_{1,2}\over\xi_{1}^{3}}=\psi_{2}\left(\xi_{1}\right)\,,$
(162) $\displaystyle{\kappa_{1,0}\over\xi_{1}^{2}}=-{d\psi_{0}\over
d\xi}(\xi_{1})-{1\over 6}\,\xi_{1}^{2}{d\hat{\rho}\over
d\xi}\left(\xi_{1}\right)\,,$ $\displaystyle
3{\kappa_{1,2}\over\xi_{1}^{4}}=-{d\psi_{2}\over d\xi}(\xi_{1})+{1\over
6}\,\xi_{1}^{2}{d\hat{\rho}\over d\xi}\left(\xi_{1}\right)\,.$ (163)

For equations (151) and (152), therefore, boundary conditions are not imposed
at the surface of the star, and instead the constants characterizing the
gravitational potential are determined through equation (161) and

$\displaystyle\kappa_{1,0}$ $\displaystyle=$
$\displaystyle-\xi_{1}^{2}{d\psi_{0}\over d\xi}(\xi_{1})-{1\over
6}\,\xi_{1}^{4}{d\hat{\rho}\over d\xi}\left(\xi_{1}\right)\,,$ $\displaystyle
c_{1,0}$ $\displaystyle=$
$\displaystyle-\psi_{0}(\xi_{1})-\xi_{1}{d\psi_{0}\over d\xi}(\xi_{1})-{1\over
6}\,\xi_{1}^{3}{d\hat{\rho}\over d\xi}\left(\xi_{1}\right)\,,$ (164)

As for equation (153), the boundary condition at the surface of the star is
given by

$3\psi_{2}\left(\xi_{1}\right)+\xi_{1}{d\psi_{2}\over
d\xi}\left(\xi_{1}\right)={1\over 6}\,\xi_{1}^{3}{d\hat{\rho}\over
d\xi}\left(\xi_{1}\right)\,.$ (165)

The constant related to the mass quadrupole moment $\kappa_{1,2}$ is
determined through

$\kappa_{1,2}=\xi_{1}^{3}\psi_{2}\left(\xi_{1}\right)\,.$ (166)

Following the assumptions given in Sec. III, we assume the polytropic equation
of state, given by

$\displaystyle p=\kappa\rho^{1+{1\over n}}\,,$ (167)

where $\kappa$ and $n$ are constants. Introducing the Lane-Emden function
$\Theta$, then, we may write $\rho$ and $p$ as

$\displaystyle\rho=\rho_{c}\Theta^{n}\,,\quad p=p_{c}\Theta^{n+1}\,,$ (168)

where $\rho_{c}$ and $p_{c}$ are values of the density and pressure at the
center of the star, respectively. The central pressure value $p_{c}$ is given
in terms of $\rho_{c}$, $\kappa$, and $n$ by $p_{c}=\kappa\rho_{c}^{1+{1\over
n}}$. Equation (151) is rewritten by

$\displaystyle{1\over\xi^{2}}{\partial\over\partial\xi}\left(\xi^{2}{\partial\over\partial\xi}\widehat{\Psi}\right)=\Theta^{n}\,,$
(169)

where $\widehat{\Psi}$ is the dimensionless quantity, defined by

$\displaystyle\widehat{\Psi}={\Psi\over 4\pi G\rho_{c}\alpha^{2}}\,.$ (170)

From equation (146), inside the star, we obtain

$\displaystyle\widehat{\Psi}=-\Theta+C\,,$ (171)

where $C$ is a constant, and we set

$\displaystyle\alpha=\sqrt{{(n+1)K\rho_{c}^{{1\over n}}\over 4\pi
G\rho_{c}}}=\sqrt{{(n+1)p_{c}\over 4\pi G\rho_{c}^{2}}}\,.$ (172)

Substituting equation (171) into equation (169) yields the Lane-Emden
equation, given by

$\displaystyle{d^{2}\Theta\over d\xi^{2}}+{2\over\xi}{d\Theta\over
d\xi}=-\Theta^{n}\,.$ (173)

At the center of the star, the boundary conditions for equation (173) are, due
to equations (155) and (168), given by

$\displaystyle\Theta=1\,,\quad{d\Theta\over d\xi}=0\,,\quad{\rm at}\ \xi=0\,.$
(174)

The function $k(\xi)$, defined in equation (154), is rewritten by

$\displaystyle k\left(\xi\right)=-n\,\Theta^{n-1}\,.$ (175)

Now that we obtain the complete set of equations for the weakly magnetized
star with mixed poloidal-toroidal fields within the framework of Newtonian
magnetohydrodynamics, which are composed of equations (173), (152), and (153),
we may construct the magnetized stars.

