Generalized r-modes of the Maclaurin spheroids

Transcription

1 PHYSICAL REVIEW D, VOLUME 59, Generalized r-odes of the Maclaurin spheroids Lee Lindblo Theoretical Astrophysics , California Institute of Technology, Pasadena, California 9115 Jaes R. Ipser Departent of Physics, University of Florida, Gainesville, Florida 3611 Received 0 July 1998; published 15 January 1999 Analytical solutions are presented for a class of generalized r-odes of rigidly rotating unifor density stars the Maclaurin spheroids with arbitrary values of the angular velocity. Our analysis is based on the work of Bryan; however, we derive the solutions using slightly different coordinates that give purely real representations of the r-odes. The class of generalized r-odes is uch larger than the previously studied classical r-odes. In particular, for each l and we find l or l1 for the 0 case distinct r-odes. Many of these previously unstudied r-odes about 30% of those exained are subject to a secular instability driven by gravitational radiation. The eigenfunctions of the classical r-odes, the l1 case here, are found to have particularly siple analytical representations. These r-odes provide an interesting atheatical exaple of solutions to a hyperbolic eigenvalue proble. S PACS nubers: Dg, Db, Sj, Jd I. INTRODUCTION During the past year the r-odes of rotating neutron stars have been found to play in interesting and iportant role in relativistic astrophysics. Andersson 1 and Friedan and Morsink showed that these odes would be driven unstable by gravitational radiation reaction in the absence of internal fluid dissipation. Lindblo, Owen, and Morsink 3 have subsequently shown that this instability will in fact play an iportant role in the evolution of hot young neutron stars. The gravitational radiation reaction force in these odes was shown to be sufficiently strong to overcoe the internal fluid dissipation present in neutron stars hotter than about 10 9 K. Hot young rapidly rotating neutron stars are expected therefore to radiate away ost i.e. up to about 90% of their angular oentu via gravitational radiation in a period of about one year. Owen et al. 4 have shown that the gravitational radiation eitted during this spin-down process is expected to be one of the ore proising potential sources for the ground based laser interferoeter gravitational wave detectors e.g., LIGO, VIRGO, etc. now under construction. To date the various analyses of the r-odes and their instability to gravitational radiation reaction have all been based on sall angular velocity approxiations. This instability is of priary iportance in astrophysics for rapidly rotating stars. The purpose of this paper is to provide the first look at the properties of these iportant odes in stars of large angular velocity. We do this by solving the stellar pulsation equations for the r-odes of the rapidly rotating unifor density stellar odels which are known as the Maclaurin spheroids. The pulsations of these odels were studied over a century ago by Bryan 5, who showed how analytical expressions for all of the odes of these stars could be found. We follow the general strategy developed by Bryan to derive analytical expressions for the r-odes of these stars. We use soewhat different coordinates than Bryan, however, in order to obtain real representations of the r-odes of priary interest to us here using purely real coordinates. We generalize the traditional definition of r-ode to include any ode whose frequency vanishes linearly with the angular velocity of the star. Such odes have as their principal restoring force the Coriolos force, and hence it is appropriate to call the rotation odes or generalized r-odes 6. We find a very large nuber of generalized r-odes in the Maclaurin spheroids 7. In particular for each pair of integers l and with l0) we find l or l1 for the 0 case distinct r-odes. The classical r-odes as studied for exaple by Papaloizou and Pringle 8 correspond here to the case l1 9. We find that any of the previously unstudied r-odes about 30% of those exained are also subject to the gravitational radiation driven instability in stars without internal fluid dissipation. These new r-odes couple to higher order gravitational ultipoles and consequently are expected to be of less astrophysical iportance than the l13 ode that is of priary iportance in the instability discussed by Lindblo, Owen, and Morsink 3. In Sec. II we review a few iportant facts about the equilibriu structures of the Maclaurin spheroids. In Sec. III we present the equations for the odes of rapidly rotating stars using the two-potential foralis 10. This foralis deterines all of the properties of the odes of rotating stars fro a pair of scalar potentials: a hydrodynaic potential U and the gravitational potential. The equations satisfied by these potentials are coupled second-order partial differential equations with suitable boundary conditions at the surface of the star and at infinity. These equations becoe extreely siple in the case of uniforly rotating unifor-density stars. In Sec. IV we introduce coordinates which allow the equations for the two potentials to be solved analytically. And following in the footsteps of Bryan 5, we present the general solutions to these equations for the generalized r-odes of the Maclaurin spheroids. In Sec. V we give expressions for the various boundary conditions in the coordinates adapted to this proble. In Sec. VI we deduce /99/594/ /$ The Aerican Physical Society

