A&A 372, 594-600 (2001)
DOI: 10.1051/0004-6361:20010556
K. Konno
Department of Physics, Hiroshima University, Higashi-Hiroshima 739-8526, Japan
Received 3 October 2000 / Accepted 6 April 200
Abstract
We consider principal moments of inertia of axisymmetric,
magnetically deformed stars in the context of general relativity.
The general expression for the moment of inertia with
respect to the symmetric axis is obtained.
The numerical estimates are derived for several
polytropic stellar models.
We find that the values of the principal moments of
inertia are modified by a factor of 2 at most
from Newtonian estimates.
Key words: stars: pulsars: general - stars: magnetic fields - stars: rotation - relativity
Various features of pulsars have been disclosed with growing observational data. Most pulsars have stable pulse shapes, and spin down steadily with typical time-scales of the order of several Myr (see e.g. Mészáros 1992). However, deviations from linear spin-down of trends have also been observed on shorter time-scales in some pulsars. These deviations convey information about the surroundings, internal structure and dynamics of neutron stars. For example, the presence of glitches (see e.g. Shemar & Lyne 1996; Wang et al. 2000) indicates that two or more poorly coupled components coexist in the internal structure of the stars, while timing noise (see e.g. D'Alessandro 1995) is believed to be due to random movements of the fluid in pulsars. Besides these short time-scale instabilities, the spin-down features indicating the precession due to stellar deformation have also been reported. Two anomalous X-ray pulsars (AXPs), 1E1048.1-5937 and 1E2259+586, were observed to indicate irregular spin-down (see e.g. Mereghetti 1995; Baykal & Swank 1996; Baykal et al. 1998; Oosterbroek et al. 1998). AXPs are a subclass of X-ray pulsars with periods in a very narrow range (6-12 s) and period derivatives in the range ( 10^{-12}-10^{-11} s s^{-1}) (see e.g. Mereghetti & Stella 1995), and which are most likely candidates of magnetars (Duncan & Thompson 1992; Thompson & Duncan 1993, 1995, 1996) along with soft gamma repeaters (SGRs) (see e.g. Kouveliotou et al. 1998, 1999). As discussed by Melatos (1999), the wobbles in their spin-down rates may be interpreted by an effect called radiative precession (Melatos 2000), which is related to an oscillating component of electromagnetic torque. Another precessing pulsar PSR B1828-11 (Stairs et al. 2000) was reported very recently. This gives the very clear observation of free precession of an isolated neutron star. These deformed, precessing objects are analyzed by solving the Euler equation of motion in the form , where I_{ij} is the inertia tensor, is the angular velocity and N_{i} is the torque acting on the object. When we take into account the electromagnetic torque by a rotating magnetic dipole (Davis & Goldstein 1970; Goldreich 1970), the above-mentioned radiative precession can be found for magnetically deformed stars (Melatos 1999; 2000). Furthermore, since magnetically deformed, rotating stars emit gravitational waves, we can consider the gravitational radiation reaction torque (Bertotti & Anile 1973; Cutler & Jones 2000). The gravitational backreaction damps the wobbles on a time-scale proportional to for wobbling, axisymmetric rigid bodies, where I_{i} is the principal moments of inertia. Thus, the moments of inertia play a significant role in the analyses of pulsar precession. The estimates of the moments of inertia have been done in the context of Newtonian gravity so far. However, neutron stars are fully general relativistic objects, and it is important to take into account general relativistic effects. Therefore, we now discuss the principal moments of inertia of magnetically deformed stars in the context of general relativity.
In Newtonian gravity, once we have the mass distribution of an object , the moment of inertia with respect to any axis can be calculated from , where denotes the length from the axis. In general relativity, only for axisymmetric objects, the moment of inertia with respect to the symmetric axis can be well defined. For this definition, we need slow rotation of the objects. The slow rotation ensures that the angular momentum J is linearly related to the angular velocity , i.e. . Here, I defines the general relativistic version of the moment of inertia. The moments of inertia of relativistic, spherically symmetric stars were discussed by Hartle (1967) and Chandrasekhar & Millar (1974). The approximate expression valid for various realistic equations of state was also derived by Ravenhall & Pethick (1994).
In this paper, we discuss the principal moments of inertia of magnetically deformed stars which are axisymmetric with respect to the magnetic axis. Deformation of relativistic magnetized stars was studied both in the numerical approach (Bonazzola et al. 1993; Bocquet et al. 1995) and in the analytic approach (Konno et al. 1999). Based on the latter, analytic approach developed by Konno et al. (1999), we develop the subsequent discussion. In the previous work (Konno et al. 1999), stellar magnetic fields were regarded as corrections to non-rotating, spherical stars, and the magnetic deformation of stars was formulated using a perturbative approach. As mentioned above, we now take into account the slow rotation of the deformed stars on the symmetric axis to define the moment of inertia. The formulation of this configuration is given in Sect. 2. The general expression for the principal moment of inertia with respect to the symmetric axis is also derived. The numerical estimates are obtained for several stellar models in Sect. 3. In Sect. 4, we discuss the other principal moments of inertia. Finally, we give our conclusions in Sect. 5. Throughout this paper, we use units in which c=G=1.