In order to investigate properties of equilibrium solutions of the magnetized
star, global quantities are frequently used. The mass of the star $M$ is given
by

$\displaystyle M$ $\displaystyle=$ $\displaystyle 2\pi\int\rho r^{2}\sin\theta
drd\theta\,,$ (176) $\displaystyle=$ $\displaystyle M_{0}\left(1+{\Delta
M\over M_{0}}\right)+O\left(\varepsilon_{t}^{2}\right)\,,$

where $M_{0}$ is the mass of the non-magnetized star, given by

$\displaystyle M_{0}=-4\pi\alpha^{3}\rho_{c}\left.\xi_{1}^{2}{d\Theta\over
d\xi}\right|_{\xi=\xi_{1}}\,,$ (177)

and $\displaystyle{\Delta M\over M_{0}}$ is the normalized dimensionless
change in the mass of the star, given by

$\displaystyle{\Delta M\over M_{0}}={\displaystyle{\Omega_{A}^{2}\over 2\pi
G\rho_{c}}\left\\{\left.{d\psi_{0}\over d\xi}\right|_{\xi=\xi_{1}}+{1\over
6}\left.\xi_{1}^{2}{d\hat{\rho}_{0}\over
d\xi}\right|_{\xi=\xi_{1}}\right\\}\over\displaystyle\left.{d\Theta\over
d\xi}\right|_{\xi=\xi_{1}}}\,.$ (178)

The internal thermal energy of the star $E_{\rm int}$ is given by

$\displaystyle E_{\rm int}$ $\displaystyle=$ $\displaystyle
2\pi\int\rho\varepsilon r^{2}\sin\theta drd\theta\,,$ (179) $\displaystyle=$
$\displaystyle\left(E_{\rm int}\right)_{0}\left(1+{\Delta E_{\rm
int}\over\left(E_{\rm
int}\right)_{0}}\right)+O\left(\varepsilon_{t}^{2}\right)\,,$

where $\left(E_{\rm int}\right)_{0}$ is the internal thermal energy of the
non-magnetized star, given by

$\displaystyle\left(E_{\rm int}\right)_{0}={4\pi\over\Gamma-1}{4\pi
G\rho_{c}^{2}\alpha^{5}\over n+1}\int_{0}^{\xi_{1}}\Theta^{n+1}\xi^{2}d\xi\,,$
(180)

and $\displaystyle{\Delta E_{\rm int}\over\left(E_{\rm int}\right)_{0}}$ is
the normalized dimensionless change in the internal thermal energy of the
star, given by

$\displaystyle{\Delta E_{\rm int}\over\left(E_{\rm
int}\right)_{0}}={\displaystyle{\Omega_{A}^{2}\over 4\pi
G\rho_{c}}2\left(n+1\right)\int_{0}^{\xi_{1}}\Theta^{n}\psi_{0}\left(r\right)\xi^{2}d\xi\over\displaystyle\int_{0}^{\xi_{1}}\Theta^{n+1}\xi^{2}d\xi}\,.$
(181)

Here, the gamma-law equation of state, given in equation (113), is used. An
average change in the radius of the star $\left(\Delta r\right)_{0}$, defined
in equation (105), is given by

$\displaystyle{\left(\Delta r\right)_{0}\over
R}=-2\Omega_{A}^{2}{\psi_{0}\left(\xi_{1}\right)\over\displaystyle 4\pi
G\rho_{c}\xi_{1}\left.{d\Theta\over d\xi}\right|_{\xi=\xi_{1}}}\,.$ (182)

