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PHYSICAL REVIEW D, VOLUME 64, 123001
R-mode instability of slowly rotating nonisentropic relativistic stars
Shijun Yoshida and Toshifumi FutamaseAstronomical Institute, Graduate School of Science, Tohoku University, Sendai 980-8578, Japan
~Received 17 June 2001; published 30 October 2001!
We investigate the properties ofr-mode instability in slowly rotating relativistic polytropes. Inside the starslow rotation and the low frequency formalism that was mainly developed by Kojima are employed to studyaxial oscillations restored by the Coriolis force. At the stellar surface, in order to take into account thegravitational radiation reaction effect, we use a near-zone boundary condition instead of the boundary condi-tion usually imposed for asymptotically flat spacetime. Because of the boundary condition, complex frequen-cies whose imaginary part represents a secular instability are obtained for discreter-mode oscillations in somepolytropic models. It is found that such discreter-mode solutions can be obtained only for some restrictedpolytropic models. The basic properties of the solutions are similar to those obtained by imposing the boundarycondition for asymptotically flat spacetime. Our results suggest that the existence of a continuous part of thespectrum cannot be avoided even when its frequency becomes complex due to the emission of gravitationalradiation.
DOI: 10.1103/PhysRevD.64.123001 PACS number~s!: 97.10.Sj, 04.40.Dg, 97.10.Kc, 97.60.Jd
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I. INTRODUCTION
Andersson@1# and Friedman and Morsink@2# discoveredthat all r modes, which are quasitoroidal modes mainlystored by the Coriolis force, in all rotating stars becomestable due to the gravitational radiation reaction if other dsipative processes are not considered. This instabilityclearly understood by the so called ChandrasekhFriedman-Shutz~CFS! mechanism@3–5#. As shown first byLindblom, Owen, and Morsink@6#, this instability stillstrongly affects the stability of typical neutron star modeven if viscous dissipation of the neutron star matter, whtends to stabilize the CFS instability, is taken into accouSince then many studies of oscillation modes restored byCoriolis force in rotating stars have been done to prove thpossible importance in astrophysics~for recent reviews, seee.g., Refs.@7–10#!.
The influence of ther mode on the stability of rotatingneutron stars is one of the most important and interesphenomena in astrophysics. In oscillations of neutron stthe relativistic effect must be important because such sare sufficiently compact. But most studies have been dwithin the framework of Newtonian gravity so far, althougour understanding ofr modes has been improved by thoinvestigations. As forr modes studied within the frameworof general relativity, Kojima@11# derived master equationfor r-mode oscillation in the lowest order slow rotation aproximation and found the possible existence of a continupart of the spectrum in his equations. Beyer and Kokko@12# generally verified the existence of a continuous partthe spectrum in Kojima’s equation. Kojima’s formalism wdeveloped to include higher order rotational effectsKojima and Hosonuma@13,14#. Lockitch, Andersson, andFriedman@15# obtained discreter-mode solutions in uniformdensity stars as well as the continuous part of the spectby solving Kojima’s equation. Recently, Yoshida@16# andRuoff and Kokkotas@17# discussed that such discreter-modesolutions are not simply allowed to appear in compressstellar models. Their results showed that, for typical neut
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star models, Kojima’s equations do not have such a discr-mode solution.
These recent developments in understanding of relativir modes have shown that basic properties ofr-mode oscilla-tions in relativistic stars are significantly different from thoin Newtonian stars. As for nonisentropic stars, most previstudies have shown the existence of a continuous part ofspectrum. This is a great contrast with the Newtonian caFor Newtonian cases, there are discrete mode solutionsno continuous parts of the spectrum forr modes in all uni-formly rotating stars as long as their rotation velocitysmall enough. However, such a drastic change in the beior of the solutions is not likely to occur due to the inclusioof even a small relativistic effect. Therefore, most authhave considered that a continuous part of the spectrum dnot appear if some effects that were omitted in previous sties are taken into account. One such effect is the dissipaeffect due to the gravitational radiation reaction. In mostudies onr modes, a slow motion approximation has beemployed in the analysis because of the slow rotationproximation. The slow motion approximation changes twave type equation into a Laplace type equation. Thus, Kma’s equations do not have solutions with wave characand the frequency is a real number if asymptotically flspacetime is assumed. In other words, the influence of grtational radiation on relativisticr modes has not been takeinto account so far. In this case, Kojima’s equation becomthat of a singular eigenvalue problem for some frequenrange, and hence has a continuous part of the spectrumsuggested by Lockitchet al. @15# ~see also Ref.@12#!, how-ever, Kojima’s equation may become that of a regular eigvalue problem if the frequency has a nonzero imaginary pTherefore, it is hoped that the existence of a continuousof the spectrum might be avoided if the effect of the emsion of gravitational radiation is considered.
