A&A 375, 680-690 (2001)
Y. Sobouti1,2 - V. Rezania1,3
1 - Institute for Advanced Studies in Basic Sciences, Gava Zang, Zanjan 45195, Iran
2 - Center for Theoretical Physics and Mathematics, AEOI, PO Box 11345-8486, Tehran, Iran
3 - Department of Physics, University of Alberta, Edmonton AB, T6G 2J1 Canada
Received 1 February 2001 / Accepted 19 June 2001
An analysis of the toroidal modes of a rotating fluid, by means of the differential equations of motion, is not readily tractable. A matrix representation of the equations on a suitable basis, however, simplifies the problem considerably and reveals many of its intricacies. Let be the angular velocity of the star and () be the two integers that specify a spherical harmonic function. One readily finds the followings: 1) Because of the axial symmetry of equations of motion, all modes, including the toroidal ones, are designated by a definite azimuthal number m. 2) The analysis of equations of motion in the lowest order of shows that Coriolis forces turn the neutral toroidal motions of () designation of the non-rotating fluid into a sequence of oscillatory modes with frequencies . This much is common knowledge. One can say more, however. a) Under the Coriolis forces, the eigendisplacement vectors remain purely toroidal and carry the identification (). They remain decoupled from other toroidal or poloidal motions belonging to different 's. b) The eigenfrequencies quoted above are still degenerate, as they carry no reference to a radial wave number. As a result the eigendisplacement vectors, as far as their radial dependencies go, remain indeterminate. 3) The analysis of the equation of motion in the next higher order of reveals that the forces arising from asphericity of the fluid and the square of the Coriolis terms (in some sense) remove the radial degeneracy. The eigenfrequencies now carry three identifications (), say, of which s is a radial eigennumber. The eigendisplacement vectors become well determined. They still remain zero order and purely toroidal motions with a single () designation. 4) Two toroidal modes belonging to and get coupled only at the order. 5) A toroidal and a poloidal mode belonging to and , respectively, get coupled but again at the order. Mass and mass-current multipole moments of the modes that are responsible for the gravitational radiation, and bulk and shear viscosities that tend to damp the modes, are worked out in much detail.
Key words: stars: neutron - stars: oscillations - stars: rotation
Recent years have seen a surge of interest in the small oscillations of rotating fluid masses. The reason for the excitement is the advocation by relativists that in rapidly rotating neutron stars, the gravitational radiation drives the r-modes to become unstable, and while spinning down the star, may itself be amenable to detection (see the recent review by Andersson & Kokkotas 2000). Nevertheless, the oscillations of rotating objects is an old problem. In the past few decades it has been studied by many investigators and from various points of view. Complexity of the problem arises from the fact that a fluid can support three distinct types of motions, derived from, say, a scalar potential, from a toroidal vector potential and from a poloidal vector potential. These are the motions associated predominantly with the familiar p-, g- and toroidal- oscillations of the fluid. Each of these motions, in turn, can be given an expansion in terms of vector spherical harmonics. The modes of an actual star are a mixture of the three types mentioned above and of the different spherical harmonic components. Sorting out this mixture and classifying the modes into well-defined sequences has not been an easy task. Moreover, and more often than not, it has not been realized that g-modes of spherically asymmetric configurations are not apt for perturbation analysis as the low frequency tail of their spectrum is a fragile structure. It is driven by minute buoyancy forces and can be completely wiped out by almost any perturbing agent, such as Coriolis, asphericity and magnetic forces.
