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observations of pulsars have been accumulated since the first discovery by hewish et al .
( @xcite ) , and several new types of pulsars appeared with great surprise .
these observations have partially revealed the structure and evolution of rotating neutron stars .
their rotation periods range from @xmath0ms to several seconds .
the surface magnetic fields range from @xmath1 to @xmath2 .
the upper limit was recently raised by a factor @xmath3 by the discovery of the soft gamma - ray repeaters ( sgrs ) and anomalous x - ray pulsars ( axps ) ( see e.g. kouveliotou et al .
@xcite ; mereghetti & stella  @xcite )
. the new class of pulsars with the very strong magnetic field ( @xmath4@xmath5 ) are most likely candidates of _ magnetars _
( e.g. thompson & duncan  @xcite ) and may be worth studying further . in the future , we may find a more extreme case , that is , a rapidly rotating relativistic star with a strong magnetic field .
most stars have spherically symmetric structure .
they are however deformed due to the rapid rotation and the strong magnetic field .
it is well known that both effects produce a flattening equilibrium star .
we will examine them within general relativity .
these effects are assumed to be small and treated as perturbations to spherically symmetric stars .
the rotational axis of pulsars does not in general coincide with the axis of the dipole magnetic field .
the relativistic treatments for the case is a complicated task , because the situation is not stationary . in this paper , however , we assume that the rotational effect decouples from the magnetic effect .
thus we consider the deformation arising from each perturbation separately and estimate the ellipticity .
the estimate is important to judge which effect dominates in the rotating magnetized stars , whose rotation rate and the magnetic field are in a wide range .
our treatment is beyond the classical estimate in the newtonian gravity , and give better comparison of the rotational effect and the magnetic effect on star deformation for various pulsars . when either of them is huge , our estimate breaks down , and sophisticated numerical codes are required ( see e.g. bocquet et al .
@xcite ; bonazzola & gourgoulhon @xcite ) .
the paper is organized as follows . in sect .  2 , we briefly review deformation of stars due to the rotation ( chandrasekhar @xcite ; chandrasekhar & roberts @xcite ) and the magnetic field ( chandrasekhar & fermi @xcite ; ferraro @xcite ; galtsov et al .
@xcite ; galtsov & tsvetkov @xcite ) within newtonian gravity .
the quadrupole deformation can be evaluated by the ellipticity of the equilibrium shape . in sect .  3
, we also calculate the ellipticity based on the general relativistic perturbation theory ( see also hartle ( @xcite ) and chandrasekhar & miller ( @xcite ) for the rotational cases and konno et al .
( @xcite ) for the magnetic cases ) .
the ellipticity can be summarized in the same form as the newtonian cases , but with different numerical factors . in sect .
4 , using the ellipticity , we compare the rotational effect with the magnetic effect on star deformation numerically . in this comparison
, we keep the parameter range of known pulsars in mind . finally , we give concluding remarks in sect .
5 . throughout the paper
, we use the units in which @xmath6 .
quadrupole deformation of an equilibrium body is characterized by the ellipticity @xmath7 , which is defined by @xmath8 for the gravitational equilibrium with uniform rotation , the value is essentially related to the ratio of the rotational energy to the gravitational energy . in the slow rotation of a homogeneous star , we have a well - known result ( see e.g. chandrasekhar @xcite ) : @xmath9 where @xmath10 , @xmath11 and @xmath12 denote the radius , mass and angular velocity , respectively . for other stellar model , the numerical factor @xmath13
should be replaced by an appropriate one .
for example , the factor is @xmath14 for the model with the polytropic equation of state ( eos ) @xmath15 with index @xmath16 ( see table 1 in chandrasekhar & roberts @xcite ) .
we therefore generalize the expression with a dimensionless factor @xmath17 as @xmath18 and will discuss how the factor @xmath17 depends on the stellar models .    in a similar way ,
the effect of the magnetic stress is also expressed by the energy ratio of the magnetic field to the gravitational field . introducing a dimensionless factor @xmath19
, we have the ellipticity @xmath20 arising from magnetic field as @xmath21 where @xmath22 is the magnetic dipole moment . the dimensionless factor @xmath19 , in general ,
depends on both the magnetic field configurations and the eos .
