A&A 381, L49-L52 (2002)
DOI: 10.1051/0004-6361:20011643
M. Ansorg - A. Kleinwächter - R. Meinel
Theoretisch-Physikalisches Institut, University of Jena, Max-Wien-Platz 1, 07743 Jena, Germany
Received 6 November 2001/ Accepted 20 November 2001
Abstract
A new spectral code for constructing general-relativistic models of rapidly
rotating stars with an unprecedented accuracy is presented.
As a first application,
we reexamine uniformly rotating homogeneous stars and compare our results with
those obtained by several previous codes. Moreover, representative relativistic
examples corresponding to highly flattened rotating bodies are given.
Key words: stars: rotation - stars: neutron - gravitation - relativity - methods: numerical
The study of relativistic, axisymmetric and stationary, uniformly rotating perfect fluid bodies is motivated by extraordinarily compact astrophysical objects like neutron stars. Several numerical codes have been developed in order to calculate the structure and the gravitational field of these bodies (Bonazzola & Schneider 1974; Wilson 1972; Butterworth & Ipser 1975,1976; Friedman et al. 1986,1989; Lattimer et al. 1990; Neugebauer & Herold 1992; Herold & Neugebauer 1992; Komatsu et al. 1989a,1989b; Eriguchi et al. 1994; Stergioulas & Friedman 1995; Bonazzola et al. 1993; for reviews see Friedman 1998, and Stergioulas 1998). While they obtain an accuracy of up to 5 digits for sufficiently smooth equations of state, these methods yield fewer than 4 digits in the case of stiff equations of state (e.g., for constant density), which is due to particular Gibbs phenomena at the star's surface. In order to avoid these Gibbs phenomena, Bonazzola et al. (1998) used a multi-domain spectral method with which they were able to achieve an accuracy of 12 digits for the Maclaurin sequence of homogeneous Newtonian bodies.
In this letter we introduce a new numerical code, which is based on a multi-domain spectral method for representing all metric functions. We intend to use this method to investigate neutron stars with realistic equations of state. In particular, our multi-domain method lends itself to considering several layers inside the star, which are characterized by different equations of state. As we will outline below, we obtain a hitherto unobtainable accuracy which permits its application even in limiting cases such as the mass-shedding limit. Moreover, we are able to study extremely flattened, homogeneous Einsteinian bodies. Such bodies were the subject of interesting conjectures made by Bardeen (1971).
As a first application of our method, we reexamine a particular example of a uniformly rotating homogeneous star that was used by Nozawa et al. (1998) to compare three different codes. We give our results, which possess a substantially higher accuracy. Moreover, we discuss representative, relativistic examples corresponding to highly flattened rotating bodies.
In what follows, units are used in which the velocity of light as well as Newton's constant of gravitation are equal to 1.
The line element for an axisymmetric, stationary, uniformly
rotating perfect fluid body assumes in Lewis-Papapetrou coordinates
the following form:
In the vacuum region, there emerge three field equations of second order to determine the potentials U, a and W. The function k follows from the other potentials by a line integral^{}.
In the interior of the body we use the metric functions valid in the comoving
frame of reference. Here, the only new coordinate is
,
where
is the angular velocity of the body.
The corresponding line element also assumes the above form with potentials
U', a', k' and W', which are given by
All potentials satisfy regularity conditions at infinity and along the axis of rotation () and possess moreover reflectional symmetry with respect to the plane (see Meinel & Neugebauer 1995).
The numerical scheme to solve the field equations with respect to boundary and transition conditions is based on a multi-domain spectral method. After imposing reflectional symmetry, the set of all relevant -values, , is divided into several subregions for physical reasons. In the simplest case, we only take two subregions, the interior and the exterior of the body. However, if we consider several layers inside the star, which are characterized by different equations of state, we will be forced to allow for more than two regions. In this letter, we will restrict ourselves to only two subregions.
Each of these subregions is mapped onto the square
I^{2}=[0,1]^{2}. In order to do
this we introduce a function
defined on the interval I=[0,1] with
We assume all potentials to be smooth functions on I^{2} such that we may approximate them well by two-dimensional Chebyshev-expansions with respect to the coordinates s and t. In the same manner, we represent the unknown boundary function as well as the boundary values and of the potentials a' and W in terms of (one-dimensional) Chebyshev-polynomials with respect to the coordinate t. If these three one-dimensional functions were given, we would have to solve a particular interior and exterior boundary value problem^{} of the respective field equations. However, we have to deal with a free boundary value problem, where these three functions are not known, but have to be determined such that the normal derivatives of the potentials U', a' and W behave continuously at the surface B ^{}. Taking only a finite number of Chebyshev-coefficients into account for the interior potentials U', a' and W, the exterior potentials U, a and W and the surface quantities , and , our numerical scheme consists in determining all unknown Chebyshev-coefficients by satisfying the interior and exterior field equations at a number of grid points in I^{2} and, moreover, requiring the above transition conditions at the surface. The total number of unknown coefficients equals the total number of equations. The system is solved by a Newton-Raphson method, where the initial guess for the entire solution in the case of constant density can be taken from the analytical Newtonian Maclaurin solution.
Nozawa et al. (1998) compared three different codes for various
choices of the
equation of state. We take one particular example in which for constant
density they prescribed
the normalized central pressure
and the
ratio
.