Here we show that a matrix representation of the equations of motions provides a set of algebraic equations that are much easier to cope with than their differential counterparts. In Sect. 2 we write down the equilibrium structure and the linearized equations of motion of a rotating star. In Sect. 3 we introduce the matrix representation of these equations. In Sects. 4 and 5 we sort out the toroidal-poloidal components and spherical harmonic constituents of the matrices. In Sects. 6 and 7 we give an ordering of the various components in powers of and sort out the equations of motion in various orders of magnitude. References and bibliographical notes relating to mode calculations in rotating stars are collected in Sect. 7.4. In Sect. 8 we discuss the numerical results. A rotating neutron star can be slowed down by gravitational radiation through the mass and mass-current multipole moments of the modes. The modes, in turn, can be damped out by bulk and shear viscosities present in the star. The time scales of relevant damping mechanisms are analyzed in Sect. 9. Calculations of matrix elements and presentation of appropriate basis sets are given in the appendices.
density, the pressure and the gravitational potential of a star
rotating with the constant angular frequency
z-axis. The equilibrium condition is
Let a mass element of the rotating fluid at position
displaced by an amount
where, for the
moment, s is the collection of all indices that specify the
displacement in question. This may include its spherical
harmonic specifications, its radial node number, and/or its
poloidal and toroidal nature. The Eulerian change in ,
and U resulting from this displacement, will be
Equation (4) in its integro-differential form is highly
complicated. We convert it into a
set of linear algebraic equations by expanding
of a complete set of known basis vectors,
The basis set can be divided into two poloidal and toroidal subsets, . The set of the eigenmodes, , of Eq. (4) in the absence of rotation also has such exact partitioning. Its poloidal subset comprises the commonly-known g- and p-modes of the fluid. The toroidal subset includes those displacements of the fluid which do not perturb the equilibrium state of the star. For them, . For the sake of mathematical completeness one might say that is a degenerate eigenfrequency and the set of all toroidal motions are its eigendisplacement vectors.
In the presence of rotation, two things happen. a) Each known
poloidal mode of the fluid acquires a small toroidal component. b) The neutral
toroidal displacements organize themselves into a new
sequence of modes and the degeneracy of
Nonetheless, one may still partition the eigensets
remembering that the subsets
unlike the no rotation case, are only
predominantly poloidal and toroidal, respectively. In view of
these considerations, Eq. (5) partitions as
We are not interested in the
poloidal modes of Eq. (8a). They are discussed in ample
detail and in a much wider scope than that of the present work
in Sobouti (1980). Here we concentrate on the
toroidal modes of Eq. (8b). The required information
comes from multiplying the block partitioned forms of all the
matrices in Eq. (7) and extracting the tt and pt
blocks of it. Thus
For a toroidal basis vector we will adopt the following spherical
In Eq. (10) we have suppressed the harmonic index m from 's because a slowly rotating star is axially symmetric. Vectors with different values of m are not mutually coupled. Vectors belonging to the same m, but different 's, however, are coupled. This feature entails a further partitioning of the basis sets into their harmonic subsets, and . Correspondingly, each of the matrices in Eqs. (9) partitions into blocks, designated by a pair of harmonic numbers , say. For example, the rs element of , say, will be obtained from Eq. (6a) by inserting the two vectors and in that equation. In the following we rewrite Eqs. (9) taking into account the new partitioning. Thus,
Expansions of ,
p, and U in powers of
in a corresponding expansion of all the matrices in Eqs. (11). Moreover, having the spherical harmonics forms of
Eqs. (10), integrations over
and dependencies in the calculation of matrix elements can be performed
analytically. These two tasks are carried out in Appendix A.
The results are quoted below:
Equations (12)-(16) allow a consistent expansion of Eqs. (11) at and orders and enable one to decipher the information contained in them.
order Eq. (11a) gives
One simplifying feature: We note that of Eq. (18a) is a constant matrix. Therefore, it will commute with any matrix carrying the same designations, such as , , etc. This feature was used in the derivation of Eq. (18c) and will be used repeatedly in what follows to simplify the matrix manipulations.
Equation (11b) at order
along with Eqs. (18) gives
The case of g-modes is different. Rotation, no matter how small, cannot be treated as a perturbation on them, as they are created by minute buoyancy forces and the low frequency tail of their spectrum gets completely wiped out by any other force in the medium, here the Coriolis forces. This, in mathematical language, means that for g-modes is not invertible and Eq. (19b) is not applicable to them. The way out of the dilemma is to consider the sum of buoyancy and other intruding forces as an inseparable entity, without dividing it to large and small components. The works of Provost et al. (1980) and of Sobouti (1977, 1980) are examples of such treatments. We leave it to the experts in the field to decide whether the scrutiny of g-modes in rotating neutron stars is a crucial or an irrelevant issue.
