Rosette modes of oscillations of rotating stars caused by close degeneracies. III. JWKB analysis

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1 Publ. Astron. Soc. Japan ), 8 4) doi:.9/pas/psu55 Advance Access Publication Date: 4 July 8- Rosette modes of oscillations of rotating stars caused by close degeneracies. III. JWKB analysis Masao TAKATA Department of Astronomy, School of Science, The University of Tokyo, 7-- Hongo, Bunkyo-ku, Tokyo -, Japan * takata@astron.s.u-tokyo.ac.p Received 4 April ; Accepted 4 May 4 Abstract An asymptotic analysis is developed to describe rosette modes of oscillations in rotating stars, which have been recently discovered in numerical computations. The name of the modes comes from characteristic rosette patterns of the distributions of the kineticenergy density on the meridional plane. As have been shown in the previous papers in this series, the physical reason for the formation of these modes is the rotation-induced interaction among several gravity modes with almost the same frequencies successive spherical degrees of the same parity. This formation process is described in the framework of the quasi-degenerate perturbation theory. In this paper, the matrix eigenvalue problem, which appears in the perturbation analysis, is solved analytically, using asymptotic expressions of eigenfunctions, which are obtained by assuming that the wavelength of oscillations in the horizontal direction is sufficiently small. By further assuming that the number of interacting modes is large, simple expressions of the eigenfunctions eigenfrequencies of rosette modes are obtained. Because these expressions can describe the essential part of characteristics of rosette modes, they are useful in understing general properties of rosette modes that are common to any equilibrium structure. Key words: methods: analytical stars: interiors stars: oscillations stars: rotation Sun: oscillations Introduction Rosette modes constitute a unique class of eigenmodes of oscillations in rotating stars. They were recently discovered in numerical computations Ballot et al. ), which were performed in a proect to investigate gravito-inertial modes in fast rotating stars Ballot et al. ). The frequencies of rosette modes are located in the range of gravity modes, but outside the inertial domain, which means that the mode amplitude is not necessarily excluded from the polar region Matsuno 966). For the polytropic model with index, a typical threshold of the rotation rate, above which rosette modes start to appear, is around percent of the breakup rate the Keplerian rotation rate at the equator). The most conspicuous property of the modes is that the kineticenergy density is concentrated on rosette patterns on the meridional plane. The physical origin of rosette modes is discussed by Takata Saio b), to which we hereafter refer as Paper I. They have shown that the modes are generated by the interaction, which is brought about C The Author 4. Published by Oxford University Press on behalf of the Astronomical Society of Japan. All rights reserved. For Permissions, please ournals.permissions@oup.com on 4 January 8

2 8- Publications of the Astronomical Society of Japan, 4, Vol. 66, No. 4 by the Coriolis force, among eigenmodes with almost the same frequencies successive spherical degrees of the same parity. Regarding the effect of the Coriolis force as a small perturbation, they apply the quasi-degenerate perturbation theory e.g., Lavely & Ritzwoller 99) to describe the eigenfunctions of rosette modes as linear combinations of the eigenfunctions in the absence of rotation. They then demonstrate that the resultant eigenfunctions can successfully reproduce the distributions of the kinetic-energy density of rosette modes on the meridional plane. Although the rosette modes are first found among axisymmetric modes about the rotation axis, they have shown in a subsequent paper Saio & Takata 4, hereafter Paper II) that nonaxisymmetric rosette modes also exist. If the rotation rate is sufficiently high, the structure of the non-axisymmetric rosette modes is essentially the same as that of the axisymmetric modes. One of the remaining problems about rosette modes is to underst the mechanism of the formation of the rosette-pattern structure in further detail. In particular, an empirical relation is found in Paper I between the mode structure the relation between the radial order, n cf. Takata ), the spherical degree, l, of the constituent modes. The primary purpose of this paper is to clarify the reason for this relation, while the secondary purpose is to elucidate general properties of rosette modes that hold for any equilibrium structure. In order to accomplish these purposes, we develop an asymptotic analysis of axisymmetric rosette modes, assuming that the spherical degrees of modes that constitute rosette modes are large l ). In this asymptotic regime, we demonstrate that it is possible to solve the perturbation equations analytically. If we additionally consider the limit of an infinite number of constituent modes, we can derive simple expressions for the eigenfunctions, hence the kinetic-energy density of rosette modes. It turns out that these expressions naturally describe rosette patterns on the meridional plane, explain the empirical relation. Preliminary results of this paper have been presented in Takata Saio 4). This paper is organized as follows: we first summarize in section the results of the quasi-degenerate perturbation theory, which have been presented in Paper I; we develop the asymptotic analysis of gravity modes for large spherical degrees in section ; we solve the matrix eigenvalue problem, which is a key equation of the quasi-degenerate perturbation theory, in section 4; the eigenfunction the eigenfrequency of rosette modes are calculated in section 5; we present discussions in section 6 a conclusion in section 7. Summary of the results of the quasi-degenerate perturbation analysis We summarize the essential part of the analysis of rosette modes of oscillations in rotating stars, based on the quasidegenerate perturbation theory. The reader should refer to Paper I for details. Assumptions that we make are summarized as follows.. The rotation rate is so slow that we can ignore deformation of the stars caused by the centrifugal force, regard the effect of the Coriolis force as a small perturbation.. Rotation is rigid with the constant angular frequency of.. The unperturbed eigenmodes are gravity modes. 