where is the rest mass density, K is a constant, and is the polytropic exponent. Instead of , one often uses the polytropic index N, defined through
For this equation of state, the quantity has units of length. In gravitational units (c = G =1), one can thus use as a fundamental length scale to define dimensionless quantities. Equilibrium models are then characterized by the polytropic index N and their properties can be scaled to different values, using an appropriate value for K . For N <1.0 (N >1.0), one obtains stiff (soft) models, while for N =0.5 - 1.0, one obtains models with bulk properties that are comparable to those of observed neutron stars.
Note that for the above polytropic EOS, the polytropic index coincides with the adiabatic index of a relativistic isentropic fluid
This is not the case for the polytropic equation of state, , that has been used by other authors, which satisfies (12) only in the Newtonian limit.
The true equation of state that describes the interior of compact stars is largely unknown. This results from the inability to verify experimentally the different theories that describe the strong interactions between baryons and the many-body theories of dense matter at densities larger than about twice the nuclear density (i.e. at densities larger than about ).
To date, many different realistic EOSs have been proposed which produce neutron stars that satisfy the currently available observational constraints (Currently, the two main constraints are that the EOS must admit nonrotating neutron stars with gravitational mass of at least and allow rotational periods at least as small as 1.56 ms, see [6, 7].). The proposed EOSs are qualitatively and quantitatively very different from each other. Some are based on relativistic many-body theories, while others use nonrelativistic theories with baryon-baryon interaction potentials. A classic collection of early proposed EOSs was compiled by Arnett and Bowers , while recent EOSs are described in Salgado et al. .
High density equations of state with pion condensation have been proposed by Migdal  and Sawyer and Scalapino . The possibility of Kaon condensation is discussed by Brown and Bethe  and questioned by Pandharipande et al. . Many authors have examined the possibility of stars composed of strange quark matter, and a recent review can be found in .
The realistic EOSs are supplied in the form of an energy density vs. pressure table, and intermediate values are interpolated. This results in some loss of accuracy because the usual interpolation methods do not preserve thermodynamical consistency. Recently however, Swesty  devised a cubic Hermite interpolation scheme that does preserve thermodynamical consistency; the scheme has been shown to indeed produce higher accuracy neutron star models in Nozawa et al. .
|Rotating Stars in Relativity
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