The ellipticity associated with the surface shape of the star $e^{*}$ is
explicitly given by

$\displaystyle
e^{*}=3\Omega_{A}^{2}{\psi_{2}\left(\xi_{1}\right)\over\displaystyle 4\pi
G\rho_{c}\xi_{1}\left.{d\Theta\over d\xi}\right|_{\xi=\xi_{1}}}\,.$ (183)

The mass quadrupole moment of the star $\Delta Q$ is, in terms of
$\kappa_{1,2}$, given by

$\displaystyle{\Delta Q\over
M_{0}R^{2}}=-{2\alpha^{2}\Omega_{A}^{2}\alpha^{3}\kappa_{1,2}\over\alpha\kappa_{0}\alpha^{2}\xi_{1}^{2}}\,.$
(184)

The toroidal magnetic energy ${}^{(t)}E_{\rm EM}$ and the poloidal magnetic
energy ${}^{(p)}E_{\rm EM}$ are, respectively, defined by

${}^{(t)}E_{\rm EM}={1\over 8\pi}\int B^{\varphi}B_{\varphi}dV\,,$
${}^{(p)}E_{\rm EM}={1\over 8\pi}\int B^{C}B_{C}dV\,.$ (185)

Then, the ratios of the toroidal and the poloidal magnetic energies to the
unperturbed gravitational energy of the star $\displaystyle{{}^{(t)}E_{\rm
EM}\over\left(\left|W\right|\right)_{0}}$ and $\displaystyle{{}^{(p)}E_{\rm
EM}\over\left(\left|W\right|\right)_{0}}$ are, respectively, given by

$\displaystyle{{}^{(t)}E_{\rm
EM}\over\left(\left|W\right|\right)_{0}}={\displaystyle{\Omega_{A}^{2}\over
3\cdot
4\pi\rho_{c}G}\int_{0}^{\xi_{1}}\hat{\rho}^{2}\xi^{4}d\xi\over\displaystyle-\int_{0}^{\xi_{1}}\hat{\rho}\widehat{\Phi}\xi^{2}d\xi}\,,$
(186)

and

$\displaystyle{{}^{(p)}E_{\rm
EM}\over\left(\left|W\right|\right)_{0}}={b^{2}\over 3\cdot 2\pi\alpha^{2}4\pi
G}$
$\displaystyle\quad\quad\times{\displaystyle\int_{0}^{\xi_{1}}\left\\{\left(\xi{d\hat{\rho}\over
d\xi}+2\hat{\rho}\right)^{2}+2\rho^{2}\right\\}\xi^{2}d\xi\over\displaystyle-\int_{0}^{\xi_{1}}\hat{\rho}\widehat{\Phi}\xi^{2}d\xi}\,,$
(187)

where the unperturbed gravitational energy of the star
$\left(\left|W\right|\right)_{0}$ is given by

$\displaystyle\left(\left|W\right|\right)_{0}$ $\displaystyle=$
$\displaystyle-{1\over 2}\int\rho_{0}\Phi_{0}dV\,,$ (188) $\displaystyle=$
$\displaystyle-2\pi\alpha^{5}\rho_{c}^{2}4\pi
G\int_{0}^{\xi_{1}}\hat{\rho}_{0}\widehat{\Phi}_{0}\xi^{2}d\xi\,.$

The dimensionless magnetic helicity of the star ${\cal H}_{M}$ is given by

$\displaystyle{\cal H}_{M}$ $\displaystyle=$ $\displaystyle{{\cal H}\over
GM_{0}^{2}}\,,$ (189) $\displaystyle=$
$\displaystyle{\displaystyle{bB_{c}\over\sqrt{2}\,3\pi
G\alpha^{2}\rho_{c}}\int_{0}^{\xi_{1}}\hat{\rho}^{2}\xi^{4}d\xi\over\displaystyle\left(\left.\xi_{1}^{2}{d\Theta\over
d\xi}\right|_{\xi=\xi_{1}}\right)^{2}}\,,$

where ${\cal H}$ is the magnetic helicity of the star, defined by ${\cal
H}=\int A_{a}B^{a}dV$.