4. We confine ourselves to axisymmetric modes. 5. The unperturbed eigenfrequencies are almost the same for several eigenmodes with successive spherical degrees of the same parity. The assumptions,, 5 are essential, whereas the assumptions 4 are made ust for simplicity. We may restate assumption more precisely. The ratio, /σ, must be much smaller than, in which σ is the angular frequency of oscillations in the corotating frame. This means that we treat eigenmodes outside the inertial domain.) In addition, for the effect of the centrifugal deformation e.g., Saio 98) to be negligible compared to the second-order effect of the Coriolis force, σ must satisfy ) ), ) K which implies σ σ K. ) Here, K is the Keplerian rotation rate at the equatorial surface, which is defined by K = GM R. ) Here, G means the gravitational constant, while M R indicate the mass radius of the equilibrium structure. The examples of the close degeneracy assumption 5) can be found among gravity modes of the polytropic model with index cf. Paper I), a realistic model of a zero-age main sequence star with five solar masses Takata & Saio a). In order to treat the close degeneracy, we invoke the quasi-degenerate perturbation theory, a kind of extension of completely) degenerate perturbation theory, which is commonly explained in textbooks of quantum mechanics e.g., Lau & Lifshitz 977). Corresponding to the on 4 January 8

3 Publications of the Astronomical Society of Japan, 4, Vol. 66, No assumption that the effect of the Coriolis force is a small perturbation, we may regard that the order of the perturbation analysis is controlled by the ratio, /σ. Practically, the order of the analysis is determined by the power of a formal parameter, ɛ, which is introduced as a coefficient of the Coriolis force term in the oscillation equation cf. equation ) of Paper I], is eventually set to after the perturbation equations are derived at each order. It is a stard technique in perturbation theory to introduce such a formal parameter.) For the purposes of this paper, we need to analyze only the leading-order corrections to the eigenfrequency the eigenfunction, meaning the secondorder perturbation to the eigenfrequency, σ ), the zeroth-order eigenfunction the displacement vector), ξ ), respectively. Note that the first-order perturbation to the eigenfrequency, σ ), is equal to zero for axisymmetric modes. The zeroth-order eigenfunction is assumed to be expressed as a linear combination of the unperturbed eigenfunctions, ξ k,as where ρ is the density, dv represents the infinitesimal volume element. The integral in equation 6) is taken over the whole stellar volume, V, while M is the operator for the Coriolis force, defined by Mξ = i ξ. 7) In the present analysis, the rotation vector,, isgivenby = e z,inwhiche z is the unit vector in the z direction of the Cartesian coordinates the direction of the rotation axis). Note that the number of rosette modes produced by equation 5) is the same as that of the closely-degenerate eigenmodes, which corresponds to the order of M k,k. In order to evaluate M k,k, we may introduce three integrals by I k,k ) = R ξ k) r ξ k ) r ρr dr, 8) ξ ) = k D α k ξ k. 4) In equation 4), the meanings of the symbols are as follows: k, which should be regarded as a pair of n l, denotes the index to each eigenmode; α k are constants, D represents the space of the close degeneracy. Therefore, the summation in equation 4) is performed over all of the closely degenerate modes. The main result of the analysis is that σ ) the coefficients α k are essentially obtained as the eigenvalues the eigenvectors of a matrix eigenvalue problem, respectively, which is given by 4 σ M ) k,k + σ k σ ) ) δ k,k σ ) α k = k D σ ) σ ) α k. 5) Here, σ k means the eigenfrequency of a mode in the absence of rotation, whose eigenfunction is given by ξ k, while σ ) represents the zeroth-order eigenfrequency. In principle, σ ) should be set a priori to a guess of the perturbed eigenfrequency Lavely & Ritzwoller 99), whereas D should be adusted so that its constituent mode frequencies σ k differ from σ ) by only second-order quantities cf. equation ) of Paper I]. However, D can practically be fixed first, while σ ) can then be set to any value, as long as its differences from σ k with k D are of the second order. The Kronecker symbol, δ k,k, in equation 5) is defined by δ n,n δ l,l, while the definition of the matrix element M k,k is given by M k,k = ξ 4 k M ) ξ k ρ dv, 6) V I k,k ) = I k,k ) = R R ξ k) h ξ k ) h ρr dr, 9) ξ k) r ξ k ) h ρr dr. ) Here, r represents the distance from the center. The functions ξ r k) ξ k) are introduced by the relation, h ξ k = ξ r k) r) Yl θ) e r + ξ k) h dy l r) dθ e θ, ) in which Yl θ) indicates a spherical harmonic, while e r e θ represent the unit vectors in r θ directions of the spherical coordinates, respectively. Note that we use the notation of I k,k ) i = I n,l;n,l ) i i =,, ) ) for k = n, l)k = n, l ). Then, we obtain the expression of M k,k as follows cf. appendix of Paper I): M k,k = δ l,l A n,n ;l + δ l l,b k,k, ) in which A n,n ;l = l + l ) l l + ) C n,l l + )l ) l l + ) C n,n ;l l + )l ) for n = n ) for n n ) 4) on 4 January 8

