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radial oscillations of non - rotating relativistic stars have been studied for over thirty years .
methods for obtaining their spectra have been well established ( bardeen , thorne & meltzer  1966 ; see also chapter 26 of misner et al .
1973 and the references therein ) , and have been applied to several equations of state ( see for example meltzer & thorne  1966 ) .
these works were mainly motivated by consideration of stellar stability because general relativistic effects tend to destabilize stellar models @xcite .    on the other hand ,
the effect of rotation on stellar oscillations is less well understood in a general relativistic context . as in the non - axisymmetric mode case ,
the slow rotation approximation has been the only accessible way for investigating the eigenmode behaviour of rotating stars @xcite .
recently , numerical relativistic hydrodynamic codes have been developed by several authors and some numerical simulations of rapidly rotating stars have been carried out .
stergioulas et al .   and font et al .
have shown that initial small perturbations around an equilibrium star evolved to a superposition of normal mode oscillations ( note that their hydrodynamic simulation is done in the fixed background spacetime .
on the other hand , shibata et al .   have solved the full system of einstein equations to investigate the dynamical stability of rapidly rotating stars . ) .
although the excitation and evolution of these modes in realistic situations should be investigated by time dependent hydrodynamic simulations , it is also important to study the mode behaviour along rotational equilibrium sequences by using linear perturbation theory .
so far we have studied a few sets of non - axisymmetric eigenmodes of rotating stars in general relativity ( f - modes by yoshida & eriguchi  1997,1999 ; p - modes by yoshida  1999 ( unpublished ; presented at the 9th yukawa international seminar black holes and gravitational waves - new eyes in the 21st century- ) ) .
these results have been obtained within the cowling approximation in which euler perturbations of the metric coefficients have been neglected ( see mcdermott et al .
1983 and finn  1988 for a definition ; see lindblom and splinter  1990 for the accuracy of the method when applied to non - radial modes of spherical stars ) .
apart from some low order modes , these results are in good agreement with those of the full perturbation theory including metric perturbations ( for comparison of the eigenfrequencies for slowly rotating stars , see yoshida & kojima  1997 ; for comparison of the neutral points of the cfs instability , see yoshida & eriguchi  1999 which compare the results with the one obtained by full computation of stergioulas & friedman  1998 and morsink et al .
1999 )    it is therefore natural to expect that the cowling approximation could also be successfully applied to the _ quasi - radial _ modes which are the smooth extensions of the radial modes of spherical stars to rotating stars . in the present paper , we study quasi - radial modes by using the cowling approximation .
contrary to the expectations , our results indicate that _ computations with this approximation can not reproduce the relativistic instability of spherical stars .
_ this is plausible because the instability is essentially caused by the loss of balance between gravity and the pressure gradient , and in calculations of it even the small corrections of gravity can not be neglected . moreover
, the phase cancellation of the perturbed gravitational potentials , which may be effective in the case of non - axisymmetric modes , can not be expected to happen for radial modes .
see the appendix of the present paper for the comparison of two methods in the case of radial modes of non - rotating stars .
although the validity of cowling approximation for rotating stars is not fully assessed , we here _ expect _ that at least a qualitative picture of the eigenmode dependence on stellar rotation could be studied by this approximation .
the equation of state used here is the polytropic one , @xmath0 where @xmath1 , @xmath2 and @xmath3 are the rest mass density , energy density and pressure of the stellar matter , respectively .
geometrized units , @xmath4 , are adopted in this paper as well as @xmath5 , following font et al .  .
the constant @xmath6 is the polytropic index .
the adiabatic exponent of the perturbed matter is assumed to coincide with @xmath7 .
the factor @xmath8 is another constant .
each equilibrium sequence is computed with @xmath8 and @xmath6 fixed .    in the present study
polar - like coordinates are used and the metric components are written as : @xmath9 the rotational axis is located at @xmath10 .
the coordinates used in the actual numerical computation are surface - fitted ones @xmath11 which are defined as : @xmath12 where @xmath13 is the form of the stellar surface in equilibrium .
the numerical method used here is basically the same as that in yoshida & eriguchi   where non - axisymmetric modes were investigated .
a minor modification is needed to obtain the axisymmetric modes . in the case of non - axisymmetric modes ,
the eulerian variable @xmath14 is explicitly set to zero at the centre of the star ( @xmath15 is the eulerian variation of the pressure ) . in the case of axisymmetric modes , however , this is not the case since the regularity of the solution requires @xmath16 to be zero at the stellar centre .
therefore we simply modify the finite - difference scheme at the innermost grid points in our numerical code .
moreover to avoid the coordinate singularity on the rotation axis , points on the axis are excluded from the computational region .
most of the results shown in this paper are computed with a resolution of 40 uniformly distributed gridpoints in the radial @xmath17 direction and 10 in the angular @xmath18 one .
the computational region is a quarter of the meridional section of stars , thus the range of the radial and the angular coordinates are @xmath19 , @xmath20 .    in figs .
[ n05freq ] and [ n15freq ] the eigenfrequencies of the axisymmetric modes are plotted against the rotational frequency of the equilibrium model .
the model parameters are tabulated in table [ eqmodel ] .