2 LEE LINDBLOM AND JAMES R. IPSER PHYSICAL REVIEW D the eigenvalue equation that deterines when the boundary conditions ay be satisfied. In Sec. VI we also present explicit solutions to this eigenvalue equation for a large nuber of generalized r-odes. In the liit of sall angular velocity we tabulate a coplete set of solutions for all generalized r-odes with l6. We also present graphically the angular velocity dependence of each of these r-odes which is unstable to the gravitational radiation instability. In Sec. VII we analyze the analytical expressions for the eigenfunctions of these odes. We show that in the case of the classical r-odes, the l1 case here, the eigenfunctions U and have particularly siple fors. In particular, each of these eigenfunctions is siply r 1 Y 1 (,) where r, and are the standard spherical coordinates ultiplied by soe angular velocity dependent noralization. We also show that these classical r-ode eigenfunctions have the unexpected property that p, the Lagrangian pressure perturbation, vanishes identically throughout the star. In Sec. VIII we discuss soe of the interesting iplications of this analysis. And finally, in the Appendix we explore in soe detail the properties of the rather unusual bi-spheroidal coordinate syste needed in Sec. IV to solve the pulsation equations for the hydrodynaic potential U. II. THE MACLAURIN SPHEROIDS The uniforly rotating unifor-density equilibriu stellar odels are called Maclaurin spheroids. The structures of these stars are deterined by solving the tie independent Euler equation 0 a 1 x y p..1 In this equation p is the pressure, is the density, is the angular velocity, and is the gravitational potential of the equilibriu star. Using the expression for the gravitational potential of a unifor-density spheroid 11, it is straightforward to show that the solution to Eq..1 for the pressure is pg o 1 o 1 o cot 1 o a x y z, 1 o o. where a is the focal radius of the spheroid, G is Newton s constant, and o is related to the eccentricity e of the spheroid by e 1/(1 o ). Siilarly, it follows that the angular velocity of the star is related to the shape of the spheroid by G o 13 o cot 1 o 3 o..3 We note that sall angular velocities,, correspond to sall eccentricities e and large o. The surfaces of these stellar odels are the surfaces on which the pressure vanishes: x y o 1 z o a..4 This surface is an oblate spheroid. Let us denote the equatorial and polar radii of this spheroid as R e and R p respectively. We see fro Eq..4 that R e a o 1, R p a o..5.6 If we consider a sequence of uniforly rotating spheroids having the sae total ass, then the volue of each of the spheroids in the sequence is the sae since the density is constant. Let R denote the average radius of the spheroid: R 3 R e R p. It follows that R is constant along this sequence since the volue of a spheroid is V4R 3 /3. Thus the angular velocity dependence of the focal radius a is deterined by R3 a 3 o o 1,.7 since o is related to the angular velocity of the spheroid by Eq..3. This expression deterines then the angular velocity dependencies of the equatorial and polar radii of the spheroid: R e R o 1 o 1/6,.8 1/3 R p R o..9 o 1 In the work that follows we will need the quantity n a a p, where n a is the outward directed unit noral to the surface of the star, in order to evaluate certain boundary conditions associated with the stellar pulsations. Since a p is also noral to the surface of the star, we ay use the expression (n a a p) a p a p and Eq.. to obtain n a a p4g o 1 o 1 o cot 1 o x 1/ o 4 y z 1 o,.10 where (x,y,z) are to be confined to the surface defined by Eq..4. III. THE PULSATION EQUATIONS The odes of any uniforly rotating barotropic stellar odel are deterined copletely in ters of two scalar potentials U and 10. The potential is the Newtonian gravitational potential, while U is a potential that priarily describes the hydrodynaic perturbations of the star: U p, 3.1 where p is the Eulerian pressure perturbation, and is the unperturbed density of the equilibriu stellar odel. We assue here that the tie dependence of the ode is e it and

3 GENERALIZED r-modes OF THE MACLAURIN SPHEROIDS PHYSICAL REVIEW D that its aziuthal angular dependence is e i, where is the frequency of the ode and is an integer. The velocity perturbation v a is deterined by solving Euler s equation. This reduces in this case to an algebraic relationship between v a and the potential U: v a iq ab b U. 3. The tensor Q ab depends on the frequency of the ode, and the angular velocity of the equilibriu star : 1 Q ab 4 ab 4 za z b i a v b. 3.3 In Eq. 3.3 the unit vector z a points along the rotation axis of the equilibriu star, ab is the Euclidean etric tensor the identity atrix in Cartesian coordinates, and v a is the velocity of the equilibriu stellar odel. The potentials U and are deterined then by solving the perturbed ass conservation and gravitational potential equations. In this case, these reduce to the following syste of partial differential equations 10 a Q ab b U d U, 3.4 dp a a 4G d dp U. 3.5 In addition these potentials are subject to the appropriate boundary conditions at the surface of the star for U and at infinity for. The stellar pulsation Eqs. 3.4 and 3.5 siplify considerably for the case of uniforly rotating unifor-density stellar odels. In this case d/dp(p) where (p) is the Dirac delta function. Thus the right sides of Eqs. 3.4 and 3.5 vanish except on the surface of the star. Further, the density that appears on the left side of Eq. 3.4 ay be factored out. The resulting equations then in the unifordensity case are siply a a U4z a z b a b U0, a a 4G pu, where is related to the frequency of the ode by These equations are equivalent to those used by Bryan 5 in his analysis of the oscillations of the Maclaurin spheroids. Next we wish to consider the boundary conditions to which the functions U and are subject. Let denote a function which vanishes on the surface of the star, and which has been noralized so that its gradient, n a a, is the outward directed unit noral vector there, n a n a 1. First, the function U ust be constrained at the surface of the star, 0, in such a way that the Lagrangian perturbation in the pressure vanishes there, p0. This condition can be written in ters of the variables used here by noting that pp va 3.9 i a p. Then using Eqs. 3.1 and 3. the boundary condition can be written in ters of U and as 0UQ ab a p b U The perturbed gravitational potential ust vanish at infinity, li r 0. In addition ust as a consequence of Eq. 3.7 have a finite discontinuity