for example , in the case of an incompressible fluid body with a dipole magnetic field treated by ferraro ( @xcite ) , we derive @xmath23 .
in this section , we will review quadrupole deformation due to the slow rotation and weak dipole magnetic field .
the deformation can be expressed by the second - order quantities with respect to the rotation rate or the magnetic field strength . in order to calculate the shape
, we also have to calculate the space - time metric , which is axisymmetric stationary or static one .
the line element can be written in the form @xmath24
dt^2   \nonumber \\     & & + \ : e^{\lambda ( r ) } \left [ 1 + \frac{2 e^{\lambda ( r)}}{r }             \left\ { m_{0}(r ) + m_{2}(r )             p_{2}\left ( \cos \theta \right)\right\ }             \right ] dr^2             \nonumber \\     & & + \ : r^2 \left ( 1 + 2 k_{2}(r )             p_{2}\left ( \cos \theta \right ) \right ) \nonumber \\     & & \qquad \times \ ;              \left [ d \theta^2              + r^2 \sin^2 \theta ( d \phi - \omega ( r ) dt)^2 \right],\end{aligned}\ ] ] where @xmath25 is the legendre s polynomial of degree @xmath26 , and @xmath27 are the second - order quantities for the rotation rate or the magnetic field strength .
the quantity @xmath28 is the angular velocity acquired by an observer falling freely from infinity to a point @xmath29 , and is equal to zero for the static magnetic field deformation .
the stress - energy tensor of the perfect fluid body is described by @xmath30 when we consider the magnetic field deformation , we further take into account the stress - energy tensor arising from the magnetic field , i.e. , @xmath31 solving the einstein - maxwell equations , we can obtain the second - order metric functions mentioned above .    the ellipticity of the relativistic star can be calculated from the definition ( [ def - e ] ) as @xmath32 where @xmath33 represents the displacement of quadrupole deformation .
since the displacement of the surface can be determined by the hydrostatic equilibrium condition , the ellipticity of the slowly rotating star is expressed as ( chandrasekhar & miller @xcite ) @xmath34    in order to compare it with the newtonian results , we normalize the ellipticity in the same form as eq .
( [ e - newton ] ) .
there are several possibilities of the normalization factors for @xmath11 and @xmath10 in the relativistic calculation .
we use natural choices , i.e. , gravitational mass for @xmath11 and circumferential radius for @xmath10 in this paper .
this normalization is useful to extrapolate from the newtonian results .
chandrasekhar & miller ( @xcite ) used a different normalization , which causes a prominent peak ( see fig .  5 in their paper ) .
we note that the peak is due merely to a less convenient choice of the normalization .
the resultant dimensionless factor @xmath17 is calculated for the polytropic eos with @xmath35 .
[ fig1 ] displays the variation of the dimensionless quantity @xmath17 with respect to the relativistic factor @xmath36 .
this figure shows that the correct relativistic calculations give smaller values of the ellipticity than those of the newtonian calculations with fixed @xmath37 .
the factor in the typical relativistic case decreases down to 0.7 of the newtonian case .    as for the weakly magnetized star ,
the lorentz force plays a role on the equilibrium .
the rotational term @xmath38 is replaced by the magnetic term in the ellipticity . from the hydrostatic condition , we have ( konno et al .
@xcite ) @xmath39 where the subscript ` 0 ' denotes the background quantities , and @xmath40 and @xmath41 are the @xmath42-components of the 4-current and 4-potential respectively . the ellipticity can also be written in the form of eq .
( [ e - mag ] ) .
we also use the gravitational mass @xmath11 and circumferential radius @xmath10 .
[ fig2 ] displays the variation of the ellipticity with respect to the relativistic factor @xmath36 . in these calculations
, we have used the current distribution @xmath43 which is a simplest case derived by the integrability condition ( see e.g. ferraro @xcite ) and gives the direct general - relativistic extension of ferraro ( @xcite )
. using the normalization , the residual factor @xmath19 is almost independent of the relativistic factor @xmath36 .
in order to compare the rotational effect with the magnetic effect on star deformation , we now consider the ratio of @xmath44 to @xmath45 , @xmath46    first , we investigate the dependence of the ratio of the two dimensionless factors , @xmath47 , on the relativistic factor @xmath36 .