4 8-4 Publications of the Astronomical Society of Japan, 4, Vol. 66, No. 4 B k,k = l l + ) l + ) l ) l + )] / { I k,k ) l + ) l ) I k,k ) + ] I k,k ) + I k,k) 4 l + ) l l ) ] } I k,k ) I k,k). 5) In equation 5), l is given by l = l + l, 6) while C n,n ;l C n, l in equation 4) are defined by C n,n ;l = I n,l;n,l ) + I n,l;n,l) + I n,l;n,l) 7) C n,l = C n,n;l = I k,k) + I k,k), 8) respectively. It is an important assumption of this paper that we neglect the second term in the parenthesis on the left-h side of equation 5), which describes the lift of degeneracy. As has been shown in Paper I, the rosette-pattern structure starts to emerge when the first term becomes dominant over the second term. We therefore consider the first term as responsible for the formation of rosette modes. Because the main purpose of the present analysis is to underst the properties of the rosette pattern, the consideration ustifies the neglect of the second term. Under this assumption, we solve equation 5) in section 4, using the asymptotic formula of the eigenfunctions, ξ k, which is given in section. Asymptotic analysis of the unperturbed eigenmodes Asymptotic analysis of the Jeffreys Wentzel Kramers Brillouin JWKB) type has been extensively applied to the problem of linear adiabatic oscillations of stars in the literature e.g., Tassoul 98; Unno et al. 989), which we basically follow here. By neglecting the Eulerian perturbation to the gravitational potential Cowling 94), which is valid for highdegree modes, we can cast the governing equations into a second-order ordinary differential equation. Because we are interested in gravity modes, we assume that, except near the surface, the frequency of oscillations, σ, is much smaller than the Lamb frequency, L l, which is defined in terms of the sound speed, c, by l l + )c L l =. 9) r Then, if we follow Unno et al. 989), the essential part of the equation is reduced to d W dr + k r W =. ) Here, W is defined in terms of the radial displacement, ξ r, as W = ρ / r ξ r, ) while k r indicates the wavenumber in the radial direction, defined by k r = l l + ) r N ) σ, ) in which N means the Brunt Väisälä frequency. Note that, unlike the stard asymptotic treatment of gravity modes, we do not assume that σ is much smaller than N, because it is not always a very good approximation for rosette modes, as we see in figure 9 of Paper I. We instead assume in the present analysis that l, so that k r is very large when σ<n). Correspondingly, l should be regarded as large from this point forward. As the stard asymptotic analysis tells us, the character of the solutions of equation ) sensitively depends on the sign of kr. In fact, the solutions oscillate very rapidly about zero for kr > propagative regions), whereas they change exponentially for kr < evanescent regions). We assume for simplicity that the equilibrium structure has a single propagative region that is swiched between two evanescent regions. It is also assumed that the eigenfunctions have negligible amplitude in the evanescent regions. The inner outer bounds in radius) of the propagative region are called the inner turning point, r, the outer turning point, r, respectively. In the propagative region, the asymptotic solution of equation ) is given by ρ / rξ r = Acos LZr) π ]. ) 4 The meanings of the symbols in equation ) are as follows: A is the amplitude function defined by N A = A r / ) /4 σ, 4) on 4 January 8

5 Publications of the Astronomical Society of Japan, 4, Vol. 66, No in which A is a normalization constant; the parameter L is introduced by L = l l + ) l +, 5) Zr) is defined by Zr) = r r r N ) / σ dr. 6) Note that the term, π/4, in equation ) represents the phase lag that is introduced when constituent waves of the oscillation turn at the inner turning point. If we also take account of another source of the phase lag at the outer turning point, the interference condition provides the implicit) expression of the eigenfrequency, r N ) / r r σ κπ + dr =. 7) L Here, is determined by the total phase lag introduced at the inner outer) turning points, while κ indicates a positive integer, which is related to the radial order, n, by κ = n + κ. 8) Here, κ is an integer that generally depends on the structure of the equilibrium model. The expressions of r are dependent on the equilibrium structure as follows: If the structure has a deep) convective envelope, r is determined by the condition, N = σ. This means that r is not dependent on l, but on only σ. According to a turning-point analysis, the corresponding value of can be determined as = π/. If the envelope is radiative, r is determined by the condition that σ is equal to the Lamb frequency. Since we consider the limit of l,wemaysetr to the total) radius of the structure. On the other h, depends on the structure of the near-surface layers. If it is described by a polytrope relation with index μ,the corresponding is given by πμ/. What is essential for the discussions in this paper is that is independent of l. Because r is essentially independent of l in both cases, its dependence on l is ignored from this point forward. In the same asymptotic limit of l, the corresponding expression of ξ h is provided by ρ / rξ h = A N ) / L σ sin LZr) π ], 9) 4 which is obtained by substituting equation ) into d r ) g ξ r r dr c ξ r L r ξ h = ) cf. Unno et al. 989). In equation ), g denotes the gravitational acceleration. Following Paper I, we normalize the eigenfunctions so that the mode inertia is equal to. Based on equations ) 9), this condition can be expressed as = R = A ξ r + l l + ) ξ ] h ρr dr r N ) / N ) / ] r r σ + σ dr, ) in which we have neglected integrals of functions that are highly oscillatory about zero those proportional to cos LZr)] or sin LZr)]). By adusting a sign, we may set A = ) n Ã, ) in which à = { r N ) / N ) / ] / r r σ + dr} σ. ) Due to the factor ) n in equation ), ξ r has a constant sign at the surface. Because equation ) implies that à does not depend on L n, but only on σ, the value of à is common to all eigenmodes in a given family of the close degeneracy. 