The results of this comparison are given in Table 11 of
Nozawa et al. (1998). Here we
calculate the same quantities with an accuracy of 12 digits and list them in
Col. 2 of Table 1^{}. Columns 3-5 refer to the codes by Komatsu et al. (1989a, 1989b) and Eriguchi
et al. (1994)
[abbreviated by KEH(OR)],
Stergioulas & Friedman (1995) [abbreviated by KEH(SF)] and Bonazzola
et al. (1993) [abbreviated by BGSM] and give the relative error of
the quantity in question, i.e. for example,
.
The fact that
was not exactly 0.7 in the BGSM calculation does not affect
the comparison substantially.
In Nozawa et al. (1998),
the general-relativistic virial identities GRV2 and GRV3 (derived by Bonazzola &
Gourgoulhon 1994 and Gourgoulhon & Bonazzola 1994)
were calculated to check the
accuracy of the numerical solution. For our result, this check yields
for GRV2 and
for GRV3. Note that we used 23
Chebyshev-polynomials for each dimension in this calculation.
As an additional test of accuracy,
we calculated the angular momentum and the gravitational
mass in two different ways: (i) from the asymptotic
behaviour of the metric and (ii) by means of integrals over
the matter distribution (cf. Bardeen
& Wagoner 1971, Eqs. (II.24), (II.26) and (II.23), (II.25)
respectively).
We get a relative deviation of
for the mass and
for the angular momentum.
(1) | (2) | (3) | ||
1 | ||||
0.7 | 0.11% | |||
1.41170848318 | 1.1% | 0.32% | 0.97% | |
0.135798178809 | 2.3% | 0.19% | 0.86% | |
0.186338658186 | 0.17% | 0.32% | 1.4% | |
0.345476187602 | 0.098% | 0.053% | 0.27% | |
0.0140585992949 | 1.6% | 0.045% | 2.3% | |
1.70735395213 | 6.1% | 0.013% | 2.1% | |
-0.162534082217 | 1.7% | 1.9% | 4.4% | |
11.3539142587 | 17% | 0.10% | 8.1% |
Chandrasekhar (1967) has shown that the post-Newtonian Maclaurin spheroids become singular at the eccentricity ..., the point of the first axisymmetric secular instability. Bardeen (1971) confirmed this result and discussed the possibility of two Newtonian axisymmetric sequences bifurcating from the Maclaurin spheroid at , a first one that should finally evolve towards the Dyson-Wong rings (Dyson 1892,1893; Wong 1974; Kowalewsky 1895; Poincaré 1885a,1885b,1885c; see also Lichtenstein 1933) and a second one, which he called the "central bulge configuration'', that should end in a mass-shedding limit. Eriguchi & Sugimoto (1981) indeed found the first sequence, and called it the "one-ring sequence''. The properties of the bifurcation have been analyzed by Christodoulou et al. (1995).
By means of a Newtonian version of our code we were able to find the second sequence as well. It appears as a continuation of the one-ring sequence across the Maclaurin sequence and is characterized by a (bi-convex) "lens shape'' of the solutions.
The point of the Maclaurin sequence corresponds to the value R=R_{1}=0.27320... of the rotation parameter . Bardeen (1971) speculated that there should be a gap in the R-values of relativistic solutions around R_{1} and that the relativistic solutions should show some properties of the Newtonian "central bulge'' or Dyson ring solutions when approaching this gap from the sphere end or from the disc end of the Maclaurin sequence.
(a) | (b) | (c) | |
0.002 | 0.004 | 0.003 | |
0.2 | 0.1 | 0.1 | |
1.089468e+00 | 1.004798e+00 | 9.305304e-01 | |
8.314919e-04 | 1.099261e-02 | 5.603298e-03 | |
8.353271e-04 | 1.124879e-02 | 5.677891e-03 | |
1.028186e-01 | 2.474602e-01 | 2.249448e-01 | |
3.663135e-06 | 3.093789e-04 | 1.072577e-04 | |
1.585397e-02 | 9.132868e-02 | 5.381392e-02 | |
-9.890231e-02 | -1.912324e-01 | -1.720749e-01 | |
1.308359e-01 | 3.827610e-01 | 2.825350e-01 | |
R | 0.2444465 | 0.3001205 | 0.3522896 |
GRV2 | 1.1e-09 | 3.1e-11 | 1.9e-08 |
GRV3 | 1.6e-09 | 3.2e-11 | 2.3e-08 |
Figure 1: Meridional cross-sections of the solutions specified in Table 2. | |
Open with DEXTER |
In order to check these conjectures, we have calculated three relativistic models in the vicinity of R=R_{1}. These models are characterized by parameters as shown in Table 2 and their (coordinate) shapes are depicted in Fig. 1. Note that the high accuracy of our code is crucial when investigating the subtle behaviour of the relativistic solutions in this region - an earlier attempt by Butterworth (1979) could not finally clarify these questions. Solution (a) shows indeed a "lens shape'', whereas solution (b) has a "one-ring tendency'' as does solution (c), albeit far less pronounced. The solution (a) is rather close to the Newtonian "lens shape'' sequence. The situation becomes more and more complex when the bifurcation points R_{2}=0.36633..., R_{3}=0.43527..., etc. (corresponding to ..., ..., etc.) of the Newtonian two-ring, three-ring, etc. sequences are approached (for the two-ring sequence see Eriguchi & Hachisu 1982). Our solution (c) already has a value for R close to R_{2}, see Table 2.
A detailed analysis of highly flattened, rotating Newtonian as well as Einsteinian bodies including a discussion of stability aspects and of the route to infinitesimally thin, relativistic discs (Bardeen & Wagoner 1969,1971; Neugebauer & Meinel 1993,1995) will be the subject of a future publication.
Acknowledgements
This work was supported by the German Deutsche Forschungsgemeinschaft (DFG-project ME 1820/1).