Weber and Glendenning [318] improve on Hartle's formalism in order to obtain a more
accurate estimate of the angular velocity at the mass-shedding
limit, but their models still show large discrepancies compared
to corresponding models computed without the slow rotation
approximation [261
]. Thus, Hartle's formalism is appropriate for typical pulsar
(and most millisecond pulsar) rotational periods, but it is not
the method of choice for computing models of rapidly rotating
relativistic stars near the mass-shedding limit.
Space is truncated at a finite distance from the star and the boundary conditions there are imposed by expanding the metric potentials in powers of 1/ r . Angular derivatives are approximated by high-accuracy formulae and models with density discontinuities are treated specially at the surface. An equilibrium model is specified by fixing its rest mass and angular velocity.
The original BI code was used to construct uniform density
models and polytropic models [55,
54]. Friedman
et al.
[113,
114] (FIP) extend the BI code to obtain a large number of rapidly
rotating models based on a variety of realistic EOSs. Lattimer
et al.
[196
] used a code that was also based on the BI scheme to construct
rotating stars using ``exotic'' and schematic EOSs, including
pion or kaon condensation and strange quark matter.
Cook, Shapiro, and Teukolsky (CST) improve on the KEH scheme
by introducing a new radial variable that maps the semi-infinite
region
to the closed region [0,1]. In this way, the region of
integration is not truncated and the model converges to a higher
accuracy. Details of the code are presented in [69
] and polytropic and realistic models are computed in [71
] and [70
].
Stergioulas and Friedman (SF) implement their own KEH code
following the CST scheme. They improve on the accuracy of the
code by a special treatment of the second order radial derivative
that appears in the source term of the first order differential
equation for one of the metric functions. This derivative was
introducing a numerical error of 1-2% in the bulk properties of
the most rapidly rotating stars computed in the original
implementation of the KEH scheme. The SF code is presented
in [295] and in [293
]. It is available as a public domain code, named
rns, and can be downloaded from [292
].
In [261,
262
] the code is used to construct a large number of models based on
recent EOSs. The accuracy of the computed models is estimated
using two general relativistic virial identities, valid for
general asymptotically flat spacetimes [132
,
43
] (see Section
2.7.7).
While the field equations used in the BI and KEH schemes
assume a perfect fluid, isotropic stress-energy tensor, the BGSM
formulation makes no assumption about the isotropy of
. Thus, the BGSM code can compute stars with a magnetic field, a
solid crust, or a solid interior, and it can also be used to
construct rotating boson stars.
This can be used to check the accuracy of computed numerical
solutions. In general relativity, a different identity, valid for
a stationary and axisymmetric spacetime, was found in [40]. More recently, two relativistic virial identities, valid for
general asymptotically flat spacetimes, have been derived by
Bonazzola and Gourgoulhon [132
,
43
]. The 3-dimensional virial identity (GRV3) [132] is the extension of the Newtonian virial identity (28
) to general relativity. The 2-dimensional (GRV2) [43
] virial identity is the generalization of the identity found
in [40] (for axisymmetric spacetimes) to general asymptotically flat
spacetimes. In [43], the Newtonian limit of GRV2, in axisymmetry, is also derived.
Previously, such a Newtonian identity had only been known for
spherical configurations [59].
The two virial identities are an important tool for checking
the accuracy of numerical models and have been repeatedly used by
several authors [47,
261,
262,
236
,
19
].
In [295], it is also shown that a large discrepancy between certain
rapidly rotating models (constructed with the FIP and KEH codes)
that was reported by Eriguchi
et al.
[95], resulted from the fact that Eriguchi
et al.
and FIP used different versions of a tabulated EOS.
Nozawa
et al.
[236] have completed an extensive direct comparison of the BGSM, SF,
and the original KEH codes, using a large number of models and
equations of state. More than twenty different quantities for
each model are compared and the relative differences range from
to
or better, for smooth equations of state. The agreement is also
excellent for soft polytropes. These checks show that all three
codes are correct and successfully compute the desired models to
an accuracy that depends on the number of grid points used to
represent the spacetime.
If one makes the extreme assumption of uniform density, the
agreement is at the level of
. In the BGSM code this is due to the fact that the spectral
expansion in terms of trigonometric functions cannot accurately
represent functions with discontinuous first order derivatives at
the surface of the star. In the KEH and SF codes, the three-point
finite-difference formulae cannot accurately represent
derivatives across the discontinuous surface of the star.
The accuracy of the three codes is also estimated by the use of the two virial identities. Overall, the BGSM and SF codes show a better and more consistent agreement than the KEH code with BGSM or SF. This is largely due to the fact that the KEH code does not integrate over the whole spacetime but within a finite region around the star, which introduces some error in the computed models.
A new direct comparison of different codes is presented by
Ansorg
et al.
[19]. Their multi-domain spectral code is compared to the BGSM, KEH,
and SF codes for a particular uniform density model of a rapidly
rotating relativistic star. An extension of the detailed
comparison in [19], which includes results obtained by the Lorene/rotstar code
in [129
] and by the SF code with higher resolution than the resolution
used in [236
], is shown in Table
2
. The comparison confirms that the virial identity GRV3 is a good
indicator for the accuracy of each code. For the particular model
in Table
2, the AKM code achieves nearly double-precision accuracy, while
the Lorene/rotstar code has a typical relative accuracy of
to
in various quantities. The SF code at high resolution comes
close to the accuracy of the Lorene/rotstar code for this model.
Lower accuracies are obtained with the SF, BGSM, and KEH codes at
the resolutions used inused in [236
].
The AKM code converges to machine accuracy when a large number of about 24 expansion coefficients are used at a high computational cost. With significantly fewer expansion coefficients (and comparable computational cost to the SF code at high resolution) the achieved accuracy is comparable to the accuracy of the Lorene/rotstar and SF codes. Moreover, at the mass-shedding limit, the accuracy of the AKM code reduces to about 5 digits (which is still highly accurate, of course), even with 24 expansion coefficients, due to the nonanalytic behaviour of the solution at the surface. Nevertheless, the AKM method represents a great achievement, as it is the first method to converge to machine accuracy when computing rapidly rotating stars in general relativity.
Table 2:
Detailed comparison of the accuracy of different numerical
codes in computing a rapidly rotating, uniform density model. The
absolute value of the relative error in each quantity, compared
to the AKM code, is shown for the numerical codes Lorene/rotstar,
SF (at two resolutions), BGSM, and KEH (see text). The
resolutions for the SF code are (angular
radial) grid points. See [236] for the definition of the various equilibrium quantities.
Going further A review of spectral methods in general relativity can be found in [42]. A formulation for nonaxisymmetric, uniformly rotating equilibrium configurations in the second post-Newtonian approximation is presented in [22].
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Rotating Stars in Relativity
Nikolaos Stergioulas http://www.livingreviews.org/lrr-2003-3 © Max-Planck-Gesellschaft. ISSN 1433-8351 Problems/Comments to livrev@aei.mpg.de |