Quasi-toroidal oscillations in slowly rotating stars are examined in the
framework of general relativity. The oscillation frequency to first order of
the rotation rate is not a single value even for uniform rotation unlike the
Newtonian case. All the oscillation frequencies of the r-modes are purely
neutral and form a continuous spectrum limited to a certain range. The allowed
frequencies are determined by the resonance condition between the perturbation
and background mean flow. The resonant frequency varies with the radius
according to general relativistic dragging effect.Comment: 4 pages, Latex file without figure, submitted to Mon. Not. R. Astron.
  So

Kojima, Yasufumi

English

arXiv

Quasi-toroidal oscillations in slowly rotating stars are examined in the framework of general relativity. The oscillation frequency to first order of the rotation rate is not a single value even for uniform rotation unlike the Newtonian case. All the oscillation frequencies of the r-modes are purely neutral and form a continuous spectrum limited to a certain range. The allowed frequencies are determined by the resonance condition between the perturbation and background mean flow. The resonant frequency varies with the radius according to general relativistic dragging effect

Kojima, Y.

MPG.PuRe

Quasi-toroidal oscillations in rotating relativistic stars

Y. Kojima

Crossref

arXiv:gr-qc/9709003v1  2 Sep 1997Mon. Not. R. Astron. Soc. 000, 000–000 (199?) Printed 24 January 2008 (MN LATEX style file v1.4)Quasi-toroidal oscillationsin rotating relativistic starsYasufumi KojimaDepartment of Physics, Hiroshima University,Higashi-Hiroshima 739, JapanAccepted ? Received ?; in original form ?ABSTRACTQuasi-toroidal oscillations in slowly rotating stars are examined in the framework ofgeneral relativity. The oscillation frequency to first order of the rotation rate is not asingle value even for uniform rotation unlike the Newtonian case. All the oscillationfrequencies of the r-modes are purely neutral and form a continuous spectrum limitedto a certain range. The allowed frequencies are determined by the resonance conditionbetween the perturbation and background mean flow. The resonant frequency varieswith the radius according to general relativistic dragging effect.Key words: oscillation, neutron star1 INTRODUCTIONIn recent X-ray observation with the Rossi X-Ray TimingExplorer (RXTE), quasi-periodic oscillations are discoveredin several sources (e.g., van der Klis et al. 1996 and the sub-sequent papers of the volume). The frequency ranges froma few Hz to kHz and may be attributed to the phenomenanear a compact object. Several models are proposed as theoscillations: beat-frequency between a magnetized neutronstar and accretion disc, stellar oscillation, and so on. For ex-ample, Strohmayer and Lee (1996) considered the excitationof the g- and r-modes as a result of the thermonuclear flashand discussed the observational possibility. Their calcula-tions are however based on the Newtonian gravity. Unlikethe spheroidal modes like f-, p-, and g-modes, the generalrelativistic effects are not clear for the r-mode, since theproblem has never been studied so far. The toroidal motionis trivial in a non-rotating star, but has non-vanishing fre-quency in a rotating star. The quasi-toroidal mode is calledr-mode and known as the Rossby wave in ocean. Papaloizouand Pringle (1978) introduced the r-mode in connection withthe variable white dwarfs. See also the subsequent study(Provost, Berthomieu & Rocca 1981; Saio 1982).In this paper, we will explore the relativistic effect onthe r-mode. We never discuss the observational implicationof the r-mode in relativistic stars, but theoretical study ofthe oscillation frequency may be a useful tool for the futureobservation. We use the slow rotation approximation andlinearized Einstein equations. The first-order effect of therotation rate is taken into account. In section 2, we presentthe perturbation equations describing the r-mode. In section3, the eigenvalue problem is solved. Finally, section 4 is de-voted to the discussion. Throughout this paper we will useunits of G = c = 1.2 PERTURBATION EQUATIONSWe assume a star with a uniform angular velocity Ω ∼ O(ε),and consider the rotational effect of order ε only. The con-figuration of the pressure p and the density ρ is the same asin the non-rotating star, since the centrifugal force deform-ing the shape is of the order ε2. The metric for the slowlyrotating