In this paper, accordingly, we will attempt to include thlowest order effect of the gravitational radiation reaction inKojima’s formalism. Because Kojima’s equation is notwave type equation, as mentioned before, we cannot ob
©2001 The American Physical Society01-1
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SHIJUN YOSHIDA AND TOSHIFUMI FUTAMASE PHYSICAL REVIEW D64 123001
any information about gravitational waves from the equatiIn order to include the gravitational radiation reaction effeinto Kojima’s equation, we will employ a near-zone bounary condition instead of the boundary condition usually usfor asymptotically flat spacetime. This boundary conditiwas introduced by Thorne@18# to include the effect of thegravitational radiation reaction on polar pulsations in a Netonian star. We start, in Sec. II, with the description of omethod of solution. A near-zone boundary condition is intduced there in order to take into account the gravitatioradiation reaction. In Sec. III, we show the properties ofr-mode solutions obtained by imposing the boundary contion. Section IV is devoted to discussion and conclusioThroughout this paper we will use units in whichc5G51,wherec andG denote the velocity of light and the gravitational constant, respectively.
II. METHOD OF SOLUTION
We consider slowly rotating relativistic stars with a unform angular velocityV, where we take account of the firsorder rotational effect inV. The geometry around the equlibrium stars can be described by the following line elem~see, e.g., Ref.@19#!:
ds252e2n(r )dt21e2l(r )dr21r 2du21r 2sin2udw2
22v~r !r 2sin2udtdw. ~2.1!
Throughout this paper, the polytropic equation of stateassumed:
p5K r111/N, ~2.2!
wherep andr denote the pressure and mass-energy denrespectively. HereN andK are constants.
We consider oscillation modes in rotating relativistic stasuch that the eigenfunctions are stationary and are compof only one axial parity component in the limitV→0. Asubclass of these modes should be the relativistic counteof r modes, which are able to oscillate in all slowly rotatinNewtonian fluid stars. According to Lockitchet al. @15#, suchmodes are allowed to exist only if the star has a nonistropic structure. Therefore, we assume the stars to beisentropic, although the effects due to deviation from istropic structure on the oscillation modes do not appear infirst order in V. According to the formalism by Lockitchet al. @15# ~see also Ref.@11#!, let us write down the pulsation equation for relativisticr modes with accuracy up to thfirst order inV. The metric perturbationdgab and the Eule-rian changes of the fluid velocitydua that do not vanish inthe limit V→0 are given by
~dgtu ,dgtw!5 ih0,l~r !S 2]wYlm~u,w!
sinu,sinu]uYlm
3~u,w! Deist, ~2.3!
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~duu,duw!5iU l~r !
r 2
3S ]wYlm~u,w!
sinu,]uYlm~u,w!
sinu Deist, ~2.4!
whereYlm(u,w) are the usual spherical harmonic functionands denotes the oscillation frequency measured in theertial frame at spatial infinity. All other perturbed quantitiebecome higher order inV. Note that this form of eigenfunction is the same as that for zero-frequency modes in a sphcal nonisentropic star, becauser modes become zerofrequency ones in the limitV→0 @20#. The metricperturbationh0,l obeys a second order ordinary differentiequation,
Dlm~r ;s̄ !Fen2ld
dr S e2n2ldh0,l
dr D2S l ~ l 11!
r 21
2212e22l
r 2
18p~p1r!D h0,l G116p~p1r!h0,l50, ~2.5!
where
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R-MODE INSTABILITY OF SLOWLY ROTATING . . . PHYSICAL REVIEW D 64 123001
h0,l~r !5A(k50
`
akr2 l 2k1B(
k50
`
bkrl 111k, ~2.11!