in its first derivative at the surface of the star. In particular the derivatives ust satisfy n a a 0 n a a 4G U n a a p The proble of finding the odes of unifor-density stars is reduced therefore to finding the solutions to Eqs. 3.6 and 3.7 subject to the boundary conditions in Eqs and IV. SOLVING FOR THE POTENTIALS In this section we find the general solutions for the potentials U and that satisfy Eqs. 3.6 and 3.7. This analysis basically follows that of Bryan 5 except for soe changes to odernize notation, and a change of coordinates to express in a purely real anner the solutions of interest to us here. Our priary concern here is to find expressions for the generalized r-odes of the Maclaurin spheroids. We first introduce a syste of spheroidal coordinates that are useful in solving for the gravitational potential. Thus we introduce the coordinates,, that are related to the usual Cartesian coordinates (x, y,z) by the transforation: xa 11 cos, ya 11 sin, za, where a is defined in Eq..4 above. These are the standard oblate spheroidal coordinates. It is straightforward to show that x y 1 z a. 4.4 Thus the surfaces of constant are oblate spheroids, with o corresponding to the surface of the star. The coordinate has the range 0 o within the star, and o outside. The surface 0 corresponds to a disk of radius a within the

4 LEE LINDBLOM AND JAMES R. IPSER PHYSICAL REVIEW D equatorial plane of the star. The nature of the surfaces of constant can siilarly be explored by noting that x y 1 z a. 4.5 Thus the constant surfaces are hyperbolas. The coordinate is confined to the range 11, with 1 corresponding to the rotation axis of the star, and 0 to the portion of the equatorial plane outside the disk of radius a. The coordinate easures angles about the rotation axis. Equation 3.7 for the gravitational potential in these spheroidal coordinates becoes , 4.6 except on the surface of the star o. The separable solutions to these equations are functions of the for P l (i)p l ()e i and Q l (i)p l ()e i. The associated Legendre functions Q l () diverge at 1, consequently only the functions P l () appear in these solutions. The associated Legendre functions P l (i) diverge as l while the Q l (i) vanish as (l1) in the liit. Thus the gravitational potential in the exterior of the star, o, ust have the for Q l i Q l i o P l e i, 4.7 for soe constant. In the interior of the star o the situation is ore coplicated. Both P l (i) and Q l (i) are bounded in the liit 0. However, we ust insure that the solution is sooth across the disk 0. The functions Q l (i) are non-zero at i0 see Batean 1 Eqs , and For the case of l odd the function Q l (i)p l () is therefore discontinuous at the disk 0, and consequently it does not satisfy Laplace s equation there. Siilarly for l even the function Q l (i)p l () is continuous but not differentiable at 0, and so it does not satisfy Laplace s equation there either. Thus, in the interior of the star, o, the solution to Eq. 3.7 for given l and is P l i P l i o P l e i. 4.8 The potentials in Eqs. 4.7 and 4.8 have been noralized so that is continuous at the surface of the star o. Following the analysis of the gravitational potential equation, we introduce a second syste of coordinates (,,) which allow the equation for the hydrodynaic potential, Eq. 3.6, to be written in a convenient for. These coordinates are related but not identical to those used by Bryan 5: xb1 1 cos, yb1 1 sin, zb Here the focal radius b is related to a by a b 4 41 o. 4.1 The paraeter b is real for frequencies, such as those of the r-odes, which satisfy 4. As we shall see, Eq. 3.6 for the potential U separates nicely in ters of these coordinates. However these new coordinates are rather unusual, so we present an in depth discussion of the in the Appendix. In suary, the coordinates and cover the interior of the star when their values are confined to the doains o 1 and o o, where o is defined as o a o b 4 o 41 o The surface 1 corresponds to the rotation axis of the star, and the surface 0 corresponds to the equatorial plane of the star. The surface of the star, o, is divided into three regions in this coordinate syste. The portions of the stellar surface nearest the two branches of the rotation axis correspond to the surfaces o, while the portion of the stellar surface that includes the equator corresponds to the surface o. The coordinate coincides with the value of the coordinate in that portion of the surface of the star where o. In the other portions of the surface of the star the value of coincides with. These facts will be essential in iposing the boundary conditions in the next section. Equation 3.6 for the potential U reduces to the following in ters of the coordinates (,,) 1 U U 1 11 U The separated solutions of this equation are associated Legendre functions of and. The coordinate includes 1 in its range, so the non-singular separated solutions to Eq are P l ()P l ( )e i and P l ()Q l ( )e i. The angular coordinate does not include 1 in its range. Thus at this stage, it is not possible to eliinate the Q l ( ) solution without iposing the boundary conditions. V. IMPOSING THE BOUNDARY CONDITIONS In order to obtain the physical solutions to the stellar pulsation equations, we ust now ipose the boundary conditions, Eqs and The siplest boundary condition is the one that involves the derivatives of, Eq