4 Asymptotic solution of the matrix eigenvalue problem 4. Close degeneracy Since r is independent of L cf. section ), the left-h side of equation 7) depends on only σ N, whereas the right-h side is asymptotically reduced to κπ/l as l. Therefore, equation 7) implies that a family of eigenmodes with an asymptotically constant value of κ/l have almost the same eigenfrequency. Prompted by the case of the polytropic model with index cf. table of Paper I), we make a slightly stronger assumption that each family of the close degeneracy is associated with a linear relation between n l, an + bl = f, 4) on 4 January 8

6 8-6 Publications of the Astronomical Society of Japan, 4, Vol. 66, No. 4 where a, b, f are constants. Note that we follow the convention that n is negative for g modes. Correspondingly, a is positive, while b is non-negative. This means that each member of a given family is uniquely specified by l only. In a given family of the close degeneracy, the effect of the Coriolis force induces interaction between different eigenmodes whose spherical degrees differ by l = ) cf. equation )]. Because the corresponding difference in n between the two modes n) must be an integer, n = b/a) l, we underst that K = b a = lim n l l 5) is equal to an non-negative) integer that characterizes each family of the close degeneracy, when we consider interaction caused by the Coriolis force. Since K is constant in a given family, n, which satisfies n = K, 6) is also constant in the family. We furthermore find from equations 6) 7) Zr ) = Kπ 7) for large l. Equation 7) can be used to determine the asymptotic value of the eigenfrequency of the unperturbed modes in a given family of the close degeneracy. 4. Evaluation of the matrix elements In order to solve equation 5), we first need to evaluate the matrix element M k,k in the asymptotic limit for eigenmodes that belong to a given family of the close degeneracy. As discussed in section, we assume that ξ r ξ h are given by equations ) 9), respectively, for r < r < r, that both of them are equal to zero elsewhere. The three integrals, which are defined by equations 8) ), can be calculated in the similar way to equation ) as follows: I k,k ) = ) n à r N ) / r r σ cos LZr)] dr, 8) I k,k ) = ) n à L r r r N ) / σ cos LZr)] dr, 9) I k,k ) = ) n à r sin LZr)] dr, 4) L r r in which we have introduced n = n n 4) L = L L l l. 4) Substituting l = l into equations 9) 4), we find, I n,l;n,l), 4) I n,l;n,l) because we assume L. We therefore underst from equation 8) C n,l, 44) which means A n,n;l =, 45) in equation 4). On the other h, if we set l = l, which implies L, we obtain from equations 9), 6), 7) I n,l;n,l ) = ) n à r L r = ) n à 4L sin dz cos Zr)] dr dr r N ) / dr] r r σ = ) n à 4L sin Kπ) = o L ). 46) The last equality in equation 46) comes from the fact that K, which is defined by equation 5), is equal to an integer. Utilizing equations 45) 46) in equation ), we get the asymptotic expression of M k,k as M k,k = δ l,l ) δ l l,s, 47) in which ] s = I n,l;n,l ) L I n,l;n,l ) I n,l ;n,l) Kπ ) = ) K à dτ dτ cos q sin q dq. 48) dq dq We have introduced q = Zr) τ = ln r to equation 48). Note that s is constant in each family of the close degeneracy. This is because K is a constant that characterizes each family cf. equation 5)], because on 4 January 8

7 Publications of the Astronomical Society of Japan, 4, Vol. 66, No à Zr), which are defined by equations ) 6), respectively, are fixed if σ is given. 4. Eigenvalues Substituting equation 47) into equation 5), neglecting the second term in the parenthesis on the left-h side, we obtain TJ + J ) I J ] α J ) =. 49) The meanings of the symbols in equation 49) are given as follows: J indicates the total number of eigenmodes that belong to the family of the close degeneracy; I J is an identity matrix of order J; T J is a J J matrix, which is defined by T J = O O vector α J ) is given by α J ) = α. α J ; 5), 5) in which we label the members of the family of the close degeneracy as,..., J in the order of l; J ) is provided by J ) = σ ) σ ) ], 5) s represents a null vector of dimension J. The problem is thus reduced to the eigenvalue problem of matrix T J. In order to solve the problem, it is crucial that the characteristic polynomial of T J can be expressed in terms of the Chebyshev polynomials of the second kind, U J,as det T J + J ) ] J ) ] I J = UJ 5) cf. equation A6)]. The reader should refer to appendix for properties of U J that are needed in this paper. Because zeros of U J can be found from equation A5), we obtain the eigenvalues, J ),as J ) = cos π J + =,,..., J ), 54) which means from equation 5) σ ) ) = π s cos σ ) J + Note that J ) satisfies =,,...,J ). 55) J ) J + = J ) =,,...,J ). 56) As demonstrated in equations 54) 55), we use index to distinguish J rosette modes that originate from the same family of the close degeneracy. 4.4 Eigenvectors In order to find the corresponding eigenvector, α J ), we start from the recurrence relation equation 49)], α J ),k + J ) α J ),k+ + αj ),k+ = k =,...,J ), 57) where we have defined α J ) ), = αj,j + =. 58) { We can regard that α J ),k } is a solution of the second-order linear finite difference equation equation 57)] that satisfies the two boundary conditions given by equation 58). On the other h, the recurrence relation of U k x) is provided by equation A). Since J ) / =,...,J) are solutions of U J x) =, it particularly holds that U J J ) ] J ) ] J + = U J =, 59) in which we have used equation 56). If we compare equation 57) with equation A), equation 58) with equations A4) 59), we find that α J ) ],k U k J ) / satisfy the same homogeneous) finite difference equation the same boundary conditions. We therefore obtain α J ),k = N U k J ) ] k =,..., J ), 6) in which N is a normalization constant. We adust N so that k= α J ),k =, 6) on 4 January 8

