The field has come a long way since then.
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Blaizot , Li Yan. Of particular interest are the so-called mass-shedding limit, which is the limit where the body is rotating so fast that it is on the verge of starting losing material, and the black hole transition, where rotating fluids are seen to approach black holes for suitable limits of their parameters.
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As the authors themselves mention, one of the emphasis of this book is placed 'on the rigorous treatment of simple models instead of trying to describe real objects with their many complex facets After discussing constant density models both in Newtonian theory the Maclaurin spheroids and in the non-rotating relativistic case the Schwarzschild interior model , the book concentrates on the so-called rigidly rotating disc of dust.
Chapter two is mainly devoted to deriving this model and presenting its physical properties. The derivation is based in the so-called inverse scattering method of integrable systems and on a thorough knowledge of the theory of integration on Riemann surfaces. The details, which take up about one fifth of the whole length, are difficult to follow for any reader without a previous mastering of the techniques involved. For the expert, however, this part of the book is very useful because it brings together all the steps required for the complete determination of the solution.
After the derivation of the disc of dust, the physical properties of the resulting one-parameter family of solutions are described, including its multipole moment structure, the existence of ergospheres, the Newtonian limit or the motion of test particles.
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Of particular interest is the transition from the disc of dust to the extreme black hole configuration corresponding to the limit when the parameter describing the fluid approaches its upper end. After this chapter devoted to exact models, the book looks at the problem from a completely different point of view, namely by using numerical methods.
This tool has proven to be fundamental for a proper study of this physical problem. This book concentrates on the so-called pseudo-spectral methods and the use of multidomains adapted to the different regions of the spacetime with qualitatively different behaviours. The presentation of the main ideas behind this method is very clear and accessible even to the non-expert. The book then is devoted to presenting both qualitative and quantitative results for a number of models with different equations of state. The case treated more in depth is the constant density case, but results for polytropic equations of state as well as a degenerate ideal gas of neutrons and strange quark matter are also presented.
The emphasis is put on the exploration of the parameter space for a fixed equation of state. This is done by studying the various limiting cases involved, namely the non-rotating limit, the Newtonian limit, the mass-shedding limit, the infinite central pressure limit, the transition from one rotating body to several bodies, the black hole limit and the disc limit. The emerging picture in the constant density case is a division of the parameter space into an infinite number of classes, all connected through the Maclaurin spheroids and approaching the limiting case of a Maclaurin disc of dust, which in turn is the Newtonian limit of the relativistic disc of dust.
If the equilibrium stars have no genus, i. For toroidal doughnut-shaped stars, these two distances are the equatorial distances to the inner and outer boundaries respectively, and the ratio is designated to have negative values.
Our approach is almost the same as the one taken by KEH except for having only one gravitational potential to solve. The major difference from the Newtonian approach is that r e appears explicitly in the equations and should be determined during the numerical calculation. These steps are repeated iteratively, as described below in detail.
For the initial model of a non-rotating star before the iteration procedure, we use the modified Toleman—Oppenheimer—Volkoff TOV equation adapted to our configuration. Since this equation is an ordinary differential equation that depends only on r , we solve it using the fourth-order Runge—Kutta method. We used the iterative method for the determination of the gravitational potential at the centre, since we do not know its exact value.
After obtaining the non-rotating model, we decrease the axis ratio for the rotating stellar model. The following is a brief summery of the iteration procedure. See equation Calculation of enthalpy H from the value obtained in step 3 and equation 9.
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The initial guess for the ring solution comes from the Newtonian solution to prevent failure in obtaining solutions using the Newton—Raphson method while solving equation 12 , because its solution procedure depends sensitively on the initial guess. All procedures are carried out until mass-shedding occurs, when the centrifugal force is so large that the gravitational force cannot overcome it. In that case, no stable solution for the rotating star exists.
With this choice, the length, mass and time units can be automatically determined. In Table 1 , we summarize the units of these quantities. Our figures are usually expressed in these units. In this section, we compare our result the pseudo-Newtonian relativistic hydrodynamics approach, pNRH hereafter with three other methods: purely Newtonian, Newtonian relativistic hydrodynamics NRH hereafter , in which special relativity is taken into account with Newtonian gravity, and general relativistic GR hereafter.
This code can calculate the equilibrium solutions of rotating stars for various equations of state. We have not implemented the toroidal configuration in the Whisky code yet. Hence, we compare only spheroidal uniform rotation and quasi-toroidal differential rotation stellar solutions.
The density profiles obtained by these four different methods for a uniformly rotating star with axis ratio of 0. Our proposed method of pNRH solid provides a solution that agrees well with the general relativistic solution dashed. The equatorial radius of the Newtonian result dot—dashed is larger by about 50 per cent compared with the general relativistic result and with ours. For comparison, we also have plotted the result from NRH dotted. Evidently, the introduction of the active mass further improves NRH.
It is evident that our approach of pNRH gives a density profile very similar to that of the GR solution. Our pNRH result for the equatorial radius agrees with that of general relativity within 5 per cent. On the other hand, the Newtonian solution is significantly different from the general relativistic one. For example, the the equatorial radius of the Newtonian solution is about 50 per cent larger than that of the GR method.
NRH that does not take into account the active mass gives a solution closer to the GR one compared with the Newtonian approach, but we can see further improvement with pNRH. The relative accuracy of our pseudo-Newtonian approach over the NRH would be even more appreciable for models with larger R.
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We took the same parameter set as in Fig. Same as Fig. The shape of such a model is quasi-toroidal, so that the maximum density does not occur at the centre see Section 4. A similar trend to the uniformly rotating model of Fig.
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Our result of pNRH solid for the density distribution along the major axis shows good agreement with the GR solution dashed line. The Newtonian solution dot—dashed has an equatorial radius about twice as large as that of the GR one. The location of maximum density is even more different.
Not shown in the figures, we have calculated the equilibrium solutions for the Newtonian hydrostatic equation coupled with the modified Poisson's equation that takes the active mass into account. The solutions are not very different from the purely Newtonian case, since the active mass differs from the rest mass only slightly.
We conclude that relativistic treatment of the hydrodynamics is crucial for better agreement. Also, it could be improved significantly by taking into account the active mass density for relativistic cases. Newly born neutron stars rotate differentially and are near critical rotation, but turn into uniformly rotating stars within a relatively short time-scale because of the shear viscosity and magnetic tension.
This means that old neutron stars tend to rotate uniformly. We now present the detailed characteristics of the pNRH solutions for rotating compact stars. We discuss our results separately for uniformly and differentially rotating cases.
Examples of our equilibrium solutions for uniformly rotating stars are shown in Fig. The top and bottom panels show spheroidal and toroidal solutions, respectively. Our results show that the physical size is about We note here that the axis ratios for toroidal models are denoted by negative numbers following the convention by Hachisu a , b. The size and the rotation frequency obtained are The physical size r e and the rotation frequency f rot are The physical size is As expected, our solutions show the typical features of polytropic rotating stars.
The density falls off more rapidly, the size increases and the rotation speed becomes smaller as N increases. In Fig. We also show the normalized angular speed versus normalized ellipticity in the inset to each panel. These figures show that the functional behaviour of the angular speed at the rotational axis mostly depends on the equation of state.
However, our model shows somewhat different results.