5 GENERALIZED r-modes OF THE MACLAURIN SPHEROIDS PHYSICAL REVIEW D In the coordinates,, used to find the solution for in Eqs. 4.7 and 4.8, the unit noral vector to the surface of the spheroid n a has only one nonvanishing coponent, n : n 1 a o 1 o. 5.1 Thus, the noral derivatives that appear in the boundary condition, n a a, can be expressed siply as n. The gradient of the pressure, n a a p that appears in Eq is given by Eq..10. When evaluated at the surface of the spheroid this reduces to n a a p4g a o o 1 o 1 o cot 1 o. 5. Thus, using Eqs. 4.7, 4.8, 5.1 and 5. the boundary condition Eq on is equivalent to o 1 dq l i o Q P l i o d l e i o 1 dp l i o P P l i o d l e i P l e i U o 1 o cot 1 o. 5.3 The first iediate consequence of this boundary condition is that the potential U ust be proportional to P l () on the surface of the star. In the last section we found that the potential U was soe linear cobination of P l ()P l ( )e i and P l ()Q l ( )e i. As we show in the Appendix, the surface of the star is soewhat coplicated in the (,,) coordinate syste. For the portion of the surface of the star that includes the equator, we found that o and. This fixes the angular dependence of U. Therefore, throughout the star U ust have the for U P l P l o P l e i, 5.4 where is an arbitrary constant. On the portion of the surface of the star that includes the equator, this expression reduces to UP l ( )e i P l ()e i. On the portion of the surface that includes the positive rotation axis, o, the expression for U reduces to U P l ()e i P l ()e i since here. Finally, on the portion of the surface that includes the negative rotation axis, o, the expression for U also reduces to U P l ()e i P l ()e i since here. Consequently, the potential U reduces to the expression U P l ()e i everywhere on the surface of the star. Thus, the boundary condition on reduces to o 1 dq l i o Q o 1 dp l i o l i o d P l i o d o 1 o cot 1 o. 5.5 We now see why it was necessary to obtain the solutions for U in the strange and coplicated (,,) coordinate syste. These unusual coordinates have two essential properties: first they allow the solutions to Eq. 3.6 to be found in separated for, and second they have the property that one of the coordinates reduces on the surface of the star to the angular coordinate. This last property was needed to allow us to satisfy the boundary conditions using siple separated solutions for both U and. The boundary condition, Eq. 3.10, on the potential U is unfortunately soewhat ore coplicated. The tensor Q ab that is used in Eq is ost siply expressed in cylindrical coordinates see Eq Therefore it is siplest to consider the boundary conditions on U in these coordinates. Let x y denote the cylindrical radial coordinate. Then, the boundary condition Eq can be expressed as 4n z z U n U n U 4 U n a a p. 5.6 The coponents of the unit noral vector to the surface of the spheroid, n a, that appear in Eq. 5.6 can be obtained by taking the gradient of the function that appears on the left side of Eq..4: n o 1 o, 5.7 n z o 1 o. 5.8 The partial derivatives z U and U that appear in Eq. 5.6 are ore difficult to evaluate. To do this we ust evaluate the Jacobian atrix that deterines the coordinate transforation defined in Eqs. 4.9 through The needed partial derivatives are given in the Appendix as Eqs. A13 through A16. These expressions can now be used to transfor the derivatives z and needed in the boundary condition into expressions in ters of and. When evaluated on the portion of the surface of the star with o and using the relationships in Eqs. 4.1 and 4.13, we obtain the following: 4n z z U n U 4 a4 o o 1 1 o U

6 LEE LINDBLOM AND JAMES R. IPSER PHYSICAL REVIEW D Siilarly, when evaluated on the portions of the surface with o and we find 4n z z U n U 4 a4 o o 1 1 o U Finally then, we ay cobine the results of Eqs. 4.7, 5.4, 5.9, and 5.10 to obtain the following representation of the boundary condition Eq. 5.6: o 1 dp l o P l o 4 o 1 d 13 o cot 1 o 3 o 1 o cot 1. o o We point out that Eq is valid on the entire surface of the star. For the portion of the surface where o and on the surface, it is helpful to reeber that P l ( o )P l ()P l ( o )P l (), while P l ()dp l ( o )/dp l ()dp l ( o )/d. In suary then the boundary conditions, Eqs. 5.5 and 5.11 are given by TABLE I. The eigenvalues o of the r-odes of the Maclaurin spheroids in the liit of low angular velocities. The frequencies of these odes are related to o by ( o ) in the low angular velocity liit. Those frequencies denoted with * have the property that ()0. These * odes are subject to a gravitational radiation driven secular instability l l * l * * * l * 1.053* * * l * 1.340* 0.979* * * * VI. THE EIGENVALUES The boundary conditions Eqs. 5.1 and 5.13 can be satisfied for only certain values of the eigenvalue. It is easy to see that the necessary and sufficient condition that there exists a solution to the boundary conditions is B o o 1 o cot 1 o, 5.1 A, o 13 o cot 1 o 3 o 1 o cot 1, o 5.13 where A(, o ) and B( o ) are defined by o 1 dp l o A, o o P l o 4 o 1 d 4, 5.14 B o o 1 dq l i o Q o 1 dp l i o l i o d P. l i o d 5.15 Note that o that appears on the right side of Eq is an iplicit function of and o as given in Eq F, o A, o B o A, o o B o 13 o cot 1 o 3 o o 1 o cot 1. o 6.1 We have verified that Eq. 6.1 is exactly equivalent i.e. up to changes in notation to Eq. 60 of Bryan 5. We now wish to evaluate this eigenvalue equation analytically for the generalized r-odes of slowly rotating stars. The paraeter o deterines the angular velocity of the star through Eq..3. Large values of o correspond to sall angular velocities, and so we ay expand our equations in inverse powers of o. The leading order ters in such an expression for the angular velocity are G 8 15 o 1 6 O 4 7 o. o 6. We now wish to obtain solutions to the eigenvalue equation F(, o )0 for large values of o. We do this by expanding the expressions on the right side of Eq. 6.1 in inverse powers of o :