star is given by (Hartle 1967)ds2 = −eνdt2 + eλdr2 + r2(dθ2 + sin2 θdφ2)−2ωr2 sin2 θdtdφ, (1)where ω ∼ O(ε) is a radial function describing the draggingof the inertial frame. Introducing a function ̟ = Ω−ω, wehave a differential equation as(jr4̟′)′− 16π(ρ+ p)eλjr4̟ = 0, (2)where a prime means a derivative with respect to r, andj = e−(λ+ν)/2. (3)The function ̟ inside the star is monotonically increasingfunction of r, so that the range is limited to̟0 ≤ ̟ ≤ ̟R, (4)where ̟0 and ̟R are the values at the center and surface(r = R), respectively.The perturbations describing non-radial oscillationswith the small amplitude can be given by the density per-turbation δρ, pressure perturbation δp, and three compo-nents of the velocity (U,R,V ). The metric perturbationsc© 199? RAS2 Y. Kojimacan be expressed by the ten functions, but the numberis reduced to six (h0, h1, H0,H1,H2,K) by the gauge fix-ing. We here use the same notation for these perturba-tion functions as in Kojima (1992), but the explicit formsare not necessary for most of the following discussion. Inthis way, the equations governing the oscillations are onethermo-dynamic relation and ten components of the lin-earized Einstein equations for these eleven functions. In thecase of non-rotation, two sets are completely decoupled. Oneset (U,h0, h1) is called axial perturbation (or “odd-parity”mode), while the other set (δρ, δp,R, V,H0,H1,H2,K) polarperturbation (or “even-parity” mode). Notice that the axialperturbation U describes the toroidal motion, and has zerofrequency in the non-rotating star (Thorne & Campolattro1967). We expect that with rotation the toroidal oscillationsof the fluids have finite frequencies of the order of Ω likein the Newtonian pulsation theory (Papaloizou & Pringle1978; Provost, Berthomieu & Rocca 1981; Saio 1982). Therealso exists gravitational wave mode with non-vanishing fre-quency in the axial perturbation (Chandrasekhar & Ferrari1991; Kokkotas 1994). The mode can be distinguished fromthe r-mode in the non-rotating limit. We never discuss thegravitational wave mode any more here.We look for the r-mode oscillations in the relativisticrotating stars. The perturbation functions are expanded byappropriate sets of spherical harmonics with index l,m andexp[−i(σt −mφ)]. The linearized Einstein equations in theslowly rotating star are schematically given byAlm + EA × P˜l±1m = 0, (5)Plm + EP × A˜l±1m = 0, (6)where A and A˜ represent some sets of the axial perturbationfunctions, while P and P˜ represent those of the polar per-turbation functions (Kojima 1992). The symbols, EP and EAare some operators of the order ε. It is clear that the presenceof the rotation induces the couplings between the axial andpolar modes. The coupling is subject to the selection rule:the axial mode with l,m is coupled with the polar modeswith l ± 1,m and vice versa. This rule is easily understoodif we notice that the slow rotation perturbation correspondsto the odd-parity perturbation with l = 1 (Campolattro &Thorne 1970).In the previous papers (Kojima 1992, 1993a,b), the pul-sation equations in the slowly rotating stars are examined,assuming that the oscillation frequencies in the non-rotatingstars are regarded as non-zero values. This is true for thespheroidal modes like f-, p-, g-modes and gravitational wavemodes. The eigenvalue problems are solved for the non-rotating stars, and the rotational corrections are calculatedfor these oscillation modes. We instead assume that the fre-quency σ is of the order of ε. Different manipulation is there-fore necessary, since the rotation should be included at thelowest order to obtain the oscillation frequency of the r-mode. The perturbation functions should be ordered in themagnitude as,h0 ∼ O(U), h1 ∼ O(ε1U),δρ ∼ O(ε1U), δp ∼ O(ε1U),H0 ∼ O(ε1U), H2 ∼ O(ε1U), K ∼ O(ε1U),H1 ∼ O(ε2U), R ∼ O(ε2U), V ∼ O(ε2U).