.parameters of the stellar model . here
@xmath21 is the rest mass density at the stellar centre , which is fixed as constant along the sequence . @xmath22 and @xmath23 are the gravitational mass and the mass - to - radius ratio , where @xmath24 is the circumferential radius . [ cols="^,^,^,^,^,^ " , ]     [ compare1d ]      comparing with the eigenfunctions of non - axisymmetric modes , we notice that the eigenfunctions of quasi - radial modes change their shapes significantly along rotational equilibrium sequences .
for example , the radial distributions of the eulerian pressure perturbation and the radial component of the velocity perturbation change considerably near the equatorial plane of the star . the number of radial nodes of these functions increases as the stellar rotation rate increases . on the other hand ,
the overall radial dependence of the @xmath25-component of the velocity perturbation changes little as the star spins up .
therefore we can use the shape of this function as a tracer of the selected mode along the equilibrium sequence .
the nomenclature of mode sequences is based upon the behaviour of the mode of a non - rotating star to which the sequence is continued smoothly .
in addition to the polytropic case presented here , we have tried to compute quasi - radial modes of realistic neutron stars by using some of the candidate zero temperature equations of state .
however it was rather difficult to obtain full sequences of these modes with our method .
as the star begins to rotate , the convergence to the quasi - radial modes suddenly becomes much more difficult
. this may partly be due to the fact that these modes are sensitive ( as compared with the non - radial modes ) to the surface condition of the equilibrium star .
unfortunately , in the case of more realistic eos , the adiabatic exponent is subject to large oscillations , becoming negative in some parts .
these large variations decrease considerably the accuracy of our method which then becomes inadequate .
we thank luciano rezzolla for detailed comments on the manuscript , and john miller for his help to improve it .
we also thank kji ury for useful discussions .
bardeen j. m. , thorne k. s. , meltzer d. w. , 1966 , apj , 145 , 505 chandrasekhar s. , 1964 , phys .
, 12 , 114 , 437 clement m. j. , 1986 , apj , 301 , 185 finn l. s. , 1988 , mnras , 232 , 259 font j. a. , stergioulas n. , kokkotas k. , 2000 , mnras , 313 , 678 fowler w. a. , 1964 , rev .
36 , 549 and 1104 hartle j. b. , friedman j. l. , 1975 , apj , 196 , 653 lindblom l. , splinter r. j. , 1990 , apj , 348 , 198 mcdermott p. n. , van horn h. m. , sholl j. f. , 1983 , apj , 268 , 837 meltzer d. w. , thorne k. s. , 1966 , apj , 145 , 514 misner c. w. , thorne k. s. , wheeler j. a. , 1973 , gravitation .
freeman , san francisco morsink s. , stergioulas n. , blattnig s. r. , 1999 , apj , 510 , 854 shibata m. , baumgarte t. w. , shapiro s. l. , 2000 , phys . rev . , d61 , 44012 stergioulas n. , font j. a. , kokkotas k. , 2000 , in aubourg , e. , montmerle , t. , paul , j. peter , p. , eds , proceedings of the 19th texas symposium on relativistic astrophysics , nucl
, 80 ( in cd - rom version of the proceedings , numbered 07/24 ) stergioulas n. , friedman j. l. , 1998 , apj , 492 , 301 unno w. , osaki y. , ando h. , saio h. , shibahashi h. , 1989 , nonradial oscillations of stars , second edition , university of tokyo press , tokyo yoshida si .
, eriguchi y. , 1997 , apj , 490 , 779 yoshida si . , eriguchi y. , 1999 , apj , 515 , 414 yoshida sj . ,
kojima y. , 1997 , mnras , 289 , 117
to test the validity and accuracy of the cowling approximation for a spherical configuration , we here present a comparison between the results obtained by the full perturbation theory and by the cowling approximation for low order radial modes .
the metric coefficients , stellar pressure @xmath3 and energy density @xmath2 are obtained by integrating the standard set of equations with regular boundary conditions at the stellar centre : @xmath27 @xmath28 @xmath29 where @xmath30 is defined from @xmath31 .
the equation of motion of the displacement ( @xmath35 ) in the full perturbation theory is : @xmath36 where @xmath37 and @xmath38 are defined as : @xmath39 and @xmath40 .
\eqno ( a7)\ ] ] the prime after a variable refers its derivative with respect to @xmath41 . here
@xmath42 is the frequency and the adiabatic exponent @xmath43 is defined by : @xmath44 where @xmath45 represents the lagrangian perturbation . in general the adiabatic exponent need not coincide with @xmath7 .
these equations can be solved by the matching method : i.e. , we have to search for @xmath52 which makes the wronskian of the solutions , obtained by integrations from the stellar centre and from the surface , vanish at some matching point inside the star .              in figs .
a1 and a2 we show the typical sequences of the three lowest order modes . for all of the sequences ,
the eigenfrequency obtained by the cowling approximation is larger than that from the full theory : i.e. , the cowling approximation _ overestimates the stability _ of the star . before the turning point
, the two curves are nearly parallel . as expected , the relative error of the results obtained by the cowling approximation becomes smaller for higher overtones .
by using the cowling approximation , quasi - radial modes of rotating general relativistic stars are computed along equilibrium sequences from non - rotating to maximally rotating models . 
 the eigenfrequencies of these modes are decreasing functions of the rotational frequency . 
 the eigenfrequency curve of each mode as a function of the rotational frequency has discontinuities , which arise from the _ avoided crossing _ with other curves of axisymmetric modes .    
 relativity  stars : rotation  stars : oscillations