A&A 395, 201-208 (2002)
DOI: 10.1051/0004-6361:20021270
S. Yoshida^{1} - U. Lee^{2}
1 - Centro Multidisciplinar de Astrofísica - CENTRA,
Departamento de Física, Instituto Superior Técnico,
Av. Rovisco Pais 1, 1049-001 Lisboa, Portugal
2 -
Astronomical Institute, Graduate School of Science,
Tohoku University, Sendai 980-8578, Japan
Received 25 June 2002 / Accepted 30 August 2002
Abstract
Nonradial oscillations of relativistic neutron stars with a solid
crust are computed in the relativistic Cowling approximation, in
which all metric perturbations are ignored. For the modal analysis,
we employ three-component relativistic neutron star models with a
solid crust, a fluid core, and a fluid ocean. As a measure for the
relativistic effects on the oscillation modes, we calculate the
relative frequency difference defined as
,
where
and
are, respectively, the relativistic and
the Newtonian oscillation frequencies. The relative difference
takes various values for different oscillation
modes of the neutron star model, and the value of
for a given mode depends on the physical properties of the models.
We find that
is less than 0.1 for most of
the oscillation modes we calculate, although there are a few exceptions
such as the fundamental (nodeless) toroidal torsional modes in the crust,
the surface gravity modes confined in the surface ocean, and the core
gravity modes trapped in the fluid core. We also find that the modal
properties, represented by the eigenfunctions, are not strongly affected
by introducing general relativity. It is however shown that the mode
characters of the two interfacial modes, associated with the core/crust
and crust/ocean interfaces, have been interchanged between the two
through an avoided crossing when we move from Newtonian dynamics to
general relativistic dynamics.
Key words: relativity - stars: neutron - stars: oscillations
After the discovery of the r mode instability in neutron stars driven by gravitational radiation reaction (Andersson 1998; Friedman & Morsink 1998), nonradial oscillations of relativistic neutron stars has attracted much wider interest than before in astrophysics (see a recent review, e.g., by Andersson & Kokkotas 2001). The relativistic formulation of nonradial oscillations of fluid neutron stars was first given by Thorne and his collaborators (e.g., Thorne & Campolattaro 1967; Price & Thorne 1969; Thorne 1969a,b), and later on extended to the case of neutron stars with a solid crust in their interior (Schumaker & Thorne 1983, see also Finn 1990). Since one of the main concerns of these studies was gravitational waves generated by the stellar pulsations (e.g., Thorne 1969a), it was essential to include the perturbations in the metric in order to obtain a consistent description of the gravitational waves. However, it was the metric perturbations that made it extremely difficult to treat both analytically and numerically the oscillation modes in relativistic stars.
McDermott et al. (1983) introduced a relativistic version of the Cowling approximation, in which all the Eulerian metric perturbations are neglected in the relativistic oscillation equations derived by Thorne & Campolattaro (1967). Under the relativistic Cowling approximation, they calculated the p-, f-, and g-modes of relativistic fluid neutron star models to examine how the oscillation modes depend on the model properties. The relativistic Cowling approximation employed by McDermott et al. (1983) was shown to be good enough to calculate the p-modes for non-rotating polytropic stars by Lindblom & Splinter (1990), who compared the fully relativistic oscillation frequencies to those obtained in the relativistic Cowling approximation. Assuming slow rotation, Yoshida & Kojima (1997) have also carried out similar computations for the f- and p-modes of polytropic models and they confirmed the good applicability of the relativistic Cowling approximation. Quite recently, Yoshida & Lee (2002) have shown in the relativistic Cowling approximation the existence of the relativistic r modes^{} with l=m, which are regarded as a counter part of the Newtonian r modes. We believe that the relativistic Cowling approximation is quite useful to investigate the oscillation modes of relativistic stars, although we understand that under the approximation we cannot discuss inherently relativistic oscillation modes like w-modes (see, e.g., Andersson et al. 1996).
In their Newtonian calculations, McDermott et al. (1988) have shown that cold neutron stars with a solid crust can support a rich variety of oscillation modes (see also Strohmayer 1991; Lee & Strohmayer 1996; Yoshida & Lee 2001). The purposes of this paper are to calculate the various oscillation modes of relativistic neutron stars with a solid crust in the relativistic Cowling approximation, and to discuss the effects of general relativity on the modal property of the oscillation modes. We regard this paper as an extension of the studies by McDermott et al. (1983) and McDermott et al. (1988). In Sect. 2, we derive relativistic oscillation equations for the solid crust in the relativistic Cowling approximation following the formulation developed by Schumaker & Thorne (1983) and Finn (1990). Numerical results are discussed in Sect. 3 for nonradial modes of three relativistic neutron star models with a solid crust. Sect. 4 is devoted to discussion and conclusion. In this paper, we use units in which c=G=1, where c and G denote the velocity of light and the gravitational constant, respectively.