whereA and B are arbitrary constants. Hereak and bk areconstants determined from recurrence relations althoughexplicit expressions are omitted here. Note thatak andbk donot depend on the frequencys̄. In most previous studies thconditionB50 has been chosen as the boundary conditiospatial infinity because spacetime must be regular evwhere. In this paper we call this condition the ‘‘propboundary condition.’’ This condition means the spacetimeasymptotically flat. Therefore, solutions satisfying the contion B50 cannot describe any gravitational radiation emsion from a star at all. In order to include the effect of gratational radiation emission in the solutions, we must nrequire zero forB. This was first shown and used in a stuof the post-Newtonian approximation by Thorne@18#.
Next, let us consider the boundary conditions for quanormal mode solutions, following the considerationsThorne @18# ~see also Ref.@21#!. In the derivation of ourbasic equations the slow motion approximation is necessaused as well as the slow rotation one because the frequof oscillation restored by the Coriolis force is of the samorder as the stellar rotation frequency. This slow motionproximation must work nicely near the star as long as lfrequency oscillations are considered. Therefore, higherder time derivatives of perturbed quantities are neglectethe governing equations~2.5!. If an oscillating star emitsgravitational radiation, however, some of such omitted termust become important in the radiation zone because asuch assr becomes dominant among all the terms in tgoverning equations. In addition, rotational effects duestellar rotation fall off faster than 1/r 2 asr→`. Thus, such aslow motion approximation is not good far from the star,gravitational waves are radiated from the stars. The RegWheeler equations with correction terms due to the sterotation, in fact, govern the axial perturbations sufficienfar from the star even when low frequency oscillations likrmodes are induced. Sincesr @1 in the radiation zone, theRegge-Wheeler equations can be approximately written
r 2d2Xl~r !
dr21s2r 2Xl~r !2 l ~ l 11!Xl~r !1OS M
r D50,
~2.12!
whereXl(r ) are the Regge-Wheeler functions. HereM de-notes the gravitational mass of the star. The metric functih0,l are determined from the functionsXl by the equation
h0,l5d@rXl~r !#
dr1OS M
r D ~2.13!
~see, e.g., Refs.@22,23#!. Here we have considered the limiting case whenM /r !1 for simplicity. The general solutionto Eqs.~2.12! can be given analytically:
Xl~sr !5sr @C jl~sr !1D nl~sr !#, ~2.14!
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where j l andnl are spherical Bessel functions andC andDare arbitrary constants. The asymptotic forms in the radiazone, that is, whensr @1, are given by
Xl~sr !;C cosFsr 21
2~ l 11!p G1D sinFsr 2
1
2~ l 11!pG .
~2.15!
Now we are interested in the quasinormal modes of a sThus, the no incoming radiation conditions are requiredmetric perturbations. From Eq.~2.15! it is found that thiscondition becomes the relationD52 i C. For this choice ofconstants the asymptotic solutions reduce to the form
Xl~sr !eist;C expF is~ t2r !1i
2~ l 11!pG as sr→`.
~2.16!
Since the frequencies of the oscillations restored by theriolis force are proportional to the rotational frequencyV,s2R2'V2R25e2M /R, wheree denotes a small parametefor stellar rotation defined ase5V/(M /R3)1/2. Here,R is theradius of the star. Thus, the surface of the star is apprmately in the near zone, that is,sr !1, if the stellar rotationis sufficiently slow or the stellar gravity is sufficiently weaIn this approximation, the solutions~2.14! can be written as
Xl~sr !;2 i C~2l 21!!!
~sr ! l F11 i~sr !2l 11
~2l 21!!! ~2l 11!!! G ,~2.17!
where the constraintD52 iC for the no incoming radiationhas been used. The approximate solutions above mayvalid near the stellar surface becauseM /r ,1 is well satis-fied at the stellar surface and outside the star for typneutron star models. Thus, in the near zone, the expressfor metric perturbationsh0,l outside the star can be given bEqs.~2.13! and ~2.17! as follows:
h0,l;C81
r l F11 i~ l 12!~sr !2l 11
~ l 21!~2l 11!@~2l 21!!! #2G , ~2.18!
where C8 is an arbitrary constant. As the outer boundacondition we require this solution to connect smoothly to tinterior solution obtained by solving Eq.~2.5! at the stellarsurface. Thus, the boundary condition is explicitly given
F11 i~ l 12!~sR!2l 11
~ l 21!~2l 11!@~2l 21!!! #2G rdh0,l
dr~R2x!
