To order , the structure of a star changes only by quadrupole terms and the equilibrium equations become a set of ordinary differential equations. Hartle’s [143, 148] method computes rotating stars in this slow rotation approximation, and a review of slowly rotating models has been compiled by Datta [82]. Weber et al. [317, 319] also implement Hartle’s formalism to explore the rotational properties of four new EOSs.
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 [260]. 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.
The BI scheme [55] solves the four field equations following a Newton–Raphson-like linearization and iteration procedure. One starts with a nonrotating model and increases the angular velocity in small steps, treating a new rotating model as a linear perturbation of the previously computed rotating model. Each linearized field equation is discretized and the resulting linear system is solved. The four field equations and the hydrostationary equilibrium equation are solved separately and iteratively until convergence is achieved.
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 . 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. [193] 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.
In the KEH scheme [183, 184], the same set of field equations as in BI is used, but the three elliptic-type field equations are converted into integral equations using appropriate Green’s functions. The boundary conditions at large distance from the star are thus incorporated into the integral equations, but the region of integration is truncated at a finite distance from the star. The fourth field equation is an ordinary first order differential equation. The field equations and the equation of hydrostationary equilibrium are solved iteratively, fixing the maximum energy density and the ratio of the polar radius to the equatorial radius, until convergence is achieved. In [183, 184, 95] the original KEH code is used to construct uniformly and differentially rotating stars for both polytropic and realistic EOSs.
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 . 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 [294] and in [292]. It is available as a public domain code, named rns, and can be downloaded from [291].
In the BGSM scheme [47], the field equations are derived in the 3+1 formulation. All four chosen equations that describe the gravitational field are of elliptic type. This avoids the problem with the second order radial derivative in the source term of the ODE used in BI and KEH. The equations are solved using a spectral method, i.e., all functions are expanded in terms of trigonometric functions in both the angular and radial directions and a Fast Fourier Transform (FFT) is used to obtain coefficients. Outside the star a redefined radial variable is used, which maps infinity to a finite distance.
In [260, 261] 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.
Bonazzola et al. [46] have improved the BGSM spectral method by allowing for several domains of integration. One of the domain boundaries is chosen to coincide with the surface of the star and a regularization procedure is introduced for the divergent derivatives at the surface (that appear in the density field when stiff equations of state are used). This allows models to be computed that are nearly free of Gibbs phenomena at the surface. The same method is also suitable for constructing quasi-stationary models of binary neutron stars. The new method has been used in [133] for computing models of rapidly rotating strange stars and it has also been used in 3D computations of the onset of the viscosity-driven instability to bar-mode formation [129].
A new multi-domain spectral method has been introduced in [18, 19]. The method can use several domains inside the star, one for each possible phase transition. Surface-adapted coordinates are used and approximated by a two-dimensional Chebyshev expansion. Requiring transition conditions to be satisfied at the boundary of each domain, the field and fluid equations are solved as a free boundary value problem by a Newton–Raphson method, starting from an initial guess. The field equations are simplified by using a corotating reference frame. Applying this new method to the computation of rapidly rotating homogeneous relativistic stars, Ansorg et al. achieve near machine accuracy, except for configurations at the mass-shedding limit (see Section 2.7.8)! The code has been used in a systematic study of uniformly rotating homogeneous stars in general relativity [264].
Equilibrium configurations in Newtonian gravity satisfy the well-known virial relation
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 [39]. 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 [39] (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, 260, 261, 235, 18].
The accuracy of the above numerical codes can be estimated, if one constructs exactly the same models with different codes and compares them directly. The first such comparison of rapidly rotating models constructed with the FIP and SF codes is presented by Stergioulas and Friedman in [294]. Rapidly rotating models constructed with several EOSs agree to 0.1 – 1.2% in the masses and radii and to better than 2% in any other quantity that was compared (angular velocity and momentum, central values of metric functions, etc.). This is a very satisfactory agreement, considering that the BI code was using relatively few grid points, due to limitations of computing power at the time of its implementation.
In [294], 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. [235] 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 10^{–3} to 10^{–4} 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 10^{–2}. 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. [18]. 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 [18], which includes results obtained by the Lorene/rotstar code in [129] and by the SF code with higher resolution than the resolution used in [235], 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 2 × 10^{–4} to 7 × 10^{–6} 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 in [235].
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.
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].
AKM | Lorene/ | SF | SF | BGSM | KEH | |
rotstar | (260 × 400) | (70 × 200) | ||||
1.0 | ||||||
0.7 | 1 × 10^{–3} | |||||
1.41170848318 | 9 × 10^{–6} | 3 × 10^{–4} | 3 × 10^{–3} | 1 × 10^{–2} | 1 × 10^{–2} | |
0.135798178809 | 2 × 10^{–4} | 2 × 10^{–5} | 2 × 10^{–3} | 9 × 10^{–3} | 2 × 10^{–2} | |
0.186338658186 | 2 × 10^{–4} | 2 × 10^{–4} | 3 × 10^{–3} | 1 × 10^{–2} | 2 × 10^{–3} | |
0.345476187602 | 5 × 10^{–5} | 3 × 10^{–5} | 5 × 10^{–4} | 3 × 10^{–3} | 1 × 10^{–3} | |
0.0140585992949 | 2 × 10^{–5} | 4 × 10^{–4} | 5 × 10^{–4} | 2 × 10^{–2} | 2 × 10^{–2} | |
1.70735395213 | 1 × 10^{–5} | 4 × 10^{–5} | 1 × 10^{–4} | 2 × 10^{–2} | 6 × 10^{–2} | |
–0.162534082217 | 2 × 10^{–4} | 2 × 10^{–3} | 2 × 10^{–2} | 4 × 10^{–2} | 2 × 10^{–2} | |
11.3539142587 | 7 × 10^{–6} | 7 × 10^{–5} | 1 × 10^{–3} | 8 × 10^{–2} | 2 × 10^{–1} | |
4 × 10^{–13} | 3 × 10^{–6} | 3 × 10^{–5} | 1 × 10^{–3} | 4 × 10^{–3} | 1 × 10^{–1} | |
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