For modal analysis of neutron stars with a solid crust, we solve general relativistic
pulsation equations derived under the relativistic Cowling approximation, in which
all the metric perturbations in the matter equations are neglected.
We assume that the solid crust is in a strain-free state in the equilibrium
unperturbed state
and the strain in the crust are generated by
small amplitude perturbations superposed on the unperturbed state
(see, e.g., Aki & Richards 1980).
For the background unperturbed state,
it is therefore possible to assume a static and spherical symmetric state,
for which the geometry is given by the following line element:
(2) |
In the relativistic Cowling approximation, the basic equations
for pulsations are obtained from the energy and momentum conservation laws:
(5) |
For relativistic analysis of vibration of the solid crust in neutron stars,
we follow the formulation given by Schumaker & Thorne (1983) and Finn (1990).
If we define the rate of shear
and the
rate of expansion
as
(6) |
(7) |
To discuss nonradial oscillations of a star, we employ a Lagrangian perturbation
formalism (see, e.g., Friedman & Schutz 1975; Friedman 1978), in which
a Lagrangian displacement vector is introduced to connect fluid elements in
the equilibrium state to the corresponding elements in the perturbed state.
In this formalism, the Lagrangian change
in a quantity is related to
its Eulerian change
by
(10) |
(11) |
(12) |
(13) |
(15) |
= | (16) |
p= | (17) |
(18) |
(19) |
By substituting Eqs. (9), (15)-(17),
together with the perturbations defined by Eqs. (21) to (24),
into Eqs. (3) and (4), and collecting the terms proportional to
the small expansion parameter ,
we obtain a system of perturbation
equations for the solid crust:
z_{1}=S_{l} (r) , | (31) |
z_{2}= | |
(32) | |
z_{3}=H_{l} (r) , | (33) |
z_{4}= | (34) |
z_{5}=T_{l'} (r) , | (35) |
z_{6}= | (36) |
In the fluid regions, the oscillation equations to be solved are given by
(44) |
For the spheroidal modes the outer boundary condition is given
at the stellar surface (r=R)
by
,
which reduces to
y_{1} - y_{2} = 0 , | (45) |
(46) |
(47) |
N_{t}= | |
N_{r}= | (48) |
= | |
= |
y_{1} = z_{1} , V_{1}(y_{1}-y_{2}) = z_{2} , z_{4} = 0 . | (49) |
Neutron star models that we use in this paper are the same as those used in the modal analysis by McDermott et al. (1988). These models are taken from the evolutionary sequences for cooling neutron stars calculated by Richardson et al. (1982), where the envelope structure is constructed by following Gudmundsson et al. (1983). They are composed of a fluid core, a solid crust and a surface fluid ocean. The interior temperature is finite and is not constant as a function of the radial distance r. The models are not barotropic and the Schwarzschild discriminant has finite values in the interior of the star. The models we use are called NS05T7, NS05T8, and NS13T8, and their physical properties such as the total mass M, the radius R, the central density , the central temperature and the relativistic factor GM/c^{2} R are summarized in Table 1 (for other quantities, see McDermott et al. 1988).
Model | R (km) | (g ) | (K) | GM/(c^{2} R) | |
NS05T7 | 0.503 | 9.839 | |||
NS05T8 | 0.503 | 9.785 | |||
NS13T8 | 1.326 | 7.853 |
To classify the various oscillation modes of the three component models, we use almost the same nomenclature as that employed by McDermott et al. (1988). We let p_{k} refer to the acoustic modes of the kth overtone. The internal gravity modes confined in the surface ocean are denoted as and those in the fluid core as , where k indicates the overtone number. The eigenfrequencies of the f modes are usually found between the p_{1} and g_{1} modes. Associated with the fluid-solid interfaces in the models, there are two interfacial modes, which we denote as i_{1} and i_{2} such that . The s_{k} modes are spheroidal shear dominated modes of the k th overtone, the amplitudes of which are strongly confined in the solid crust. The t_{k} modes are toroidal shear dominated modes of the k th overtone propagating only in the solid crust. Note that the p_{k}-, g_{k}-, f-, i_{1(2)}, and s_{k} modes are classified as spheroidal modes while the t_{k} modes as toroidal modes.