1F l 2 i~ l 11!~ l 12!~sR!2l 11
~ l 21!~2l 11!@~2l 21!!! #2Gh0,l~R2x!
50 as x→0, ~2.19!
where we need actual values of the rotation frequencyV toobtain eigensolutions. In this paper we assume the valuV asV5(pr̄)1/2, wherer̄ is the average density defined b
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SHIJUN YOSHIDA AND TOSHIFUMI FUTAMASE PHYSICAL REVIEW D64 123001
r̄5M /(4pR3/3). This V is an approximate value for thmaximum rotation frequency to be possible that settles douniformly rotating stars in hydrostatic equilibrium. In thpaper, we call the condition~2.19! the ‘‘near-zone boundarycondition.’’ This method is a crude version of matchasymptotic expansions. If we consider a nonrotating limthat is, thes50 limit, the boundary condition~2.19! be-comes an approximation of the proper boundary conditiin which only the lowest order terms inM /R are included.As shown by Lindblomet al. @21#, a boundary conditionsimilar to this, by using the obtained asymptotic soluti~2.18! can give a good approximate value of the eigenfquency even forf-mode oscillation although thef mode isnot a low frequency oscillation mode.
III. NUMERICAL RESULTS
As shown in previous studies@11,12,15#, we should dis-tinguish two cases in treating Eq.~2.5! when the properboundary condition is imposed. One case is the regulargenvalue problem and the other is the singular one. Forregular eigenvalue problem Eq.~2.5! may have discretemode frequencies in the range
2mv̄~R!
l ~ l 11!,s̄<
2mv̄~`!
l ~ l 11!5
2mV
l ~ l 11!. ~3.1!
On the other hand, Eq.~2.5! becomes the singular eigenvaluproblem if s̄ is in the region
2mv̄~0!
l ~ l 11!<s̄<
2mv̄~R!
l ~ l 11!. ~3.2!
Notice that the range~3.2! is the continuous part of the spetrum of Eq.~2.5!. As pointed out by Lockitchet al. @15# ~seealso Ref.@12#!, Eq. ~2.5! becomes that of the regular eigevalue problem when the corresponding frequency has a nzero imaginary part. Thus the frequency ranges abovenot have clear mathematical meanings when the frequehas a nonzero imaginary part. According to the frequeranges above, however, we will divide the eigensolutiointo three classes: The first and the second class solutioncharacterized by the real parts of their frequencies satisfyinequalities~3.1! and ~3.2!, respectively. The third class icomposed of a compensatory set of the first and secclasses.
First of all, we concentrate our attention onr-mode solu-tions with frequencies whose real parts are in the range~3.1!.We compute frequencies of mode solutions for several ptropic stellar models. In the present study, only the funmentalr modes, whose eigenfunctionUm has no node in theradial direction except at the stellar center, are obtained. Tis similar to the result in studies for the proper boundacondition case@16,17#. In Figs. 1 and 2, the real and imagnary parts of the scaled eigenfrequenciesk[s̄/V of the rmodes are, respectively, given as functions ofM /R. Theeigenfrequencies for stars with four different polytropic ind
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n
,
s
-
i-e
n-aycyysareg
nd
--
isy
cesN50, 0.5, 0.75, and 1 are shown in the different panin both figures. Only the frequency curves for the modwith l 5m52 are depicted in dependence on the relativisfactor M /R because they are considered to be the mostportant modes forr-mode instability.
The real parts of the frequency illustrated in Fig. 1 aregood agreement with Fig. 1 of Ref.@16#, in which the fre-
FIG. 1. Real parts of scaled frequenciesk5s̄/V of the r modeswith l 5m52 plotted as functions ofM /R. In each panel, the fre-quencies of modes for polytropic models withN50, 0.5, 0.75, and1 are shown. The labels indicating their polytropic indicesN areattached in the corresponding panels. Vertical dotted lines showmaximum values ofM /R for equilibrium states:M /R50.444 forN50, M /R50.385 for N50.5, M /R50.349 for N50.75, andM /R50.312 forN51.0.