Since the effects of magnetic fields on the structure of the star are treated
as perturbations of the non-magnetized star, the solutions constructed in this
study are inherently independent of the magnetic-field strength. However, the
representation for the global and physical quantities defined before are
dependent on the magnetic-field strength. In order to remove their field-
strength dependence, following the treatment used in Sec. IV, we introduce the
two dimensionless quantities representing magnetic-field strength, given by

$\displaystyle{}^{(t)}{\cal R}_{M}$ $\displaystyle=$
$\displaystyle{{}^{(t)}B_{\rm max}^{2}R^{4}\over 4GM^{2}}\,,$ (190)
$\displaystyle=$ $\displaystyle{\left({\rm
max}\left[\hat{\rho}\xi\right]\right)^{2}\over\displaystyle
8\left(\left.{d\Theta\over
d\xi}\right|_{\xi=\xi_{1}}\right)^{2}}{\Omega_{A}^{2}\over 4\pi G\rho_{c}}\,,$
$\displaystyle{}^{(p)}{\cal R}_{M}$ $\displaystyle=$
$\displaystyle{{}^{(p)}B_{c}^{2}R^{4}\over 4GM^{2}}\,,$ (191) $\displaystyle=$
$\displaystyle{b^{2}\over\displaystyle
G\left(4\pi\right)^{2}\alpha^{2}\left(\left.{d\Theta\over
d\xi}\right|_{\xi=\xi_{1}}\right)^{2}}\,,$

where ${}^{(t)}B_{\rm max}$ is the maximum absolute value of the toroidal
magnetic field inside the star, ${\rm max}[f]$ means the maximum value of the
function $f$, and ${}^{(p)}B_{c}$ is the absolute value of the poloidal
magnetic field at the center of the star. ${}^{(t)}{\cal R}_{M}$ and
${}^{(p)}{\cal R}_{M}$ are as large as the ratios of the toroidal and the
poloidal magnetic energies to the gravitational energy, respectively.

Table 3:  Global and physical quantities.  $(\Gamma,M/R)$ | $\displaystyle{\Delta M\over M{}^{(t)}{\cal R}_{M}}$ | $\displaystyle{\Delta E_{\rm int}\over E_{\rm int}{}^{(t)}{\cal R}_{M}}$ | $\displaystyle{(\Delta r)_{0}\over R{}^{(t)}{\cal R}_{M}}$ | $\displaystyle{\Delta Q\over MR^{2}{}^{(t)}{\cal R}_{M}}$ | $\displaystyle{e^{*}\over{}^{(t)}{\cal R}_{M}}$ | $\displaystyle{E_{\rm EM}^{(t)}\over|W|{}^{(t)}{\cal R}_{M}}$ | $\displaystyle{E_{\rm EM}^{(p)}\over|W|{}^{(p)}{\cal R}_{M}}$ | $\displaystyle{{\cal H}_{M}\over\sqrt{{}^{(t)}{\cal R}_{M}{}^{(p)}{\cal R}_{M}}}$
---|---|---|---|---|---|---|---|---
(2.05,0) | $-0.2532$ | $-0.8757$ | $0.3692$ | $-0.2107$ | $-0.3161$ | 0.2358 | 0.1134 | 1.386

In this appendix, we numerically obtain the magnetized star model assuming
that $n=1.05$ and $\Gamma=2.05$, which is the Newtonian version of the weakly
magnetized general relativistic star model calculated in this study. Table 3
lists global and physical quantities characterizing the magnetized stars
constructed within the framework of Newtonian magnetohydrodynamics; the
changes in the mass $\Delta M$, the internal energy $\Delta E_{\rm int}$, the
mean radius $(\Delta r)_{0}$, the mass quadrupole moment $\Delta Q$, the
ellipticity $e^{*}$, the toroidal magnetic energy $E_{\rm EM}^{(t)}$, the
poloidal magnetic energy $E_{\rm EM}^{(p)}$, and the dimensionless magnetic
helicity ${\cal H}_{M}$. In this table, all the quantities are normalized to
be nondimensional, as given in the first row.

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