The gravitational mass, equatorial radius, and rotational
period of the maximum mass model constructed with one of the
softest EOSs (EOS B) (, 9.3 km, 0.4 ms) are a factor of two smaller than the
mass, radius, and period of the corresponding model constructed
by one of the stiffest EOSs (EOS L) (
, 18.3 km, 0.8 ms). The two models differ by a factor
of 5 in central energy density and by a factor of 8 in the moment
of inertia!
Not all properties of the maximum mass models between proposed
EOSs differ considerably, at least not within groups of similar
EOSs. For example, most realistic hadronic EOSs predict a maximum
mass model with a ratio of rotational to gravitational energy
T
/
W
of
, a dimensionless angular momentum
of
, and an eccentricity of
[112]. Hence, within the set of realistic hadronic EOSs, some
properties are directly related to the stiffness of the EOS while
other properties are rather insensitive to stiffness. On the
other hand, if one considers strange quark EOSs, then for the
maximum mass model
T
/
W
can become a factor of about two larger than for hadronic
EOSs.
Compared to nonrotating stars, the effect of rotation is to increase the equatorial radius of the star and also to increase the mass that can be sustained at a given central energy density. As a result, the mass of the maximum mass rotating model is roughly 15-20% higher than the mass of the maximum mass nonrotating model, for typical realistic hadronic EOSs. The corresponding increase in radius is 30-40%. The effect of rotation in increasing the mass and radius becomes more pronounced in the case of strange quark EOSs (see Section 2.9.8).
The deformed shape of a rapidly rotating star creates a
distortion, away from spherical symmetry, in its gravitational
field. Far from the star, the dominant multipole moment of the
rotational distortion is measured by the quadrupole-moment tensor
. For uniformly rotating, axisymmetric, and equatorially
symmetric configurations, one can define a scalar quadrupole
moment
Q, which can be extracted from the asymptotic expansion of the
metric function
at large
r, as in Equation (10
).
Laarakkers and Poisson [188] numerically compute the scalar quadrupole moment Q for several equations of state, using the rotating neutron star code rns [292]. They find that for fixed gravitational mass M, the quadrupole moment is given as a simple quadratic fit,
where
J
is the angular momentum of the star and
a
is a dimensionless quantity that depends on the equation of
state. The above quadratic fit reproduces
Q
with remarkable accuracy. The quantity
a
varies between
for very soft EOSs and
for very stiff EOSs, for
neutron stars. This is considerably different from a Kerr black
hole, for which
a
=1 [305].
For a given zero-temperature EOS, the uniformly rotating
equilibrium models form a 2-dimensional surface in the
3-dimensional space of central energy density, gravitational
mass, and angular momentum [295], as shown in Figure
1
for EOS L. The surface is limited by the nonrotating models (J
=0) and by the models rotating at the mass-shedding (Kepler)
limit,
i.e.
at the maximum allowed angular velocity (above which the star
sheds mass at the equator). Cook
et al.
[69
,
71
,
70
] have shown that the model with maximum angular velocity does
not coincide with the maximum mass model, but is generally very
close to it in central density and mass. Stergioulas and
Friedman [295] show that the maximum angular velocity and maximum baryon mass
equilibrium models are also distinct. The distinction becomes
significant in the case where the EOS has a large phase
transition near the central density of the maximum mass model;
otherwise the models of maximum mass, baryon mass, angular
velocity, and angular momentum can be considered to coincide for
most purposes.
Going further Although rotating relativistic stars are nearly perfectly axisymmetric, a small degree of asymmetry (e.g. frozen into the solid crust during its formation) can become a source of gravitational waves. A recent review of this can be found in [165].
where
In differentially rotating stars, even a small amount of differential rotation can significantly increase the angular velocity required for mass-shedding. Thus, a newly-born, hot, differentially rotating neutron star or a massive, compact object created in a binary neutron star merger could be sustained (temporarily) in equilibrium by differential rotation, even if a uniformly rotating configuration with the same rest mass does not exist.
In the Newtonian limit the maximum angular velocity of
uniformly rotating polytropic stars is approximately
(this is derived using the Roche model, see [268]). For relativistic stars, the empirical formula [142,
114,
109]
gives the maximum angular velocity in terms of the mass and radius of the maximum mass nonrotating model with an accuracy of 5-7%, without actually having to construct rotating models. A revised empirical formula, using a large set of EOSs, has been computed in [141].
The empirical formula results from universal proportionality relations that exist between the mass and radius of the maximum mass rotating model and those of the maximum mass nonrotating model for the same EOS. Lasota et al. [193] find that, for most EOSs, the coefficient in the empirical formula is an almost linear function of the parameter
The Lasota et al. empirical formula
with
, reproduces the exact values with a relative error of only
1.5%.