8 8-8 Publications of the Astronomical Society of Japan, 4, Vol. 66, No. 4 which implies that the mode inertia of ξ ) is equal to one. In order to evaluate the left-h side of equation 6), we first note U k J ) ] ) sin kπ π = U k cos = J + J + sin π, 6) J + which is obtained from equation A5). Substitution of equation 6) a mathematical identity, sin k= kπ J + = J + cf. equation A4)], into equation 6) yields N = 6) J + sin π J +, 64) from which we finally find α J ),k = J + sin kπ J +, k =,,..., J ). 65) We can easily confirm the following properties: α J ),k = αj ) k, 66) α J ),J k+ = ) + α J ) k,. 67) It is remarkable that α J ),k depends on only J, the number of closely-degenerate modes, is totally independent of the other properties of the equilibrium structure. This point can be understood from equation 49), in which the model dependence of the problem except for J) is confined in the expression of J ) cf. equation 5)]. 5 Asymptotic expressions of the eigenfunctions eigenfrequencies of rosette modes 5. Calculation of the asymptotic structures We evaluate the zeroth-order eigenfunctions of axisymmetric rosette modes, which are given by equation 4), based on the asymptotic expressions of ξ r equation )], ξ h equation 9)], α k equation 65)], regarding σ = σ ) in these expressions. We particularly consider the limit of J an infinite number of closely degenerate modes). We put subscript k to quantities that are associated with the member of the family of the close degeneracy with the kth smallest value of the spherical degree. If we denote the eigenfunction of the th rosette mode by ρ / rξ ) = ) r e r + ) θ e θ, 68) then r ) ) θ are calculated in the asymptotic limit of l as ) r ) θ = ρ / r = Ã π = ρ / r = Ã π k= r sin θ α J ),k ξ nk,lk) r r) Y l k θ) ) / N ) /4 σ R r r,θ; q,j, J ) 69) α J ),k ξ nk,lk) h k= r sin θ r) dy l k dθ ) / N ) /4 σ R θ r,θ; q,j, J ), 7) respectively, in which q, J is defined by q,j = J +, 7) while the definitions of R r R θ are provided by R r r,θ; q, J ) = k= J + ) nk sin k qπ) cos L k Zr) π ] cos L k θ π ) 4 4 R θ r,θ; q, J ) = k= J + 7) ) nk sin k qπ) sin L k Zr) π ] sin L k θ π ), 4 4 7) respectively. In equations 7) 7), we have used the asymptotic expressions for large l, Yl θ) π sin θ cos Lθ π 4 ) 74) dyl dθ L Lθ π sin θ sin π ), 75) 4 on 4 January 8

9 Publications of the Astronomical Society of Japan, 4, Vol. 66, No respectively e.g., Gradshteyn & Ryzhik 7), in which the range of θ is given by ɛ θ π ɛ with ɛ being any positive number smaller than π/). Following the assumptions about the close degeneracy in subsection 4., we can express n k l k as linear functions of k, whicharegivenby n k = n + k ) n = kk + n + K 76) cf. equation 6)] l k = l + k ) = k + l, 77) respectively. Utilizing equations 76) 77), we can convert the products of the trigonometric functions in equation 7) into sums of single cosine functions based on the formula, ) kk sin k qπ) cos kzr) + a ] cos kθ + b ) = cos k{ Zr) + θ] + qπ Kπ} + a + b π ) 4 + cos k{ Zr) + θ] qπ Kπ} + a + b + π ) + cos k{ Zr) θ] + qπ Kπ} + a b π ) + cos k{ Zr) θ] qπ Kπ} + a b + π ) ], 78) in which a b are real constants. Then, equation A) allows us to perform the summation with respect to k in equation 7). By taking the limit of J, introducing q as the corresponding limiting value of q, J hence, q ), we find from equation A) that the amplitude of R r concentrates on curves that are described by ] Kπ Zr) ± θ ± qπ = kπ double sign in any order ), 79) in which k is an arbitrary integer. Although the range of θ is given by θ π, it is convenient to extend the range to <θ< formally. Considering that change in k by in equation 79) is equivalent to that in θ by ±π, we find that only the following curves can essentially be different from each other: ] Kπ Zr) ± θ ± qπ = 8) ] Kπ Zr) ± π θ) ± qπ =. 8) If we draw the curves given by equation 8) on the meridional plane, the curves with a + sign chosen at the first double sign in front of θ) are the mirror images of those with a sign with respect to the rotation axis. We also find that the curves given by equation 8) are the mirror images of those by equation 8) with respect to the equatorial plane. Because, for q q, equation 8) represents four different curves, equations 8) 8) describe eight different curves in total. On the other h, if q = orq =, all of the eight curves are doubly degenerate, so that there exist only four different curves. The end points of each curve in equation 8) are found by setting Zr) = Kπ/Zr) =. In the former case, the corresponding values of r θ are given by r,θ) = r, ± qπ ), 8) whereas the latter case is realized when r,θ) = r, ± q ± K) π ), 8) in which the first double sign corresponds to that of equation 8). Equations 8) 8) also tell us that, in the case of q, each of the eight or four when q = ) curves is obtained by rotating the corresponding curve at q = about the center by ±qπ/. A completely analogous argument based on equation 7) leads to the same expressions for R θ as equations 8) 8). We thus underst that the two expressions represent the asymptotic structure of rosette modes. 