In the Newtonian limit of , we calculate various oscillation modes with l=2for the three neutron star models NS05T7, NS05T8, and NS13T8, and tabulate the Newtonian eigenfrequencies, which we denote as , in Tables 2 through 4. These tables confirm that the Newtonian frequencies obtained in this paper are in good agreement with those computed by McDermott et al. (1988). Assuming , we also compute the corresponding relativistic oscillation modes with l=2, and tabulate the relativistic eigenfrequencies, which we denote as , in Tables 2 through 4, in which the relative frequency differences defined as are also given. The relative difference takes various values for different oscillation modes, and the value of of a given mode is also dependent on the physical properties of the neutron star models. We find for most of the oscillation modes we calculate. The elastic s_{k} and t_{k} modes with have particularly small values of , which are at most a few percent. However, there are some exceptions, and the values of become as large as -0.45 for the t_{0} modes and -1 for the - and -modes for the model NS13T8, which is the most compact one among the three. If we consider the gravitational redshift as one of the general relativistic effects that are responsible for the difference from the Newtonian oscillation frequency, we may have and hence , which is negative since . Although this estimation gives a consistent result for the surface modes confined in the surface ocean and for the core modes trapped in the fluid core, it does not always give good estimations of the difference as suggested by the tables. In fact, we have positive for some oscillation modes. This is understandable, however, since the eigenfunctions usually possess large radial extent in the interior of stars, and the eigenfrequencies are dependent on the equilibrium quantities which contain terms and factors which are absent in the Newtonian oscillation equations.
mode | |||
( Spheroidal) | |||
g^{c}_{1} | -0.230 | ||
g^{s}_{2} | -0.158 | ||
g^{s}_{1} | -0.159 | ||
i_{1} | -0.104 | ||
i_{2} | -0.059 | ||
s_{1} | 0.021 | ||
s_{2} | -0.001 | ||
f | -0.034 | ||
p_{1} | -0.015 | ||
p_{2} | -0.006 | ||
( Toroidal) | |||
t_{0} | -0.111 | ||
t_{1} | -0.002 | ||
t_{2} | -0.004 | ||
t_{3} | 0.001 |
mode | |||
( Spheroidal) | |||
g^{c}_{1} | -0.234 | ||
g^{s}_{2} | -0.177 | ||
g^{s}_{1} | -0.169 | ||
i_{1} | 0.093 | ||
i_{2} | -0.085 | ||
s_{1} | 0.014 | ||
s_{2} | -0.002 | ||
f | -0.035 | ||
p_{1} | -0.019 | ||
p_{2} | -0.004 | ||
( Toroidal) | |||
t_{0} | -0.111 | ||
t_{1} | -0.003 | ||
t_{2} | -0.005 | ||
t_{3} | -0.002 |
mode | |||
( Spheroidal) | |||
g^{c}_{1} | -1.109 | ||
g^{s}_{2} | -0.956 | ||
g^{s}_{1} | -0.933 | ||
i_{1} | 0.210 | ||
i_{2} | -0.114 | ||
s_{1} | 0.004 | ||
s_{2} | -0.002 | ||
f | -0.148 | ||
p_{1} | -0.306 | ||
p_{2} | -0.197 | ||
( Toroidal) | |||
t_{0} | -0.449 | ||
t_{1} | -0.000 | ||
t_{2} | -0.004 | ||
t_{3} | -0.004 |
For the various oscillation modes of NS13T8, we display the displacement vector versus the radial coordinates r/R or in Figs. 1 through 8, where the amplitude normalization of the eigenfunctions is given by y_{1}=1 at r=R for the spheroidal modes and z_{5}=1 at for the toroidal modes with being the radius of the ocean-crust interface. Notice that for the modes, the normalization condition is adapted because the modes are very insensitive to the behavior of the eigenfunctions in the surface ocean, where is the radius at the bottom of the crust. The eigenfunctions obtained in the Newtonian calculations are also plotted for comparison in each of the figures. From Figs. 1 through 8, we can see that the basic properties of the eigenfunctions of the relativistic modes are not very much different from those of the corresponding Newtonian modes, except for the interfacial modes (see the next paragraph). In this sense, we can say that general relativity does not bring about any essential changes in the modal properties represented by the eigenfunctions.