We consider a slowly rotating star which is subject to
quadrupole deformation due to a dipole magnetic field.
As described by Konno et al. (1999),
magnetic fields can be treated as corrections to a
background, spherically symmetric star,
i.e.
.
We now assume that the star slowly rotates on the magnetic axis
with a uniform angular velocity
,
which is also regarded as a perturbation.
In this paper, we take into account the rotational
corrections up to first order in
.
The metric describing such a star can be given by
W_{1} | = | (2) | |
W_{3} | = | W_{3}^{(2)} , | (3) |
The stress-energy tensor of the star which is
composed of a perfect fluid endowed with a
dipole magnetic field has the form
= | (5) | ||
p | = | (6) |
(7) |
(8) |
The above-mentioned functions except W_{1} and W_{3} were
investigated in detail by Konno et al. (1999) and
Konno & Kojima (2000)
(see Appendix A for a brief summary).
The differential equations which W_{1} and W_{3} obey can be
obtained from the -component of the Einstein equation,
j= | (11) |
= | (12) |
The differential equations (9) and (10)
can be solved numerically by imposing
boundary and junction conditions.
The boundary conditions are summarized as
(15) |
= | (17) | ||
= | (18) |
Before deriving numerical solutions, it is worthwhile investigating the behavior of W_{1} and W_{3} at large r in detail using Eqs. (9), (10), (13) and (14), because this inspection gives the expressions for the angular momentum and the moment of inertia of the star.
At large r, we have
(19) |
(20) |
(21) |
(24) |
Figure 1: The magnetic correction of the principal moment of inertia I^{(2)}_{z} plotted as a function of M/R. The values are normalized by , and the polytropic index is denoted by n. | |
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Using the boundary and junction conditions mentioned in the last section, we can derive the numerical solutions for W_{1} and W_{3} for any stellar models. The numerical results for the magnetic correction of the principal moment of inertia I^{(2)}_{z} can also be obtained by using Eq. (25). We now show the results of I^{(2)}_{z} obtained for several polytropic stellar models. In these calculations, we adopted a different condition from that of the previous study (Konno et al. 1999), in which sequences with constant central density were investigated. In the current case, in order to clarify the correspondence between non-magnetized and magnetized stars having same mass, we use the condition in which the total mass of the star does not change through the perturbative approach. This is also because the moment of inertia can be modified significantly by the mass shift rather than the magnetic deformation. In our formulation, this condition is accomplished by imposing the boundary condition that m_{0}vanishes at infinity.
Figure 1 displays the magnetic correction I^{(2)}_{z} as a function of the general-relativistic factor M/R. The values are normalized by the typical value . As a simple example, we now discuss quadrupole deformation of a fluid body in the Newtonian limit. If the deformed body has constant density, which corresponds to n=0, then we can derive the result from the estimate of ellipticity (Ferraro 1954; Konno et al. 2000). However, as seen in Fig. 1, our numerical result for n=0 is different from this simple estimate. This is because the perturbed star does not have constant effective density by the added perturbation, even though we assume the background star with constant density. This reason can be understood by seeing the differential equation for m_{0}, i.e. Eq. (A.3). The derivative of m_{0} is related to the effective density including electromagnetic energy. In the case of n=0, although the first term on the right-hand side in Eq. (A.3) vanishes, the remaining terms do not vanish and are non-trivial functions. It follows that the effective density is not a constant. Thus, our results include the inertia of electromagnetic fields as well as mass. Therefore, our result shown in Fig. 1 cannot be compared with the above simple estimate.
Concentrating on the general relativistic effects on the magnetic correction I_{z}^{(2)}, we can find from Fig. 1 that the values of I_{z}^{(2)}for each stellar model become large with the general relativistic factor M/R. The increments are 50% at most. However, the rates of increase may be neglected except for the case of n=1.5.
Figure 2: Comparison between and for n=1. and are normalized by and plotted as a function of M/R. | |
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Next, we discuss the other principal moments of inertia, i.e. I_{x}^{(2)} and I_{y}^{(2)}. In the context of general relativity, the definition of these quantities is associated with a difficulty in the concept. As seen from Eqs. (22) and (23), the notion of moments of inertia is related to quantities at infinity (see also Geroch 1970; Hansen 1974). If we consider the rotation of the star on the x-axis or the y-axis, it produces the radiation of electromagnetic and gravitational fields. Thus, the exterior space-time is not stationary, but radiative. There is no rigorous way to define the moments of inertia in such a radiative space-time. The concept of I_{x}^{(2)} and I_{y}^{(2)} itself may be meaningless to a considerable extent.
Therefore, only approximate expressions are available.
It would be useful to extend the Newtonian expressions,
since it seems that most Newtonian features
survive for neutron stars specified by
the general relativistic factor
.
We cannot prove the validity mathematically, but
will test some empirical relations.
For this purpose, let us recall relations using the principal
moments of inertia and the ellipticity which hold for an
incompressible fluid in the context of Newtonian gravity.