FIG. 2. The same as Fig. 1 but for imaginary parts of frequen
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R-MODE INSTABILITY OF SLOWLY ROTATING . . . PHYSICAL REVIEW D 64 123001
quencies were obtained by imposing the proper boundcondition for asymptotically flat spacetime. The relative dferences are less than 0.1%. This shows that our approxtion works nicely, and higher order effects ofM /r on theouter boundary condition are not so important for the demination of the real parts of the frequency. We also find tthe frequency curves in Fig. 1 are terminated at some vaof M /R beyond which equilibrium states can still exist. Hethe maximum values ofM /R for polytropic equilibrium starshaving N50, 0.5, 0.75, and 1.0 are given by 4/9, 0.380.349, and 0.312, respectively. It is also found thatlengths of the frequency curves tend to become shorter apolytrope indexN increases. This feature is similar to that fthe case where the proper boundary condition is used. Athe terminal points of the frequency curves appear at almthe same values of relativistic factors as those for the proboundary condition case~see Ref.@16#!. Beyond the value ofthe relativistic factors corresponding to those terminal poinwe can obtain many eigensolutions with a singular eigfunction but not with a regular one. Furthermore, the rpart of the corresponding frequency belongs to the ra~3.2! but not the range~3.1!.
From Fig. 2, we can see that ther modes obtained in thisstudy are all unstable. It is also found that the curves forimaginary parts ofk have one relative minimum near thterminal point of the frequency curves. These minimum vues are given by Im@k(M /R50.425)#522.931023,Im@k(M /R50.344)#526.431024, Im@k(M /R50.258)#521.331024, and Im@k(M /R50.095)#522.231026, forstars havingN50, 0.5, 0.75, and 1.0, respectively. Thvalue of Im(k) also approaches zero as the relativistic facM /R gets closer to a value corresponding to that forterminal point of the frequency curves. Those behaviorsIm(k) can be understood from the distribution of eigenfuntions Ul(r ) because Im(k) is approximately proportional tothe square of the current multipole moment. In Figs. 3 anthe distributions of the eigenfunctionUm are shown forN
FIG. 3. Absolute value of eigenfunctionU2 of an r mode withl 5m52 for an N50.5 polytrope withM /R50.1 is given as afunction of r /R, where normalization of the eigenfunction is giveby U2(R)51.
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50.5 polytropic models havingM /R50.1 andM /R50.37,respectively. As seen from these figures the motion of pturbed fluid elements is strongly confined near the stesurface when the mode frequency becomes closer to theminal frequency, which satisfiess̄'2mv̄(R)/@ l ( l 11)#. Wecan easily understand this behavior from Eq.~2.9!. Conse-quently, the values of current multipole moments of sumodes may become small when the value of the relativifactor increases. On the other hand, the efficiency ofgravitational radiation emission becomes good with increof the value of the relativistic factor. Due to both effecabove a relative minimum of the imaginary part of frequenmay appear. We should note that in theN50 case the valueof Im(k) does not approach zero even whenM /R;4/9, aswe can see from Fig. 2. The reason is that the eigenfunctUm do not have so strong a peak at the stellar surface ewhen M /R;4/9 because the relations̄'2mv̄(R)/@ l ( l11)# is not satisfied in this case.
Next let us estimate the instability time scale for tgravitational radiation driven instability ofr-mode solutionswith frequency in the range~3.1!. Here a typical neutron stamodel whose mass and radius are, respectively, 1.4M ( and12.57 km is considered forN50, 0.5, and 0.75 polytropicmodels. When the star rotates with angular frequencyV5(3/43GM/R3)1/258377 s21, the growth time scalest j ofthe r-mode instability are given byt j51.29 s, 2.04 s, and2.97 s for theN50, 0.5, and 0.75 models, respectively. Nothat for theN51 case the growth time scale cannot be emated because we cannot find a discreter-mode solution forthat model. These time scales are similar to those obtafrom the Newtonian estimate~see, e.g., Ref.@6#!.