(7)The velocity perturbations and the metric perturbationswith htj are ’anti-symmetric’ with respect to time and oth-ers like density perturbations are ’symmetric’. The formershould therefore have even power of ε, while the latter oddpower of ε. The polar perturbation functions should beof higher order in the quasi-toroidal oscillations. From theabove ordering, the linearized Einstein equations correspondto those for the axial part, Alm = 0 at the lowest order. Thepolar part is induced through the coupling, equation (6) athigher order level. The corrections to the axial part are alsoinduced at higher order level. In proceeding to the higherorder, the higher order rotational corrections for the equi-librium states are necessary, but the corrections of the orderε are sufficient at the lowest order.We now solve Alm = 0 for U, h0, h1. The quasi-toroidalvelocity can be expressed as(σ −mΩ+2m̟l(l + 1))U = −4π(σ −mΩ)(ρ+ p)e−νh0. (8)The relation between the metric perturbations ish1 = −ir4e−ν(l − 1)(l + 2)[(σ −mΩ)Φ′ +2mω′l(l + 1)Φ], (9)whereΦ =h0r2. (10)The master equation governing the quasi-toroidal oscilla-tions can be written as(̟ − µ)[1jr4(jr4Φ′)′− vΦ]= qΦ, (11)wherev =eλr2[l(l + 1)− 2] , (12)q =1jr4(jr4̟′)′(13)= 16π(ρ+ p)eλ̟, (14)and the eigenvalueµ = −l(l + 1)2m(σ −mΩ). (15)In equation (14), we have used the relation (2).3 SINGULAR EIGENVALUE PROBLEMThe basic equation (11) is not a regular eigenvalue problem.The coefficient (̟− µ) becomes singular inside the star fora certain value of µ. The coefficient also vanishes outside thestar, but the singularity can be removable because q = 0.This equation is very analogous to the Rayleigh’s equationfor the incompressible shear flow (e.g., Lin 1955). The per-turbation propagating with the wave number k and speed cin the mean flow with velocity u can be described as(u− c)[Φ′′ − k2Φ]= u′′Φ. (16)The similar singular eigenvalue problems appear in manyother fields, e.g., differential rotating fluid discs and plasmaoscillations. See e.g., Balmforth & Morrison (1995) for themethods of solving the singular eigenvalue problem. The sin-gular point is called as critical layer in the fluid dynamics,or co-rotating point in the rotating discs. The studies ofc© 199? RAS, MNRAS 000, 000–000Quasi-toroidal oscillations 3the singular eigenvalue problem indicate that unless thereis an inflection point, u′′ = 0, somewhere within the flow,the eigenvalue is not discrete, but continuous and neutralagainst the stability. All neutral modes must have criticallayers (co-rotation points) that lie within the flow, and there-fore form a continuous spectrum of intrinsically irregulareigen-functions.The parallel argument holds for our problem. The es-sential points are that the potential v is positive definite forl > 2 and that there is no inflection point (q > 0) inside thestars. We can conclude that the eigenvalue of equation (11)is real number and the range is limited to̟0 < µ = −l(l + 1)2m(σ −mΩ) < ̟R, (17)where the range of ̟, (4) is used.We shall simply show the conclusion by reductio ad ab-surdum. If there is a non-trivial solution of which the eigen-value µ is not located within the domain (17), then we havethe integral relation0 =∫∞0(∣∣Φ′∣∣2 + v |Φ|2) jr4dr+∫ R01̟ − µq|Φ|2jr4dr, (18)where we have assumed that the function Φ tends to zeroboth at the center and the infinity. The imaginary part ofequation (18) gives0 = ℑ(µ)∫ R01|̟ − µ|2q|Φ|2jr4dr. (19)Since q is positive definite for 0 ≤ r < R as seen in equation(14), we have ℑ(µ) = 0, except the trivial case Φ = 0. Thatis, the eigenvalue µ must be real number. In a similar way,we introduce Φ = (̟−µ)f to have another integral relation,0 =∫∞0(̟ − µ)2(∣∣f ′∣∣2 + v |f |2) jr4dr. (20)The function within the integral is positive at least for 0 ≤r < R. We therefore have the contradiction.If the eigenvalue is located within the domain (17), theeigen-function has the singular point, say, r∗, inside thestar. The function is approximated by the delta-functionas f ∼ δ(r − r∗). The quasi-toroidal fluid velocity has theform U ∼ δ(r − r∗) from the equation (8). The functionrepresents a steep resonance between the perturbation andthe mean flow. The resonance may be more clear, if we con-sider so-called the Cowling approximation. In the Newtonianpulsation theory, gravitational perturbations are sometimesneglected in the oscillations. This trick gives good resultsfor the spheroidal mode as well as the r-modes. The rela-tivistic Cowling approximation is given by δTαβ;β = 0 withδgαβ = 0. One component of the equations is reduced in theslow rotation case to(̟ − µ)U = 0. (21)It is clear that the solution of this equation is U ∼ δ(r−r∗),and that the range of eigenvalue is given by equation (17).Finally, we consider the Newtonian limit, in which ̟ →Ω. The frequency therefore corresponds to a single value asσ =(1−2l(l + 1))mΩ. (22)This is the frequency of the r-mode oscillation measured byinertial frame.4 DISCUSSIONIn this paper, the r-mode oscillation is examined as theconsistent first-order solution to the quasi-toroidal motion.The frequency forms a continuous spectrum. The