Weber and Glendenning [32] attempt to 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 show large discrepancies
compared to corresponding models computed with fully rotating
schemes [9]. Thus, Hartle's formalism cannot be used to compute models of
rapidly rotating relativistic stars with sufficient accuracy.
The 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 [21,
33]. Friedman et al. [34,
35] extend the BI code to obtain a large number of rapidly rotating
models based on a variety of realistic EOSs. Lattimer et al. [36] used a code which was also based on the BI scheme to construct
rotating stars using recent ``exotic'' and schematic EOSs,
including pion or Kaon condensation and self-bound strange quark
matter.
Cook, Shapiro and Teukolsky (CST) improve on the KEH scheme by
introducing a new radial variable which 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 [38
] and polytropic and realistic models are computed in [39
] and [40
].
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
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 [41
] and in [42
]. It is available as a public domain code, named
rns, and can be downloaded from [43
].
In [49] 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 space-times, that were discovered by
Gourgoulhon and Bonazzola [50,
51
].
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 magnetic field,
solid crust or solid interior, and it can also be used to
construct rotating boson stars.
Since it is based on the 3+1 formalism, the BGSM code is also suitable for providing high-accuracy, unstable equilibrium models as initial data for an axisymmetric collapse computation.
In [41], 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. [37
], was only due to the fact that a different version of a
tabulated EOS was used in [37] than by FIP.
Recently, Nozawa et al. [16] 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
excellent for soft polytropes, which shows that all three codes
are correct and 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 due to Gourgoulhon and Bonazzola [50, 51]. 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.
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Rotating Stars in Relativity
Nikolaos Stergioulas http://www.livingreviews.org/lrr-1998-8 © Max-Planck-Gesellschaft. ISSN 1433-8351 Problems/Comments to livrev@aei-potsdam.mpg.de |