When we concentrate our attention on solutions whfrequency has a real part in the frequency range~3.2!, weobtain a large number of solutions whose eigenfunctionsingular behavior in its real part at a radius determined bysolution of Dlm@r ;Re(s̄)#50. In addition, those solutionshave severe truncation errors due to the finite differemethod. A similar behavior of solutions appears in osciltions of differentially rotating disks~see, for example, Ref
FIG. 4. The same as Fig. 3 but forM /R50.37.
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SHIJUN YOSHIDA AND TOSHIFUMI FUTAMASE PHYSICAL REVIEW D64 123001
@24#!. As discussed by Schutz and Verdaguer@24#, this isconsidered a sign of the existence of continuous parts ofspectrum, although the existence has to be proved by omathematical techniques because an exact continuoustrum is never obtained from a simple numerical analysisthe present case, we are sure that the appearance of atinuous spectrum is plausible because the imaginary parthe frequency are too small to drastically change the chater of the solutions derived from the proper boundary contion. Thus, our numerical results suggest the existencecontinuous part of the spectrum in Kojima’s equation evwhen the frequency becomes a complex number. Asregular solutions with frequency in the range~3.2!, we can-not obtain such a solution at all. Finally, we consider tthird class of solutions, for which the real part of the frquency is in neither the range~3.1! nor ~3.2!. In this regionof real parts of the frequencies, we cannot obtain any stions at all.
IV. DISCUSSION AND CONCLUSION
In this paper, we have investigated the propertiesr-mode instability in slowly rotating relativistic polytropesInside the star, slow rotation and the low frequency formism that was mainly developed by Kojima@11# and Lockitchet al. @15# is employed to study axial oscillations restoredthe Coriolis force. At the stellar surface, in order to take inaccount the gravitational radiation reaction effect, we usnear-zone boundary condition, which was devised by Tho@18# and recently developed for relativistic pulsationsLindblom et al. @21#, instead of the usually imposed bounary condition for asymptotically flat spacetime. Due to tboundary condition, complex frequencies whose imaginpart represents secular instability are obtained forr-modeoscillations in some polytropic models. It is found that sudiscreter-mode solutions can be obtained only for somestricted polytropic models. The basic properties of modelutions of Eq.~2.5! that are obtained in this study are similto those with the boundary condition for asymptotically flspacetime although their frequency becomes complexcause of the near-zone boundary condition.
As suggested by Lockitchet al. @15# ~see also Refs
tt.
pl
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@12,16,17#!, when an eigenfrequency becomes a compnumber, which expresses the damping of the oscillationto energy dissipation such as the gravitational radiation emsion, there is the possibility that the existence of the contious part of the spectrum in the eigenfrequency of Eq.~2.5! isavoided. In this study, we consider the complex frequencorresponding to the quasinormal mode as the mode soluof Eq. ~2.5! by imposing the lowest order near-zone bounary condition. Our numerical results suggest the existenca continuous part of the spectrum in Kojima’s equation evwhen the frequency is allowed to be a complex numbHowever, we still think that the existence of a continuopart of the spectrum in axial oscillations restored by the Criolis force is not plausible because such a property doesappear in Newtonianr modes. The existence of a continuopart of the spectrum in this study might be an artifact duethe approximation because our treatment is the lowest oapproximation. In other words, the inclusion of the full effeof gravitational radiation emission might avoid the existenof a continuous part of the spectrum. Another possibilityprevent the appearance of a continuous spectrum might bavoid a singular reduction in the order of the equation.Kojima and Hosonuma@14# showed, the basic equations for-mode oscillations become a fourth order ordinary differetial equation for the metric perturbationh0,l when rotationaleffects up to the third order ofV/(M /R3)1/2 are consistentlyconsidered. In this equation, the extra two degrees of frdom of solutions may be used to avoid singular behaviorthe eigenfunction. Because of these extra boundary cotions, all eigenfrequencies may become discrete. Verificaof those possibilities remain for future studies.
ACKNOWLEDGMENTS
The authors would like to thank Professor B. F. Schutzhis hospitality at the Albert Einstein Institute, where a partthis work was done. S.Y. is grateful to Professor Y. Kojimand M. Hosonuma for useful discussions. He also would lto thank Dr. U. Lee and Y. Itoh for valuable comments. Sacknowledges support from the Japan Society for the Protion of Science.
c.
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