oscilla-tion is caused by a certain resonance between the perturba-tion and the background rotating flow. The resonance con-dition is that the co-rotating frequency, (σ − mΩ)e−ν/2 ofthe wave should be −2m/(l(l+1)) times the angular veloc-ity, ̟e−ν/2 measured by ZAMO (zero-angular-momentum-observer). The angular velocity depends on the position ofthe local inertia frame due to the dragging effect. In this way,the r-mode oscillations in relativistic stars are much analo-gous to those in the differential rotating discs, although theangular velocity, Ω is uniform. The mechanism works every-where within the rotating star, but the resultant frequency,σ measured at infinity is not identical. This is the physicalmeaning of the continuous spectrum of the r-mode.The eigen-function of the Newtonian r-mode is not de-termined to first-order of the rotation, since any functionsfor the same µ satisfy the equation governing the oscillation,(µ− Ω)U = 0. (23)The modes are degenerated in this sense. In order to deter-mine the radial structure of the r-modes, calculation of thenext order is necessary. The higher order corrections to thefrequency will remove the degeneracy. As for the relativis-tic r-mode, the frequencies are distinguished correspondingto the resonance points. All the positions are on an equalfooting to the first-order of the rotation. Therefore, the nor-mal frequency forms a continuous spectrum. We expect thatsome favored resonance points are selected as a result of thehigher order corrections. That is, the axial part of the firstorder drives the density and pressure perturbations at thesecond order. The gravitational radiation may be also as-sociated with the density perturbations. The polar pertur-bations react on the frequency at the third order. The in-ternal structure will strongly affect the modes through thecoupling. The relevant second-order rotational corrections∼ O(ε2) like rotational deformation are of course necessaryto solve the problem. The study beyond the first-order ofthe rotation is very important not only for the radial struc-ture, but also for the stability, although the calculations aresignificantly complicated.The frequency at the first order is a real number, andthe mode represents standing ripple in the rotating flow. Thewave will decay or grow due to the dissipation. The grav-itational radiation reaction and/or the viscosity cause theinstability of spheroidal modes in the rotating star. Simi-lar instability may set in for the r-mode, according to thegeneral argument (Friedman & Schutz 1978; Friedman &Morsink 1997). Recent numerical calculation suggested theinstability of the r-mode (Andersson 1997). However, theseworks are not in agreement as for the growth rate, which ishigher order consequence of εn(n ≥ 2).c© 199? RAS, MNRAS 000, 000–0004 Y. KojimaIn conclusion, the second-order effect to the r-mode os-cillation in the relativistic star is complicated, but quite in-teresting problem.ACKNOWLEDGMENTI would like to acknowledge the hospitality of the AlbertEinstein Institute in Potsdam where most of this work wasdone. I also thank N. Andersson, K.D. Kokkotas, and B.F.Schutz for their discussion. This was supported in part bythe Grant-in-Aid for Scientific Research Fund of the Min-istry of Education, Science and Culture of Japan (08640378).REFERENCESAndersson N., 1997, submitted to ApJ, preprint gr-qc/9706075Balmforth N.J., Morrison P.J., 1995, in Waves in Astrophysics ,Annals of the New York Academy of Science, Vol. 773, NewYork, p.80Campolattro A., Thorne K.S., 1970, ApJ, 159, 847Chandrasekhar S., Ferrari V., 1991, Proc. R. Soc. Lond. A434,449.Friedman J.L., Schutz B.F., 1978, ApJ, 222, 281Friedman J.L., Morsink S.M., 1997, submitted to ApJ, preprintgr-qc/9706073Hartle J.B., 1967, ApJ, 150, 1005Kojima Y., 1992, Phys. Rev. D46, 4289Kojima Y., 1993a, ApJ, 414, 247Kojima Y., 1993b, Prog. Theo. Phys., 90, 977Kokkotas K.D, 1994, MNRAS, 268, 1015Lin C.C., 1955, The Theory of Hydrodynamic Stability, Cam-bridge University PressPapaloizou J., Pringle J.E., 1978, MNRAS, 182, 423Provost J., Berthomieu G., Rocca A., 1981, A&A, 94, 126Saio H., 1982, ApJ, 256, 717Strohmayer T.D., Lee, U., 1996, ApJ, 467, 773Thorne K.S., Campolattro A., 1967, ApJ, 149, 591van der Klis M. et al., 1996, ApJ, 469, L1c© 199? RAS, MNRAS 000, 000–000

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Quasi-toroidal oscillations in rotating relativistic stars

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