7 GENERALIZED r-modes OF THE MACLAURIN SPHEROIDS PHYSICAL REVIEW D dp F, o l1 o l / P 4 In the case of the classical r-odes with l1, this expression reduces to: l / d 4 1O o. 6.3 Setting the coefficient of this lowest-order ter to zero, we obtain an equation for the lowest-order expression for the eigenvalue, o : dp l o / 4 P d 4 l o /. o 6.4 Using the Rodrigues forula for the associated Legendre functions, this equation can be transfored for the 0 odes into the for 0 d P l o / d o 1 d1 P l o / d 1, 6.5 where P l is the Legendre polynoial of degree l. Equation 6.5 is equivalent to Bryan s Eq This equation adits l or l1 for the 0 case distinct roots all of which lie in the interval o 13. For the case l 1 the single root of Eq. 6.5 is o /(1). This agrees with the frequency of the classical r-ode of order as found for exaple by Papaloizou and Pringle 8. The odes with 0 are equivalent to those with 0: if o is a solution to Eq. 6.4 for soe, then o is a solution for. In Table I we present nuerical solutions to Eq. 6.4 for a range of different values of l and. We see that for each value of there exist solutions of this equation for each value of l1. For each value of l and there are l different solutions. Thus, there exist a vast nuber of odes whose frequencies vanish linearly with the angular velocity of the star. We indicate with a * those frequencies in Table I that satisfy the condition ()0. The odes satisfying this condition would be driven unstable by gravitational radiation reaction in the absence of internal fluid dissipation i.e. viscosity 3. Next we wish to extend the forula for the eigenvalue to higher angular velocity. We define the next ter in the expansion of the eigenvalue as o o O o Using this definition and Eq. 6.4 for o we find the next order ter in F(, o )tobe F, o l1 oll1 4 5 l1 o O o. ll1 4 o 6.7 We can now deterine the second order correction to the eigenvalue of the ode by solving F(, o )0 for : o4 o ll1 ll1 o 4 5 l1 l Thus, for the classical r-odes the frequency of the ode is given by G O For rapidly rotating stars we ust solve the eigenvalue Eq. 6.1 nuerically. This presents certain unusual nuerical difficulties. In particular the associated Legendre functions P l (i) and Q l (i) that appear in Eq are not coonly encountered. Thus, we digress briefly here to describe how these functions ay be evaluated nuerically. First we note that these functions are essentially real. In particular the functions P l () and Q l () defined by P l ii l P l, 6.11 Q l ii l1 Q l, 6.1 are real for real values of. The functions P l () can be evaluated nuerically using essentially the sae algoriths used to evaluate their counterparts on the real axis. In particular P 1!! 1 /, P 1 1P These expressions can be used as initial values for the recursion forula, lp l l1p l1 l1p l, 6.15 fro which the higher order functions can be deterined. This approach does not work for the Q l however. The proble is that we need to evaluate these functions over a fairly wide range of their arguents, e.g For large values of the recursion for the Q l involves a high degree of cancellation aong the various ters. Standard coputers siply do not have the nuerical precision to perfor these calculations to sufficient accuracy. Instead we rely on an integral representation of Q l. Based on Batean s Eq , wefind

8 LEE LINDBLOM AND JAMES R. IPSER PHYSICAL REVIEW D FIG. 1. Angular velocity dependence of the eigenvalues of the classical r-odes, i.e. those odes with l1 for l6. The frequencies of these odes are related to by (). Q l 1l1 l! l1 l! 1 / 1 t l1/ 1t l 1 cosl1tan 1 t/dt FIG.. Angular velocity dependence of the frequency ( ) of the classical r-ode. The solid curve gives corresponding to the exact solution of Eq. 6.1, while the dotdashed and dashed curves correspond to the first and second order approxiations respectively fro Eq The integrand in this expression is well behaved for all values of 0, and the integrals ay be deterined nuerically quite easily. Using these nuerical techniques, then, it is straightforward to solve for the eigenvalues of Eq. 6.1, F(, o )0, over the relevant range of the paraeter 0.5 o 75. Figures 1,, and 3 display the angular velocity dependence of the eigenvalue for a nuber of r-odes. Figure 1 depicts for the classical r-odes, l1, with l6. This l 13 ode is the one found by Lindblo, Owen, and Morsink 3 to be sufficiently unstable due to gravitational radiation eission that it is expected to cause all hot young rapidly rotating neutron stars to spin down to low angular velocities within about one year. We have verified that our nuerical solutions of Eq. 6.1 agree with those of Eq. 6.10, up to ters that scale as 5. Figure presents the angular velocity dependence of the frequency of the classical l13 r-ode as easured in an inertial frae. The solid curve corresponds to the exact solution to Eq. 6.1, while the dot-dashed and dashed curves represent the first and second order approxiations respectively given in Eq The units in Fig. for both the vertical and horizontal axis scale as G. The value of g/c 3 chosen here represents a typical average density for a neutron star. Figure illustrates three interesting features about the frequencies of this ode in large angular velocity stars. First, the frequency is only about /3 that predicted by the first-order forula for stars rotating near their axiu angular velocity. This eans that the gravitational radiation reaction force, which scales as ( ) 5, could be about 1/5 of that predicted by the first order forula. Unless the ass and current ultipoles of the rapidly rotating odels are uch larger than their nonrotating values. Second, the frequencies of these odes first increase and then decrease as the angular velocity of the star is reduced. This eans that the tie evolution of the gravitational radiation signal fro these sources will be ore coplicated than had been anticipated by Owen et al. 4 on the basis of the first order expression for the frequency. In particular it appears that the evolution of the frequency will not be onotonic for the ost rapidly rotating stars. Third, the accuracy of the second-order forula for the frequency is in fact considerably worse than that of the siple first-order forula for very rapidly rotating stars. This suggests that any application of low angular velocity expansions of the r-odes to the study of rapidly rotating stars is soewhat suspect. Figure 3 depicts the angular velocity dependence of for several other previously unstudied r-odes. The odes depicted in Fig. 3 all have the property that ()0, and hence these odes would all be subject to the gravitational radiation secular instability in the absence of internal fluid dissipation i.e. viscosity. These additional odes all couple to higher order gravitational oents than the classical l13 ode. Thus, these additional odes probably do not play a significant role in the astrophysical process which spins down hot young neutron stars. FIG. 3. Angular velocity dependence of the eigenvalues of r-odes with 5l1 which are unstable to the gravitational radiation instability. The frequencies of these odes are related to by ()