A systematic extraction of
order terms of the
-block of Eq. (11a) with the help of Eqs. (12)-(16) and elimination of
appearing in it by Eq. (19b) gives
The presence of
in Eqs. (13) and (14c) indicates
that two toroidal motions belonging to
are mutually coupled. Likewise, the
in Eqs. (13b) and (14b)
shows the coupling of toroidal and poloidal motion of
symmetry. This brings in an additional
coupling between two toroidal motions of
and symmetries through the intermediary of poloidal motions.
Therefore, the only unexplored blocks of
Eq. (11a) are those with (
designations. As in Sect. 7.3.1 above, we extract the
order terms of Eq. (11a), but this time with
appearing in it by Eq. (19b) and arrive at
Papaloizou & Pringle (1978) have studied the low frequency g- and r-modes of Eq. (4) in an equipotential coordinate system with their applicability to the short period oscillations of cataclysmic variables in mind.
Sobouti (1980) has studied the problem primarily with the goal of analyzing the perturbative effects of slow rotations on p-modes and demonstrating that rotation, no matter how small, cannot be treated as a perturbation on g-modes. He argues that the g-modes are fragile structures and their low frequency tail of the spectrum, below the rotation frequency of the star, will be completely wiped out by Coriolis and asphericity forces of the star. The criterion for the validity of perturbation expansion is that the perturbing operator should be smaller than the initial unperturbed operator everywhere in the Hilbert space spanned by the eigenfunctions of the unperturbed operator, (Rellich 1969). This condition is not met by g-modes when exposed to rotation, magnetic, tidal forces, etc., as they have vanishingly small eigenfrequencies.
Provost et al. (1981) present an analysis of what they call the "quasi-toroidal modes of slowly rotating stars''. Their work should be noted for the consistency of mathematical manipulations exercised throughout the paper. They noted that in neutrally convective rotating stars one cannot have modes with predominantly toroidal motions. They get mixed with the neutral convective displacements.
Lockitch & Friedman (1999) also address the hybrid modes with comparable toroidal and poloidal motions. Their work should be noted for the emphasis put on the () parities of the hybrid components that get coupled through asphericity forces.
Yoshida & Lee (2000) study Eq. (4) for a) those modes that are predominantly toroidal in their nature and b) for those that have comparable poloidal and toroidal components. Their latter modes are the same as those of Provost et al. (1981).
The bulk properties of some observed neutron stars seem to approximate those of a polytrope of index 1. See Sterigioulas (1998). It has become fashionable to categorize neutron stars as stiff or soft, depending on whether their density distributions are similar to those of polytropes of index smaller than 1.5 or larger, respectively. To have an example of each category, sample calculations are given for polytropes of indices 1 and 2.
The structure of rotating polytropes, taken from Chandrasekhar (1933), is summarized in Appendix C. The ansatz for the scalars , and appearing in Eqs. (10) are given in Appendix B. The required matrix elements are reduced in Appendix A. For a given , the matrices are numerically integrated. The eigenvalues and are calculated from Eqs. (18a) and (20b). Once the eigenvalues are known, the various components of the eigenmatrices , and are calculated from Eqs. (20), (22) and (19b). The results are given in Tables 1 to 3.
In Table 1, to show the convergence of the variational calculations, the eigenvalues are displayed for a polytrope of index 2, , and for N=1,2,3,4,5. This table should be considered as a basis for judging the accuracy of the numerical values. With only five variational parameters, the first, second and third eigenvalues are produced with an accuracy of a few parts in 105, 104 and 102, respectively. Likewise, the numerical values in the remaining tables should be trusted to the same degree of accuracy and for the first few modes.