Figure 1: Displacement vectors r S_{2}/R and r H_{2}/R of the mode for the model NS13T8, given as a function of r/R. Here, the normalization of the eigenfunction is chosen as , where is the radius at the bottom of the crust. The Newtonian eigenfunctions are shown as well as the relativistic ones. | |
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Figure 2: Displacement vectors r S_{2}/R and r H_{2}/R of the mode for the model NS13T8, given as a function of . Here, the normalization of the eigenfunction is chosen as S_{2}(R)=1. The Newtonian eigenfunctions are shown as well as the relativistic ones. | |
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Figure 3: Same as Fig. 2 but for the i_{1} mode. | |
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Figure 4: Same as Fig. 3 but for the i_{2} mode. | |
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Figure 5: Same as Fig. 4 but for the s_{1} mode. | |
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Figure 6: Displacement vectors r S_{2}/R and r H_{2}/R of the f mode for the model NS13T8, given as a function of r/R. Here, the normalization of the eigenfunction is chosen as S_{2}(R)=1. The Newtonian eigenfunctions are shown as well as the relativistic ones. | |
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Figure 7: Same as Fig. 6 but for the p_{1} mode. | |
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Figure 8: Displacement vector r T_{2}/R of the t_{0} and the t_{1} modes of the neutron star model NS13T8, given as a function of r/R. Here, normalization of the eigenfunction is chosen as , where is the radius at the crust/ocean interface. The Newtonian eigenfunctions are shown as well as the relativistic ones. Note that two curves for r T_{2}/R nearly overlap each other. | |
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As shown by Figs. 3 and 4, the relativistic eigenfunctions of the interfacial modes i_{1} and i_{2}are at first sight quite different from the Newtonian eigenfunctions. Since the eigenfunctions of the relativistic i_{1(2)} mode are rather similar to those of the Newtonian i_{2(1)} mode, it is tempting to make a guess that the modal properties have been exchanged between the two modes when we move from Newtonian dynamics to relativistic dynamics. In their Newtonian calculation, McDermott et al. (1988) showed that the properties of the i modes are very sensitive to the changes in the physical quantities near the interfaces, and mentioned a numerical experiment in which they computed the two interfacial modes by artificially reducing the bulk modulus and found an avoided crossing between the two modes through which the modal characters are interchanged with each other. Inspired by this report, we have carried out a numerical experiment, calculating the two interfacial modes as a function of for the model NS13T8. Plotting the frequencies of the two modes versus in Fig. 9, we confirm the occurrence of an avoided crossing between the two modes at , through which the modal characters of the two have been exchanged with each other. As increases from (Newtonian limit), the effective local shear modulus in the solid crust decreases because of the gravitational redshift effects, as a result of which the two interfacial modes experience the avoided crossing when we move from Newtonian dynamics to relativistic dynamics.
Figure 9: The avoided crossing between the i_{1} and i_{2} mode of the model NS13T8, where is the parameter that represents the strength of general relativistic effects. The avoided crossing happens at . | |
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In this paper, we have calculated a variety of oscillation modes of relativistic neutron stars with a solid crust in the relativistic Cowling approximation, in which all metric perturbations are ignored. We find is less than 0.1 for most of the oscillation modes with l=2, although there are some exceptions such as the surface gravity modes, the core gravity modes, and the nodeless toroidal torsional modes. We also find that the essential modal properties represented by the eigenfunctions are not strongly affected by introducing general relativity in the sense that the relativistic eigenfunctions of a mode have almost the same properties as the corresponding Newtonian eigenfunctions. An exception may be the two interfacial modes whose eigenfunctions differ from their Newtonian eigenfunctions, and we have shown that this can be explained as a result of the mode exchange between the two modes through an avoided crossing that occurs when we move from Newtonian dynamics to relativistic dynamics.
Needless to say, nonradial oscillations of neutron stars should be treated within the framework of general relativity because of their strong gravity. At present, however, we cannot investigate the pulsations of neutron stars in all the aspects by using fully general relativistic formalism because of the complexity general relativity brings about^{}. One of the simplest ways to study the oscillations of a neutron star is to treat the problems within the framework of Newtonian dynamics, that is, to solve the Newtonian oscillation equations for neutron star models constructed with the Newtonian hydrostatic equations. However, it is obvious that this Newtonian treatment cannot be fully approved for neutron stars since the Newtonian equilibrium models do not give us correct mass and radius for the stars. To reduce the distance between fully relativistic and Newtonian calculations of nonradial oscillations of the stars, we may use relativistic equilibrium models and solve the Newtonian oscillation equations, which was the strategy taken by McDermott et al. (1988). In this paper, as an extension of the study by McDermott et al. (1988), we solved relativistic oscillation equations derived in the relativistic Cowling approximation for relativistic neutron star models with a solid crust. Our calculation confirmed that the classification scheme McDermott et al. (1988) employed for the oscillation modes of neutron stars with a solid crust is valid even if we integrate relativistic oscillation equations.
Acknowledgements
S.Y. acknowledges financial support from the Portuguese FCT through a Sapiens project, number 36280/99.