These are helpful in seeking the other principal
moments of inertia.
First, if the fluid body is subject to
quadrupole deformation, then the relation between
the principal moments of inertia exists
If we assume that the relation (26) is applicable to general relativistic cases, it seems that we should simply apply relation (26) to the results shown in Fig. 1 and derive I_{x}^{(2)} and I_{y}^{(2)}. However, in order to utilize this relation, the magnetic corrections must be purely quadrupole contributions. As seen from the metric (1), the magnetic deformation includes monopole parts as well as quadrupole parts. Thus, our results in Fig. 1 also include monopole parts. Hence, we have to subtract the monopole contributions from I^{(2)}_{z} to derive I_{x}^{(2)} and I_{y}^{(2)}. It is, nevertheless, not so easy to extract monopole contributions accurately from these magnetically deformed stars.
However, there is no guarantee that all the results in Fig. 1 include monopole contributions dominantly. If there is a case in which the monopole part can be neglected, then we can estimate I_{x}^{(2)}and I_{y}^{(2)}, by assuming that the relation (26) holds approximately. That case would also provide some extrapolation for general relativistic effects on I_{x}^{(2)}and I_{y}^{(2)} in the other cases. In order to seek such a case, we now compare derived by simply using the results in Fig. 1 with calculated from the ellipticity, which was already estimated for several stellar models (Konno et al. 2000). In the case that is almost consistent with about the values and tendency, it seems that the monopole part can be neglected, and we may use the relation (26). From the comparison, we find that is almost consistent with in the case of n=1. Figure 2 displays the comparison between and in this stellar model. The two curves coincide within 10%. Hence, we can estimate I_{x}^{(2)} and I_{y}^{(2)} using Eq. (26) in this stellar model. Since the values of I_{x}^{(2)} and I_{y}^{(2)} are simply given by multiplying I_{z}^{(2)} by a factor of -1/2, the changes of the absolute values of I_{x}^{(2)} and I_{y}^{(2)} due to general relativistic effects are specified by the same factor as in the case of I_{z}^{(2)}(see Fig. 3). Consequently, we derive very similar result to that of I_{z}^{(2)}. However, I_{x}^{(2)} and I_{y}^{(2)} decrease with the general relativistic factor M/R, while I_{z}^{(2)} increases. We expect that similar features exist also in the other stellar models.
Figure 3: The other components of the principal moments of inertia I_{x}^{(2)} and I_{y}^{(2)}, which are derived for n=1. | |
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Recent observations indicate precession of some pulsars. In order to understand the dynamics of these pulsars in detail, we have to know the moments of inertia, which play a crucial role in the Euler equation of motion. Motivated by this consideration, we have discussed the general relativistic effect on the moments of inertia of magnetically deformed stars. By developing the formulation by Konno et al. (1999), we have considered the slow rotation of magnetically deformed stars on the symmetric axis in order to define the moment of inertia. The general expression for the magnetic correction I_{z}^{(2)} to the moments of inertia of spherically symmetric stars was obtained, and the numerical estimates for various polytropic stellar models were also obtained. The other components of the moments of inertia I_{x}^{(2)} and I_{y}^{(2)} were discussed from the extension of relations in Newtonian gravity. From those results, we found that each principal moment of inertia is modified by a factor of 2 at most due to the general relativistic effect. Nevertheless, it seems that the general relativistic effect does not affect precessing motion noticeably in the case of free precession of pulsars. However, it is not clear whether the electromagnetic radiation reaction torque acting on the pulsars changes due to the general relativistic effect. If this torque is modified by the general relativistic effect, then the period of the wobbles would also be modified. Therefore, this point should be made clear in future investigation.
Acknowledgements
I would like to thank Y. Kojima for careful reading of the manuscript and for fruitful discussions and suggestions. I would also like to thank M. Kasai, H. Asada and K. Ioka for valuable and useful comments, and M. Hosonuma for useful discussions. This work was supported in part by a Grant-in-Aid for Scientific Research Fellowship of the Ministry of Education, Science, Sports and Culture of Japan (No. 12001146).
First, we consider the magnetic field described by ,
which has the lowest order.
This function obeys the following equation
derived from the Maxwell equation,
(A.2) |
Next, we consider the effect of magnetic stress, which
is of second order in
.
This effect arises from the electromagnetic part of the
stress-energy tensor (4).
The second-order metric functions h_{0}, h_{2}, m_{0},
m_{2} and k_{2} can be obtained by solving the following
two sets of differential equations and one algebraic equation
derived from the Einstein equation
and the equation of motion
,
(A.7) |
(A.8) |
Once we have the above metric functions, we can derive the
second-order corrections of pressure and density from the relations
p_{20} | = | (A.12) | |
p_{22} | = | (A.13) | |
= | (A.14) | ||
= | (A.15) |
Finally, we consider the quantities a_{t0} and a_{t2} of order
,
which are related to
the electric fields induced by stellar rotation.
These functions obey the Maxwell equation.
Outside the star, we have
a_{t0} | = | (A.18) | |
a_{t2} | = | (A.19) |