Weber and Glendenning [317,
318] derive analytically a similar empirical formula in the slow
rotation approximation. However, the formula they obtain involves
the mass and radius of the maximum mass
rotating
configuration, which is different from what is involved
in (32).
When magnetic field effects are ignored, conservation of
angular momentum can yield very rapidly rotating neutron stars at
birth. Recent simulations of the rotational core collapse of
evolved rotating progenitors [151,
119] have demonstrated that rotational core collapse can easily
result in the creation of neutron stars with rotational periods
of the order of 1 ms (and similar initial rotation periods have
been estimated for neutron stars created in the accretion-induced
collapse of a white dwarf [212]). The existence of a magnetic field may complicate this
picture. Spruit and Phinney [288] have presented a model in which a strong internal magnetic
field couples the angular velocity between core and surface
during most evolutionary phases. The core rotation decouples from
the rotation of the surface only after central carbon depletion
takes place. Neutron stars born in this way would have very small
initial rotation rates, even smaller than the ones that have been
observed in pulsars associated with supernova remnants. In this
model, an additional mechanism is required to spin up the neutron
star to observed periods. On the other hand, Livio and
Pringle [213] argue for a much weaker rotational coupling between core and
surface by a magnetic field, allowing for the production of more
rapidly rotating neutron stars than in [288]. A new investigation by Heger
et al., yielding intermediate initial rotation rates, is presented
in [152]. Clearly, more detailed computations are needed to resolve this
important question.
The minimum observed pulsar period is still
1.56 ms [187], which is close to the experimental sensitivity of most pulsar
searches. New pulsar surveys, in principle sensitive down to a
few tenths of a millisecond, have not been able to detect a
sub-millisecond pulsar [52,
81,
75,
94]. This is not too surprising, as there are several explanations
for the absence of sub-millisecond pulsars. In one model, the
minimum rotational period of pulsars could be set by the
occurrence of the
r
-mode instability in accreting neutron stars in Low Mass X-ray
Binaries (LMXBs) [12]. Other models are based on the standard magnetospheric model
for accretion-induced spin-up [322] or on the idea that gravitational radiation (produced by
accretion-induced quadrupole deformations of the deep crust)
balances the spin-up torque [35
,
313]. It has also been suggested [53] that the absence of sub-millisecond pulsars in all surveys
conducted so far could be a selection effect: Sub-millisecond
pulsars could be found more likely only in close systems (of
orbital period
), however the current pulsar surveys are still lacking the
required sensitivity to easily detect such systems. The absence
of sub-millisecond pulsars in wide systems is suggested to be due
to the turning-on of the accreting neutron stars as pulsars, in
which case the pulsar wind is shown to halt further spin-up.
Going further A review by J.L. Friedman concerning the upper limit on the rotation of relativistic stars can be found in [110].
In relativistic perfect fluids, the speed of sound is the characteristic velocity of the evolution equations for the fluid, and the causality constraint translates into the requirement
(see [120]). It is assumed that the fluid will still behave as a perfect fluid when it is perturbed from equilibrium.
For nonrotating stars, Rhoades and Ruffini showed that the EOS
that satisfies the above two constraints and yields the maximum
mass consists of a high density region as stiff as possible (i.e.
at the causal limit,
), that matches directly to the known low density EOS. For a
chosen matching density
, they computed a maximum mass of
. However, this is not the theoretically maximum mass of
nonrotating neutron stars, as is often quoted in the literature.
Hartle and Sabbadini [146] point out that
is sensitive to the matching energy density and Hartle [144] computes
as a function of
:
In the case of rotating stars, Friedman and Ipser [111] assume that the absolute maximum mass is obtained by the same
EOS as in the nonrotating case and compute
as a function of matching density, assuming the BPS EOS holds at
low densities. A more recent computation [186
] uses the FPS EOS at low densities, arriving at a similar result
as in [111]:
where
is roughly nuclear saturation density for the FPS EOS.
A first estimate of the absolute minimum period of uniformly
rotating, gravitationally bound stars was computed by
Glendenning [124] by constructing nonrotating models and using the empirical
formula (32) to estimate the minimum period. Koranda, Stergioulas, and
Friedman [186
] improve on Glendenning's results by constructing accurate,
rapidly rotating models; they show that Glendenning's results are
accurate to within the accuracy of the empirical formula.
Furthermore, they show that the EOS satisfying the minimal set
of constraints and yielding the minimum period star consists of a
high density region at the causal limit (CL EOS),
, (where
is the lowest energy density of this region), which is matched
to the known low density EOS through an intermediate constant
pressure region (that would correspond to a first order phase
transition). Thus, the EOS yielding absolute minimum period
models is as stiff as possible at the central density of the star
(to sustain a large enough mass) and as soft as possible in the
crust, in order to have the smallest possible radius (and
rotational period).