9 GENERALIZED r-modes OF THE MACLAURIN SPHEROIDS PHYSICAL REVIEW D VII. THE EIGENFUNCTIONS The eigenfunctions associated with these odes are those given in Eqs. 4.7, 4.8, and 5.4. Inside the fluid we have P l i P l i o P l e i e it, U P l P l o P l e i e it, while outside we have Q l i Q l i o P l e i e it The coefficients and are deterined by solving the boundary conditions. The eigenvalue Eq. 6.1 is the consistency condition for the existence of these solutions. Given a solution of the eigenvalue equation then, the ratio of and can be deterined fro Eq. 5.1: 1 o 1 o cot 1 o B o. 7.4 It ight appear at first glance that the eigenfunctions associated with the l distinct r-odes are identical. However, the coordinates and depend on the eigenvalue. Thus, the spatial dependence of U will be different for each of these odes. Interestingly enough, however, the spatial dependence of the gravitational potential,, depends only on l and, and hence is the sae for all l distinct odes. While the expressions for the eigenfunctions are quite siple in ters of the special spheroidal coordinates used here, they are rather coplicated when expressed in ore traditional coordinates. One exception to this is the case of the classical r-odes, i.e. those with l1. In this case the needed associated Legendre function, P 1, has the siple expression P 1 1 1!!1 /. 7.5 Using the expressions for the bi-spheroidal coordinates (, ) in ters of the cylindrical coordinates (,z) given in Eqs. A6 and A7, it follows that P 1 P 1 1 1!!P 1 o z R p R e. 7.6 Thus, the hydrodynaic potential U is given by U z R p R e e i e it. 7.7 Aside fro the overall noralization then, the spatial dependence of U is copletely independent of the angular velocity of the star. A siilar expression can be obtained for the gravitational potential within the star. Using the fact that P 1 ip 1 1 1!!P 1 i o z R p R e, 7.8 the gravitational potential is given by z R p R e e i e it. 7.9 Thus the spatial dependencies of the potentials U and are identical for the classical r-odes. This spatial dependence can also be expressed in spherical coordinates as r 1 Y 1 (,), up to an overall noralization. It is interesting that this sae function satisfies both the hydrodynaic Eq. 3.6 and the gravitational potential Eq The Eulerian perturbation in the pressure for these odes is deterined fro these two potentials by Eq. 3.1: p z R p R e e i e it We note that the constants and used here have been scaled by the factor (1) (1)!! copared to their original definitions in Eqs. 4.8 and 5.4. It is also instructive to evaluate the Lagrangian perturbation of the pressure, p, as defined in Eq Using the expressions in Eqs. 7.7 and 7.9, we find that p 1 1 o cot 1 o 1 o o o 13 o cot 1 o 3 o U The ter enclosed in brackets in Eq depends only on the frequency of the ode, the angular velocity of the star through o ), and the aplitudes of the perturbations and. When the boundary condition Eq or equivalently Eq is satisfied, this ter vanishes. Thus, we find that the Lagrangian perturbation in the pressure p vanishes identically for the classical r-odes of the Maclaurin spheroids. And this result which is a consequence of the extreely siple eigenfunctions for these odes holds for stars with any angular velocity. VIII. DISCUSSION Our analysis, which follows closely in the footsteps of the rearkable analysis of Bryan, provides several interesting

10 LEE LINDBLOM AND JAMES R. IPSER PHYSICAL REVIEW D insights into the properties of the r-odes of rapidly rotating stars. It deonstrates for exaple, that these odes actually do exist in rapidly rotating barotropic stars, and are not erely unfulfilled expectations of the first-order perturbation theory as claied by soe authors 15. This analysis shows that soe properties of the r-odes are not well approxiated by the low angular velocity expansions. Figure illustrates, for exaple, that the first-order expression for the angular velocity dependence of the frequency of the classical r-odes is in fact superior to the second-order expression in the ost rapidly rotating stars. This analysis shows that the frequency evolution of the gravitational radiation eitted by the r-ode instability is likely to be ore interesting than had previously been thought 4. Figure shows that the frequency of these odes will first increase and then decrease as the angular velocity of the star is reduced by the eission of gravitational radiation. This analysis shows that there is a uch larger faily of r-odes than the classical r-odes studied for exaple by Papaloizou and Pringle 8. For each pair of integers l and which satisfy l 0) there exist l or l1 in the 0 case distinct r-odes. This analysis shows that a significant fraction of these previously unstudied r-odes are subject to the gravitational radiation driven secular instability. This analysis has derived siple analytical expressions for the eigenfunctions of the classical r-odes. Both of the potentials U and are proportional to r 1 Y 1 (,) for the classical r-odes in Maclaurin spheroids of any angular velocity. This analysis shows that the Lagrangian variation in the pressure, p, associated with the classical r-odes vanishes identically in Maclaurin spheroids of arbitrary angular velocity. Thus, the r-odes of the Maclaurin spheroids provide a copletely unsuitable odel for the study of the effects of bulk viscosity on the r-odes. This analysis also provides an interesting atheatical exaple of a hyperbolic eigenvalue proble. The r-odes studied here have the property that 4. Thus, the equation satisfied by the potential U, Eq. 3.6, is in fact hyperbolic. Nevertheless, the boundary