5. Geometrical meanings of K q The properties of the asymptotic structure of rosette modes can be visually verified in figures, which are calculated in the case of the polytropic model with index the first adiabatic index of 5/. The angular frequency, σ = σ ) ), is determined so as to satisfy equation 7) for each value of K. Since we confine ourselves to the cases of σ< K cf. equation )], each equilibrium structure has the minimum value of K, only above or equal to which the present analysis can be ustified. The minimum value is equal to for the polytropic model with index. It is observed in figure that the difference in the polar angle between the two end points of each curve gets larger as K is increased. In fact, equations 8) 8) tell us that the angle difference is equal to Kπ/ in absolute value). on 4 January 8

10 8- Publications of the Astronomical Society of Japan, 4, Vol. 66, No. 4 Fig.. Asymptotic structures of rosette modes on the meridional plane of the polytropic model with index the first adiabatic index of 5/. The amplitude of the eigenfunctions concentrates on the solid curves, which are described by equations 8) 8). Just for clarity, one segment of the curves in each panel is drawn in a thicker line, its end points are indicated by dots. The values of K σ ) are given above each panel, whereas q is fixed at. The radii of the small large circles, which are drawn in the dashed line in each panel, represent the inner turning point, r, the outer turning point, r, which coincides with the surface, respectively. Fig.. Same as figure, but for K = different values of q. Figure demonstrates how the structure varies as q is changed in the range, q, with K fixed. We confirm that, in each of the second, third, fourth panels, the curves are obtained by rotating those in the first panel for q = ) by ±qπ/. Although the two figures assume a particular profile of N that is valid only for the polytropic model with index, the geometrical meanings of K q hold irrespectively of the choice of the profile, because Zr) is generally a monotonically increasing function of r cf. equation 6)]. 5. Asymptotic formula of the eigenfrequency Taking the limit of J hence q, J = /J + ) q in equation 55), we obtain the asymptotic expression of the eigenfrequency, σ = σ ) + s cos qπ)]. 84) σ ) Equation 84) shows that the range of the eigenfrequencies of rosette modes that originate from a given family of the close degeneracy is provided by σ σ σ +, in which σ ± = σ ) + ± s ). 85) σ ) The central value of the range is determined by only σ ) as σ c = σ ) +, 86) σ ) which is equal to the frequency in the absence of the close degeneracy. In fact, equation 86) is obtained by ignoring the off-diagonal components or setting s = ) on the righth side of equation 47). Although there is no clear reason, s, which is defined by equation 48), turns out to be positive in all cases that we have checked, including modes of polytropic models some main-sequence star models. If s > for a given family of the close degeneracy, we find from equation 84) that the corresponding rosette modes with larger q have higher frequencies, because q. In figure, for example, the eigenfrequency attains the minimum maximum) value σ σ + ) in the leftmost rightmost) panel, increases from left-h panel to right-h panel. on 4 January 8

11 Publications of the Astronomical Society of Japan, 4, Vol. 66, No Discussions 6. Characteristics of the asymptotic rosette structure The asymptotic structure of axisymmetric rosette modes on the meridional plane is described by equations 8) 8), which contain an integer parameter K a real parameter q. These expressions depend on the equilibrium structure only through Zr), which depends on the profile of the Brunt Väisälä frequency, N cf. equation 6)]. The asymptotic value of the zeroth-order) eigenfrequency, σ, in equation 6) is fixed by the condition given by equation 7), which depends on N K. The asymptotic structure is particularly independent of the minimum value of the spherical degree of the constituent closely-degenerate) modes, l cf. equation 77)]. Although change in l corresponds to addition or removal of a finite number of terms or modes) in equations 7) 7), the contribution of those modes becomes negligible as J. In figures, the mode structure indicated by solid curves has cusps at the inner turning point, whereas the structure near the outer turning point shows smooth variations. This different behavior at the inner outer turning points originates from different dependence on the radius, r, of the integr on the right-h side of equation 6). Near the inner turning point, the integr is generally proportional to the square root of the distance from the turning point, whereas it scales linearly near the outer turning point with the inverse square root of the distance from the turning point, if the equilibrium structure has a radiative envelope as in the case of the polytropic model with index. However, the structure of modes near the outer turning point is different, if the equilibrium structure has a thick convective envelope. As an example, a mode structure is presented in figure for a stard solar model Christensen-Dalsgaard et al. 996). The cusp structure is observed not only at the inner turning point, but also at the outer turning point, near which the integr increases with the square root of the distance from the turning point. The structure in figure recalls the ray path of gravity waves e.g., Aerts et al. ), though the ray of eigenmodes is not closed in general.ballot etal. ) also point out that the structure of rosette modes is associated with a closed ray path in the case of the polytropic model with index. 