The absolute minimum period of uniformly rotating stars is an (almost linear) function of the maximum observed mass of nonrotating neutron stars,
and is rather insensitive to the matching density
(the above result was computed for a matching number density of
). In [186], it is also shown that an absolute limit on the minimum period
exists even without requiring that the EOS matches to a known low
density EOS,
i.e.
\ if the CL EOS,
, terminates at a surface energy density of
. This is not so for the causal limit on the maximum mass. Thus,
without matching to a low-density EOS, the causality limit on
is lowered by only 3%, which shows that the currently known part
of the nuclear EOS plays a negligible role in determining the
absolute upper limit on the rotation of uniformly rotating,
gravitationally bound stars.
The above results have been confirmed in [139], where it is shown that the CL EOS has
, independent of
, and the empirical formula (34
) reproduces the numerical result (38
) to within 2%.
In a neutron star binary merger, prompt collapse to a black hole can be avoided if the equation of state is sufficiently stiff and/or the equilibrium is supported by strong differential rotation. The maximum mass of differentially rotating supramassive neutron stars can be significantly larger than in the case of uniform rotation. A detailed study of this mass-increase has recently appeared in [215].
Cook
et al.
[69,
71,
70] have discovered that a supramassive relativistic star
approaching the axisymmetric instability will actually spin up
before collapse, even though it loses angular momentum. This
potentially observable effect is independent of the equation of
state and it is more pronounced for rapidly rotating massive
stars. Similarly, stars can spin up by loss of angular momentum
near the mass-shedding limit, if the equation of state is
extremely stiff or extremely soft.
If the equation of state features a phase transition, e.g. to quark matter, then the spin-up region is very large, and most millisecond pulsars (if supramassive) would need to be spinning up [289]; the absence of spin-up in known millisecond pulsars indicates that either large phase transitions do not occur, or that the equation of state is sufficiently stiff so that millisecond pulsars are not supramassive.
A magnetized relativistic star in equilibrium can be described
by the coupled Einstein-Maxwell field equations for stationary,
axisymmetric rotating objects with internal electric currents.
The stress-energy tensor includes the electromagnetic energy
density and is non-isotropic (in contrast to the isotropic
perfect fluid stress-energy tensor). The equilibrium of the
matter is given not only by the balance between the gravitational
force, centrifugal force, and the pressure gradient; the Lorentz
force due to the electric currents also enters the balance. For
simplicity, Bocquet
et al.
consider only poloidal magnetic fields that preserve the
circularity of the spacetime. Also, they only consider stationary
configurations, which excludes magnetic dipole moments
non-aligned with the rotation axis, since in that case the star
emits electromagnetic and gravitational waves. The assumption of
stationarity implies that the fluid is necessarily rigidly
rotating (if the matter has infinite conductivity) [47]. Under these assumptions, the electromagnetic field tensor
is derived from a potential four-vector
with only two non-vanishing components,
and
, which are solutions of a scalar Poisson and a vector Poisson
equation respectively. Thus, the two equations describing the
electromagnetic field are of similar type as the four field
equations that describe the gravitational field.
For magnetic field strengths larger than about
, one observes significant effects, such as a flattening of the
equilibrium configuration. There exists a maximum value of the
magnetic field strength of the order of
, for which the magnetic field pressure at the center of the star
equals the fluid pressure. Above this value no stationary
configuration can exist.
A strong magnetic field allows a maximum mass configuration
with larger
than for the same EOS with no magnetic field and this is
analogous to the increase of
induced by rotation. For nonrotating stars, the increase in
due to a strong magnetic field is 13-29%, depending on the EOS.
Correspondingly, the maximum allowed angular velocity, for a
given EOS, also increases in the presence of a strong magnetic
field.
Another application of general relativistic E/M theory in neutron stars is the study of the evolution of the magnetic field during pulsar spin-down. A detailed analysis of the evolution equations of the E/M field in a slowly rotating magnetized neutron star has revealed that effects due to the spacetime curvature and due to the rotational frame-dragging are present in the induction equations, when one assumes finite electrical conductivity (see [252] and references therein). Numerical solutions of the evolution equations of the E/M have shown, however, that for realistic values of the electrical conductivity, the above relativistic effects are small, even in the case of rapid rotation [336].
Going further
An
slow rotation approach for the construction of rotating
magnetized relativistic stars is presented in [137].
Hashimoto
et al.
[150] and Goussard
et al.
[134
] construct fully relativistic models of rapidly rotating, hot
proto-neutron stars. The authors use finite-temperature
EOSs [239
,
195] to model the interior of PNSs. Important (but largely unknown)
parameters that determine the local state of matter are the
lepton fraction
and the temperature profile. Hashimoto
et al.
consider only the limiting case of zero lepton fraction,
, and classical isothermality, while Goussard
et al.
consider several nonzero values for
and two different limiting temperature profiles - a constant
entropy profile and a relativistic isothermal profile. In
both [150
] and [239], differential rotation is neglected to a first
approximation.