condition iposed on the potential U, Eq. 3.10, isofthe ixed Dirchlet-Neuann type that is generally associated with elliptic probles. The analytical solutions given here illustrate that this unusual hyperbolic eigenvalue proble does nevertheless adit well behaved solutions. ACKNOWLEDGMENTS We particularly thank J. Friedan, K. Lockitch and N. Stergioulas for reading and criticizing the first draft of this anuscript. We also thank P. Brady, C. Cutler, B. J. Owen, and K. S. Thorne for helpful discussions concerning this work; and we thank S. Morsink for providing us with her notes on the non-existence of pure axial odes in the Maclaurin spheroids. This research was supported by NSF grants PHY and PHY , and by NASA grants NAG and NAG APPENDIX: BI-SPHEROIDAL COORDINATES The coordinates and defined in Eqs. 4.9 through 4.11 are rather unusual. The purpose of this appendix is to explore the geoetrical properties of these coordinates. The coordinates and cover the planes usually described with the spherical coordinates r and or equivalently the cylindrical coordinates x y and z. First, we note that it follows fro Eqs. 4.9 through 4.11 that 1 z 4 b. A1 Thus, the surfaces of constant with 1) are spheroids for the r-odes which have 4. The particular surface o, with o defined by o a o b 4 o 41 o 4 1, A is identical to the surface of the star o. Paradoxically, the constant surfaces also satisfy the equation 1 z 4 b. A3 Thus, the constant surfaces are the sae faily of spheroids as the constant surfaces. Thus the coordinates and constitute a bi-spheroidal coordinate syste. We note that the equatorial and polar radii of the spheroid, R e and R p defined in Eqs..5 and.6, are related to the constants b and o by R e b 1 o, A4 R p b o 4. A5 In order to understand how the coordinates and cover the interior of the star it is helpful to introduce two additional coordinates s and : s sin R e o, A6 s cos z R p o. A7 The coordinate s has the value 1 on the surface of the star and 0 at its center. The coordinate ranges fro the value 0 on the positive rotation axis, through / on the equatorial plane, to on the negative rotation axis. Thus, the coordinates s and ap the interior of the star into the unit sphere

11 GENERALIZED r-modes OF THE MACLAURIN SPHEROIDS PHYSICAL REVIEW D in a natural way. It will be instructive to express the coordinates and then in ters of s and. These expressions can be obtained fro Eqs. A6 and A7: the star. We do this by evaluating the Jacobian atrix i.e. the atrix of partial derivatives of the transforation: 1 uv, A8 z 1 b 4, A13 where 1 uv, A9 1 1 b, A14 u1s s o cos, A10 z 1 b 4, A15 v u 4s o cos. A11 We note that while Eqs. A6 and A7 are syetric in and, this syetry has been broken in order to obtain the expressions A8 and A9. We will now show that the coordinates and also cover the interior of the star when their values are restricted to the ranges: o 1 and o o. It is easy to see fro the second equalities in Eqs. A6 and A7 that each point (, ) in the doain o 1 and o o corresponds to a point within the star i.e. a point with s1). Proving the converse is ore difficult. We do this in three steps: First we show that the functions and are real and finite at each point within the interior of the star. Second we show that these functions have no critical points e.g. no axia or inia except on the surface of the star. Third, and last, we show that and are confined to the ranges o 1 and o o for points on the boundary. First we show that and are real and finite for each point within the star. The quantity u defined in Eq. A10 is positive for points within the star i.e. points with s1). Next we show that v defined as the positive root in Eq. A11 is real and thus positive for points within the star. We do this by re-writing v as v 1s o cos s1 o sin 1s o cos s1 o sin 1s o cos s1 o sin 1s o cos s1 o sin. A1 Each of the ters on the right side of Eq. A1 is positive, since each is 1 plus the inner product of a pair of unit vectors ultiplied by s. Thus v 0 and so v is real and positive. Further, vu and so and are positive. Thus and are finite and real for each point of the interior of the star. Second we wish to show that the transforation between the bi-spheroidal coordinates (, ) and the standard cylindrical coordinates (,z) is non-singular everywhere within 1 1 b. A16 These expressions show that the Jacobian atrix, and hence the coordinate transforation between (, ) and (,z), is non-singular except for the points within the star where 1, or o. The transforation is also singular at 1, however, these points are not in the range of interest to us here. The non-singularity of the Jacobian atrix proves that a and a are non-vanishing everywhere within the star. Thus, the axiu and iniu values of and will only occur on the surface or the rotation axis of the star. Third, and finally, we explore the values of the coordinates (, ) at specific physical locations in the star, e.g. the surface of the star, the rotation axis, etc. We begin first with the rotation axis. In the (s, ) coordinates defined in Eqs. A6 and A7, the rotation axis corresponds to the points where sin 0. It follows then that these points correspond to 1 and s o. Thus, the rotation axis is the surface 1, a singular surface of the coordinate transforation. The coordinate takes on its entire range, o o, for points along this axis. The equatorial plane of the star, cos 0, corresponds to the coordinate surface 0. The coordinate 1s s o ranges fro 1 on the rotation axis to the value o on the surface of the star. The surface of the star, s1, is unexpectedly coplicated in the (, ) coordinate syste. For points of the stellar surface near the equator, cos o, the function v defined in Eq. A11 has the value: v o cos. Thus, the surface of the star in this region has o while cos. In the regions of the stellar surface near the rotation axis, cos o, the function v defined as the positive root in Eq. A11 has the value: vcos o. Thus the stellar surface in these regions have o and cos. The role of and as radial and angular coordinates are reversed therefore in different regions of the star. In suary then, we have shown that the axiu and iniu values of are 1 and o respectively, while the

