where
K
and
are the polytropic constant and polytropic exponent,
respectively. Notice that the above definition is
different
from the form
(also due to Tooper [311]) that has also been used as a generalization of the Newtonian
polytropic EOS. Instead of
, one often uses the polytropic index
N, defined through
For the above 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 dimensionless central energy density. Equilibrium
properties can be scaled to different dimensional values, using
appropriate values for
K
. For
N
<1.0 (N
>1.0) one obtains stiff (soft) models, while for
, one obtains models with bulk properties that are comparable to
those of observed neutron star radii and masses.
Notice 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
, which satisfies (25
) only in the Newtonian limit.
Many different so-called realistic EOSs have been proposed to
date that all produce neutron star models that satisfy the
currently available observational constraints. The two most
accurate 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 [243
,
187
]). Recently, the first direct determination of the gravitational
redshift of spectral lines produced in the neutron star
photosphere has been obtained [74]. This determination (in the case of the low-mass X-ray binary
EXO 0748-676) yielded a redshift of
z
=0.35 at the surface of the neutron star, corresponding to a mass
to radius ratio of
M
/
R
=0.23 (in gravitational units), which is compatible with most
normal nuclear matter EOSs and incompatible with some exotic
matter EOSs.
The theoretically 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 [20], while recent EOSs are used in Salgado
et al.
[261
] and in [84]. A review of many modern EOSs can be found in a recent article
by Haensel [138]. Detailed descriptions and tables of several modern EOSs,
especially EOSs with phase transitions, can be found in
Glendenning's book [125
].
High density equations of state with pion condensation have
been proposed by Migdal [228] and Sawyer and Scalapino [264]. The possibility of kaon condensation is discussed by Brown and
Bethe [51] (but see also Pandharipande
et al.
[241]). Properties of rotating Skyrmion stars have been computed
in [237].
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. Swesty [301] devised a cubic Hermite interpolation scheme that does preserve
thermodynamical consistency and the scheme has been shown to
indeed produce higher accuracy neutron star models in Nozawa
et al.
[236].
Usually, the interior of compact stars is modeled as a one-component ideal fluid. When neutron stars cool below the superfluid transition temperature, the part of the star that becomes superfluid can be described by a two-fluid model and new effects arise. Andersson and Comer [9] have recently used such a description in a detailed study of slowly rotating superfluid neutron stars in general relativity, while the first rapidly rotating models are presented in [248].
The strange quark matter equation of state can be represented by the following linear relation between pressure and energy density:
where
is the energy density at the surface of a bare strange star
(neglecting a possible thin crust of normal matter). The MIT bag
model of strange quark matter involves three parameters, the bag
constant,
, the mass of the strange quark,
, and the QCD coupling constant,
. The constant
a
in (26
) is equal to 1/3 if one neglects the mass of the strange quark,
while it takes the value of
a
=0.289 for
. When measured in units of
, the constant
B
is restricted to be in the range
assuming
. The lower limit is set by the requirement of stability of
neutrons with respect to a spontaneous fusion into strangelets,
while the upper limit is determined by the energy per baryon of
Fe at zero pressure (930.4 MeV). For other values of
the above limits are modified somewhat.
A more recent attempt to describe deconfined strange quark
matter is the Dey
et al.
EOS [87], which has asymptotic freedom built in. It describes deconfined
quarks at high densities and confinement at zero pressure. The
Dey
et al.
EOS can be approximated by a linear relation of the same form as
the MIT bag model strange star EOS (26
). In such a linear approximation, typical values of the constant
a
are 0.45-0.46 [128
].
Going further A review of strange quark star properties can be found in [320]. Hybrid stars that have a mixed-phase region of quark and hadronic matter, have also been proposed (see e.g. [125]). A study of the relaxation effect in dissipative relativistic fluid theories is presented in [200].
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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 |