6. Explanation of the empirical relation about the rosette structure The empirical relation, which is established in Paper I, consists of the following two points.. The parameter K, which is defined by equation 5), is equal to a non-negative integer. Fig.. Same as figure, but for a mode of a stard solar model Christensen-Dalsgaard et al. 996) with K = 5, q =, σ =.5 K. The dotted circle corresponds to the surface.. The difference in the polar angle between the two end points of each curve that constitutes the asymptotic structure of rosette modes cf. equations 8) 8)] is equal to Kπ/. Both points have already been discussed based on the asymptotic analysis. As has been explained in subsection 4., the first point can be derived from the fact that the Coriolis force can induce the interaction only between modes whose spherical degree differ by or. On the other h, the second point is apparent from the coordinates of the end points, which are provided by equations 8) 8). Although the relation has been derived empirically in Paper I based on the lowest-frequency modes, which have the smallest values of q, the above two points hold irrespectively of q. 6. Comparison with the full perturbative calculations In Papers I II, the structure of rosette modes is calculated numerically based on the quasi-degenerate perturbation theory. We call this approach full perturbative calculation, the results of which are compared with those of asymptotic analysis developed in section 5. We discuss only axisymmetric modes, because the well-developed) nonaxisymmetric rosette modes in Paper II show no essential difference from the axisymmetric modes in the distributions of the kinetic-energy density on the meridional plane carried by the r θ components of the displacement vector. Because the spherical degrees of modes that compose rosette modes are assumed to be large in the asymptotic analysis, we pay attention to families of the close degeneracy with high-degree modes. As an example, on 4 January 8

12 8- Publications of the Astronomical Society of Japan, 4, Vol. 66, No. 4 we can pick up family C of the polytropic model with index, which is discussed in Paper I. The family is specified by the relation, n + l =, between the radial order n) the spherical degree l). Equation 5) then gives K =. The considered values of l here are all odd integers in the range 9 l 59. The minimum frequency among all of the eigenmodes in the family in the absence of rotation is equal to.645 K, which we adopt as σ ). The parameter s, which is calculated by equation 48), is equal to Distribution of the kinetic-energy density Figure 4 depicts the kinetic-energy density distributions of two representative modes with the lowest- sixthlowest frequencies on the meridional plane. The results of the full perturbative calculation are indicated as the color contours, whereas the asymptotic structure of the distributions is shown by the solid curves, which are calculated by equations 8) 8). The value of q is determined by the asymptotic formula of the eigenfrequency equation 84)], in which we set σ to the eigenfrequency of each mode obtained by the full perturbative treatment the relevant rotation rate.5 K ), respectively. The thusobtained values of q are equal to.5.7 for the lowest the sixth-lowest frequency modes, respectively. While we observe a reasonable agreement at least at first look) between the full perturbative the asymptotic calculations in the left-h panel, we can identify in the right-h panel some structures of the color contours between the two solid curves, which are not reproduced by the asymptotic expressions. These structures appear to be described by equations 8) 8) with several different values of q between.5. In other words, the asymptotic curves seem to fix the boundaries of the kinetic-energy distribution, rather than represent the distribution itself. We can also notice similar structures in the left-h panel with a closer look. After some investigation, we realize that these structures are generated by the contribution from the second term in the parentheses on the left-h side of equation 5), because they disappear if we artificially set the term to zero in the full perturbative calculation. This term, which is neglected in the asymptotic analysis cf. section ), means the lift of degeneracy in the absence of rotation. 6.. Frequency range In figure 5, the asymptotic expressions of the frequency range of rosette modes, which are given by equations 85) 86), are compared with the results of the full perturbative calculations. Except for a few high-frequency modes, the range of the frequencies obtained by the full perturbative treatment reasonably agrees with the asymptotic frequencyrange. The higher the rotation rate is, the more modes the asymptotic range includes. The poorer agreement at the low rotation rate can be interpreted as the influence of the small lift of degeneracy at =, which is not taken into account in the asymptotic analysis. We have verified that the highfrequency outliers do not have the rosette structure. 