12 LEE LINDBLOM AND JAMES R. IPSER PHYSICAL REVIEW D axiu and iniu values of are o. This concludes our deonstration that the coordinates (, ) when confined to the ranges o 1 and o o do for a non-singular coordinate syste that covers the interior of the star. The transforation between these and the usual cylindrical coordinates is singular only on the rotation axis, 1, and at the singular point o on the surface of the star. Finally, it will be useful to work out the relationship between the surface values of the (, ) coordinates with those of the oblate spheroidal coordinates,. Using the definitions of the (s, ) coordinates introduced in Eqs. A6 and A7, and the definitions of the oblate spheroidal coordinates, fro Eqs. 4.1, 4., and 4.3, it is straightforward to show that s sin 11, A17 o 1 s cos o. A18 Thus on the surface of the star, o and s1, we have cos. We have also shown that cos on the portion of the surface of the star where cos o, and that cos on the portion of the surface where cos o. Thus we find that on the portion of the surface where o, and that on the portion of the surface where o. 1 N. Andersson, Astrophys. J. 50, J. L. Friedan, and S. M. Morsink, Astrophys. J. 50, L. Lindblo, B. J. Owen, and S. M. Morsink, Phys. Rev. Lett. 80, B. J. Owen, L. Lindblo, C. Cutler, B. F. Schutz, A. Vecchio, and N. Andersson, Phys. Rev. D 58, G. H. Bryan, Philos. Trans. R. Soc. London A180, The ter r-ode traditionally refers to odes whose velocity perturbations are proportional to r Y l J. Papaloizou and J. E. Pringle, Mon. Not. R. Astron. Soc. 18, Such odes were called r-odes because they are analogous to the Rossby waves of atospheric physics. The odes studied here do not fit neatly into the traditional classification of the r- and g-odes of slowly rotating stars. The velocity perturbations of our odes are not in general purely axial as are the traditional r-odes nor purely polar as are the traditional g-odes in the sall angular velocity liit. Thus we have extended the traditional definition of r-ode by defining it in ters of the physical process that governs the ode, rotation, rather than the syetry of its velocity perturbation. Our generalized r-odes include all odes considered r-odes under the traditional definition. 7 The Maclaurin spheroids are a special case of barotropic stellar odels: stars in which the density of the perturbed fluid depends only on the pressure. In non-barotropic stars there also exist buoyancy forces that doinate the behavior of a class of odes called g-odes. The non-barotropic analogues of the generalized r-odes discussed here will alost certainly be influenced at soe level by buoyancy forces. At sufficiently sall angular velocities i.e. at angular velocities saller than the Brunt-Väisälä frequency buoyancy forces could well doinate the dynaics of soe of these odes; and such odes ight then be called generalized g-odes in nonbarotropic odels. By analogy soe ight prefer to call these odes generalized g-odes even in the barotropic case. Since we do not at present know which if any of these odes ight be doinated by buoyancy forces in the non-barotropic case, we prefer to refer to the here as r-odes. In the barotropic case the dynaics of these odes are doinated by rotational forces. 8 J. Papaloizou and J. E. Pringle, Mon. Not. R. Astron. Soc. 18, It has been traditional in the literature to discuss the r-odes in ters of a vector-spherical-haronic decoposition of the Lagrangian displaceent. In ters of that traditional decoposition, the classical r-odes correspond to the case l. The analysis presented here uses a certain scalar potential U fro which the Lagrangian displaceent is deterined. The analytic solutions that we find for U are particular spheroidal haronic functions. The classical r-odes correspond to the case l1 in ters of these spheroidal haronics. The potential U for the classical r-odes can also be expressed as a spherical haronic with l1. In general, however, the odes discussed here do not have siple finite representations in ters of spherical haronics. 10 J. R. Ipser and L. Lindblo, Astrophys. J. 355, S. Chandrasekhar, Ellipsoidal Figures of Equilibriu Yale University Press, New Haven, H. Batean, in Higher Transcendental Functions, edited by A. Erdélyi McGraw-Hill, New York, 1953, Vol. I. 13 The Legendre polynoial P l has l distinct roots, all of which lie in the interval (1,1) 1. It follows that dp l /d has l 1 roots which interleave the roots of P l, while d P l /d has l roots which interleave those of d 1 P l /d 1, etc. For the case 0 then, the roots of Eq. 6.5 are siply the roots of dp l /d. Thus, there are l1 distinct roots for o in the 0 case, and these lie in the interval o. For the case 0 the first ter in Eq. 6.5 has l distinct zeros, while the second ter has l including the one at o ) zeros that interleave those of the first. Evaluating the right side of Eq. 6.5 at these zeros of its second ter, we find that its sign alternates. Thus, the right side of Eq. 6.5 has l zeros that interleave those of its second ter. Thus, these zeros lie in the range o. These are all of the zeros of the right side of Eq. 6.5 because it is a polynoial of degree l

13 GENERALIZED r-modes OF THE MACLAURIN SPHEROIDS PHYSICAL REVIEW D We point out that Eq does not agree with the secondorder expressions for the frequency of the classical r-odes given by J. Provost, G. Berthoieu, and A. Rocca, Astron. Astrophys. 94, , and ore recently by K. D. Kokkotas and N. Stergioulas, Astron. Astrophys. to be published, astro-ph/ This is not entirely surprising since the Provost et al. calculation uses the Cowling approxiation, and self-gravitational effects do influence the frequencies of the r-odes at the 3 order. Kokkotas and Stergioulas only clai that their expression is an approxiation. 15 J. Provost, G. Berthoieu, and A. Rocca, Astron. Astrophys. 94,