[ fig3 ] displays this ratio . from this figure
, we can see that the true relativistic calculations with @xmath48 give smaller values of the ratio than those obtained by the newtonian calculations .
this fact means that the approach using the newtonian gravity overestimates the rotational effect .
the curves @xmath47 are approximately reproduced within 10% if we use @xmath49    next , we consider the comparison including other parameters @xmath12 and @xmath50 of stars .
we use the stellar model with @xmath51 and @xmath52 .
[ fig4 ] displays a critical line on which @xmath53 and the two regions divided by this line in @xmath54-@xmath12 space , where @xmath54 denotes the typical magnetic field strength on the surface , which is defined by @xmath55 .
we have plotted only one representative line of @xmath16 .
we can also derive very close results for other indices . the critical line , in general ,
can be written from eq .
( [ ratio ] ) as @xmath56   \approx 5.3 \times 10^{13 }    \sqrt{\frac{f}{g } } \left ( \frac{m / 1.4 m_{\odot } }   { r / 10 { \rm km } } \right)^{1/2 } \omega \ : [ \mbox{sec}^{-1 } ] , \ ] ] where it is useful to use the fitting formula eq .
( [ fitting ] ) for @xmath47 . in the region
, the magnetic effect dominates the rotational effect , i.e. , @xmath57 , whereas in the region vice versa , i.e. , @xmath58 . from this figure
, we find that objects having magnetic field strength @xmath4@xmath59 g and period @xmath60sec such as sgrs and axps belong to the region .
thus , the magnetic effect overwhelms the rotational effect for such observed candidates of magnetars .    on the other hand ,
well - known typical pulsars with magnetic field strength @xmath61@xmath62 and the period @xmath63@xmath64 ( see e.g. taylor et al .
@xcite ) obviously belong to the region .
millisecond pulsars also belong to the region .
the magnetic deformation is neglected .
the new classes of objects , which are candidates of magnetars , have inspired us to investigate the relation between the rotational effect and the magnetic effect on deformation of stars .
we have briefly reviewed the quadrupole deformation due to the rotation and that due to the magnetic field based on previous studies , and compared the rotational effect with the magnetic effect for various pulsars reported observationally . from our investigation , we have found that the new classes of objects such as sgrs and axps belong to the region in which the magnetic effect dominates the rotational effect , while well - known typical pulsars with magnetic field strength @xmath65@xmath66 g and millisecond pulsars vice versa .
thus , the critical line on which the ellipticity arising from the rotation equals to that arising from the magnetic field divides the new classes and the well - known pulsars .
once we obtain the parameters @xmath12 , @xmath54 , @xmath10 and @xmath11 of a pulsar , we can see whether the magnetic effect is dominant or the rotational effect is dominant using eqs .
( [ fitting ] ) and ( [ c - line ] ) and fig .
the deformation due to the magnetic field may come into play in the spin - down of the magnetars .
the spin - down history of the axps is rather irregular .
recently , melatos ( @xcite ) ascribed the irregularity to the radiative precession produced by the deformation . at present , the fit to the observational data is not so good , but will be improved by detailed models .
theoretical models for the deformed stars will be required there .
as discussed by bonazzola & gourgoulhon ( @xcite ) , the non - axisymmetric distortion is also important for the gravitational emission .
the deformation can be calculated from the magnetic field , which is estimated from the observed pulsar period and the period derivative .
the inferred amplitudes of the gravitational waves are too small for the present known pulsars . in the future
, we might find more extreme case such as the early stage of rapidly rotating magnetars .
we calculated the ellipticity of the deformed star due to the rotation or magnetic field . 
 these two effects are compared to each other within general relativity . 
 it turned out that the magnetic distortion is important for recently observed candidates of magnetars , while the magnetic effect can be neglected for well - known typical pulsars .