In Table 2, the second order eigenvalues, and coefficient matrices, , , and are given for polytrope of index 1 and for . In Table 3 we give the same calculations for polytrope of index 2. The eigenvalues are in units of . They are in agreement with those of Lindblom et al. (1999), Yoshida & Lee (2000), and Morsink (2001). Each of these authors have used their own technique which are different from that of the present paper.
A novel feature of the present analysis is the provision of much
detail on eigendisplacement vectors, information that can be
profitably used to calculate any other bulk or local parameter of
the model. For example, for modes belonging to
|Figure 1: Radial behavior of the various components of the eigenfunction of Eq. (23) for , dashed curve; , dot-dashed curve; , dotted curve. Data for Z's are taken from the first column of Table 2. Nodes in all three components are at the center and surface.|
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|Figure 2: Same as Fig. 1 for . Data for Z's are from the second column of Table 2. Note the extra node in all three components.|
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|Figure 3: Same as Fig. 1 for . Data for Z's are from the first column of Table 3. Nodes are at the center and the surface.|
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|Figure 4: Same as Fig. 1 for . Data for Z's are from the second column of Table 3. Note the extra nodes in all components.|
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In this section we study the dissipative effects of viscosity and
gravitational radiation on r-modes. Quite generally and regardless
of whether the star rotates or not, the total energy of an
In a newly born hot neutron star, K, the bulk viscosity has a dominant role in damping out the perturbations and cooling down the star. In colder stars, K and , the gravitational radiation is more important than radiation shear and bulk counterparts. While driving the r-modes to become unstable, it spins down the star. It is believed that the star loses much of its energy and angular momentum through the gravitational radiation in this stage. In a case study of Andersson & Kokkotas (2000), the rotational period increases from 2 ms to 19 ms in one year. Below T=108 K and , the shear viscosity is the dominant factor in cooling down the star.
We wish to thank Sharon Morsink for a careful reading of the manuscript and illuminating comments. We would like to acknowledge the referee for his/her valuable comments.
In calculating the elements of various matrices, the following parameters
and integrals are encountered frequently:
Finally, the spherical harmonics form of the
basis toroidal and poloidal vectors are given in Eqs. (10).
Angular integrals entering the definition of any
matrix element at any desired order are performed analytically.
Integrals in radial directions are left for numerical
Finally, we note that the upper limit of all radial integrals here
is the radius of the non-rotating star instead of that of the rotating
one. We reproduce the argument of Sobouti (1980) to show
that, in most cases, the effect arising from this difference in the limits of
integration is of far higher order in
than would influence the
analysis of this paper. Let
be the distance between two points with coordinate
situated on the surfaces of rotating and non-rotating stars.
order. In a typical error
we expand f(r) about Rand obtain
The vicinity of the center of a star is a uniform medium, in the
sense that, as r tends to zero, ,
U(r), etc. all tend to constant values. Any scalar function,
say, associated with a wave in such a nondispersive uniform and
isotropic environment should satisfy the wave equation
k=const. Furthermore, if this scalar
is associated with the spherical harmonic ,
i.e. if it is
and is finite at
the origin, should tend to zero as .
Therefore should have an expansion of the form
This is how the
solutions of Laplace's equation (k=0), the spherical Bessel function
and many other hypergeometric functions behave. A spherical harmonic vector,
belonging to ,
quite generally can be written in terms of
We adopt the following ansatz for the
1) That a power set is complete for expanding any function of r that behaves as near the origin follows from a theorem of Weiresstraus (Relich 1969; Dixit et al. 1979).
2) We have chosen the ansatz of Eq. (B.3) for their simplicity. They are not the most efficient ones for rapid convergence of variational calculations. The set of the asymptotic expressions that helioseismogists use for eigendisplacement vectors in the sun and other stars would, perhaps, give a faster convergence of the numerical computations, see Christensen-Dalsgaard (1998) and references therein.
The structure of rotating polytropes is taken from a landmark
paper of Chandrasekhar (1933). A summary of what is
needed here with slight changes in his notation is as follows
Let us take the real part of Eq. (4) and write it in the following form
Since the time dependence of
upon taking the time average
of Eq. (D.5) the second integral vanishes and we obtain