7 Conclusion An asymptotic analysis of axisymmetric rosette modes of oscillations in rotating stars has been developed. It is found that the asymptotic structure of the kinetic-energy distribution on the meridional plane is composed of eight or four) curves, each of which is described by a simple linear relation Fig. 4. Normalized distributions of the kinetic-energy density on the meridional plane of axisymmetric rosette modes of the polytropic model with index the first adiabatic index 5/. These modes originate from family C of the close degeneracy see the main text). The mode on the left-h right-h) panel is that of the lowest sixth-lowest) frequency. The rotation rate,, the dimensionless eigenfrequency, ω = σ/ K, are shown above each panel. The background color contours represent the results of the full perturbative calculation multiplied by the squared distance from the center for better visibility), whereas the solid curves indicate those of the asymptotic analysis cf. equations 8) 8)]. The parameter K is set to, while the values of q for the left right-h panels are set to.5.7, respectively. Fig. 5. Eigenfrequencies, σ, of rosette modes that come from family C of the close degeneracy of the polytropic model with index as functions of the rotation rate,. Bothσ are normalized by the Keplerian rotation rate, K ω = σ/ K ). While the results of the full perturbative calculations are represented by the solid lines, the asymptotic formulae of the frequency range are used to draw the dashed lines. The upper lower limits defined by equation 85) are indicated by ω ±, whereas the central value given by equation 86) is denoted by ω c. on 4 January 8

13 Publications of the Astronomical Society of Japan, 4, Vol. 66, No between the radial function, Zr), the polar angle, θ. The relation includes integer K real number q as parameters, both of which have clear geometrical meanings. While K is common in a given family of rosette modes, q is related to the eigenfrequency of each mode, hence serves as an index to distinguish members of the family. Based on the thus-obtained expressions, the structure of rosette modes has been successfully connected with the relation between the radial order the spherical degree, which characterizes the corresponding family of the close degeneracy. It is confirmed in the case of the polytropic model with index that the results of the asymptotic analysis capture the essential aspects of those obtained by the full perturbative calculations. Acknowledgments This work was supported by JSPS KAKENHI Grant Number 649. The author thanks H. Saio for helpful comments about the manuscript of this paper. Appendix. Mathematical formulae A... Chebyshev polynomials of the second kind We summarize some properties of the Chebyshev polynomials of the second kind of degree k, U k x). For more detail, the readers should refer to appropriate literatures e.g., Gradshteyn & Ryzhik 7). We assume for simplicity that the argument, x, is a real number. One way to define U k x) is to specify the recurrence relation, U k+ x) xu k x) + U k x) = k =,,...), A) with initial values, U x) = U x) = x. It is formally possible to replace equation A) with U x) =, A) A) A4) apply equation A) even for k =. If the argument, x, is between, U k x) can explicitly be expressed as sin k + ) θ] U k cos θ) =, A5) sin θ where θ is a real number. A formula that U k x) satisfies is given by U k x) = det T k + xi k ), A6) in which T k is defined by equation 5), I k is an identity matrix of order k. In fact, if we set the determinant on the right-h side of equation A6) to D k x), it is easy to verify D x) = x = U x) D x) = det A7) ) x = 4x = U x). A8) x Exping D k + x) with respect to the first or last) column or row), we find D k+ x) = xd k+ x) D k x) k =,,...). A9) The thus-derived recurrence relation of D k x) is identical to that of U k x) equation A)]. Because of equations A7) A8), we inductively underst D k x) = U k x) for k =,,.... A... Summation of cosine functions Suppose that ϕ ϕ are real numbers, that ϕ satisfies ϕ kπ, where k is an arbitrary integer. Then, we obtain = e i ϕ+ϕ) = eiϕ+ϕ) e Jiϕ) e iϕ = sin J ϕ sin ϕ { ]} J + ) ϕ exp i + ϕ. A) Taking the real part of equation A), we get cos ϕ + ϕ ) = sin J ϕ ] J + ) ϕ = sin ϕ cos + ϕ ϕ kπ), J cos ϕ ϕ = kπ), A) in which the case of ϕ = kπ is trivial. We find from equation A) lim J J + cos ϕ + ϕ ) = ϕ kπ or cos ϕ = ), = ϕ = kπ cos ϕ > ), ϕ = kπ cos ϕ < ). A) on 4 January 8

14 8-4 Publications of the Astronomical Society of Japan, 4, Vol. 66, No. 4 ψ = Another related formula can also be obtained. If kπ J +, then it is found that sin ψ) = cos ψ)] = = A) sin J ψ) = J cos J + ) ψ] sin ψ = J +. A4) Here, we have used equation A) J + ) ψ = kπ, A5) which implies sin J ψ) = sin kπ ψ) = ) k+ sin ψ. References A6) Aerts, C., Christensen-Dalsgaard, J., & Kurtz, D. W., Asteroseismology, Astronomy Astrophysics Library Dordrecht: Springer) Ballot, J., Lignières, F., Prat, V., Reese, D. R., & Rieutord, M., in ASP Conf. Ser. 46, Progress in Solar/Stellar Physics with Helio- Asteroseismology, ed. H. Shibahashi et al. San Fransisco: ASP), 89 Ballot, J., Lignières, F., Reese, D. R., & Rieutord, M., A&A, 58, A Christensen-Dalsgaard, J., et al. 996, Science, 7, 86 Cowling, T. G. 94, MNRAS,, 67 Gradshteyn, I. S., & Ryzhik, I. M. 7, Table of Integrals, Series, Products, 7th ed., ed. A. Jeffrey & D. Zwillinger Oxford: Elsevier) Lau, L. D., & Lifshitz, E. M. 977, Course of Theoretical Physics, Vol., Quantum mechanics: non-relativistic theory, rd ed. Oxford: Pergamon Press) Lavely, E. M., & Ritzwoller, M. H. 99, Philos. Trans. R. Soc. Lond. A, 9, 4 Matsuno, T. 966, J. Meteor. Soc. Japan, 44, 5 Saio, H. 98, ApJ, 44, 99 Saio, H., & Takata, M. 4, PASJ, 66, 58 Paper II) Takata, M., PASJ, 64, 66 Takata, M., & Saio, H. a, in ASP Conf. Ser., 479, Progress in Physics of the Sun Stars: a New Era in Helio- Asteroseismology, ed. H. Shibahashi & A. E. Lynas-Gray San Francisco: ASP), 59 Takata, M., & Saio, H. b, PASJ, 65, 68 Paper I) Takata, M., & Saio, H. 4, in Proc. IAU Symp., Precision Asteroseismology, ed. W. J. Chaplin et al. Cambridge: Cambridge University Press), 497 Tassoul, M. 98, ApJS, 4, 469 Unno, W., Osaki, Y., Ando, H., Saio, H., & Shibahashi, H. 989, Nonradial Oscillations of Stars, nd ed. Tokyo: University of Tokyo Press) on 4 January 8