The construction of numerical models with the above assumptions shows that, due to the high temperature and the presence of trapped neutrinos, PNSs have a significantly larger radius than cold NSs. These two effects give the PNS an extended envelope which, however, contains only roughly 0.1% of the total mass of the star. This outer layer cools more rapidly than the interior and becomes transparent to neutrinos, while the core of the star remains hot and neutrino opaque for a longer time. The two regions are separated by the ``neutrino sphere''.
Compared to the
T
=0 case, an isothermal EOS with temperature of 25 MeV has a
maximum mass model of only slightly larger mass. In contrast, an
isentropic EOS with a nonzero trapped lepton number features a
maximum mass model that has a considerably lower mass than the
corresponding model in the
T
=0 case and, therefore, a stable PNS transforms to a stable
neutron star. If, however, one considers the hypothetical case of
a large amplitude phase transition that softens the cold EOS
(such as a kaon condensate), then
of cold neutron stars is lower than the
of PNSs, and a stable PNS with maximum mass will collapse to a
black hole after the initial cooling period. This scenario of
delayed collapse of nascent neutron stars has been proposed by
Brown and Bethe [51] and investigated by Baumgarte
et al.
[31].
An analysis of radial stability of PNSs [127] shows that, for hot PNSs, the maximum angular velocity model
almost coincides with the maximum mass model, as is also the case
for cold EOSs.
Because of their increased radius, PNSs have a different
mass-shedding limit than cold NSs. For an isothermal profile, the
mass-shedding limit proves to be sensitive to the exact location
of the neutrino sphere. For the EOSs considered in [150] and [134
], PNSs have a maximum angular velocity that is considerably less
than the maximum angular velocity allowed by the cold EOSs. Stars
that have nonrotating counterparts (i.e.
that belong to a normal sequence) contract and speed up while
they cool down. The final star with maximum rotation is thus
closer to the mass-shedding limit of cold stars than was the hot
PNS with maximum rotation. Surprisingly, stars belonging to a
supramassive sequence exhibit the opposite behavior. If one
assumes that a PNS evolves without losing angular momentum or
accreting mass, then a cold neutron star produced by the cooling
of a hot PNS has a smaller angular velocity than its progenitor.
This purely relativistic effect was pointed out in [150] and confirmed in [134
].
It should be noted here that a small amount of differential rotation significantly affects the mass-shedding limit, allowing more massive stars to exist than uniform rotation allows. Taking differential rotation into account, Goussard et al. [135] suggest that proto-neutron stars created in a gravitational collapse cannot spin faster than 1.7 ms. A similar result has been obtained by Strobel et al. [298]. The structure of a differentially rotating proto-neutron star at the mass-shedding limit is shown in Figure 2 . The outer layers of the star form an extended disk-like structure.
The above stringent limits on the initial period of neutron stars are obtained assuming that the PNS evolves in a quasi-stationary manner along a sequence of equilibrium models. It is not clear whether these limits will remain valid if one studies the early evolution of PNS without the above assumption. It is conceivable that the thin hot envelope surrounding the PNS does not affect the dynamics of the bulk of the star. If the bulk of the star rotates faster than the (stationary) mass-shedding limit of a PNS model, then the hot envelope will simply be shed away from the star in the equatorial region (if it cannot remain bounded to the star even when differentially rotating). Such a fully dynamical study is needed to obtain an accurate upper limit on the rotation of neutron stars.
Going further The thermal history and evolutionary tracks of rotating PNSs (in the second order slow rotation approximation) have been studied recently in [300].
Nonrotating strange stars obey scaling relations with the
constant
in the MIT bag model of the strange quark matter EOS
(Section
2.6.3); Gourgoulhon
et al.
[133] also obtain scaling relations for the model with maximum
rotation rate. The maximum angular velocity scales as
while the allowed range of
implies an allowed range of
. The empirical formula (32
) also holds for rotating strange stars with an accuracy of
better than 2%. A derivation of the empirical formula in the case
of strange stars, starting from first principles, has been
presented by Cheng and Harko [62], who found that some properties of rapidly rotating strange
stars can be reproduced by approximating the exterior spacetime
by the Kerr metric.
Since both the maximum mass nonrotating and maximum mass rotating models obey similar scalings with B, the ratios
are independent of
(where
is the radius of the maximum mass model). The maximum mass
increases by 44% and the radius of the maximum mass model by 54%,
while the corresponding increase for hadronic stars is, at best,
and
, correspondingly. The rotational properties of strange star
models that are based on the Dey
et al.
EOS [87] are similar to those of the MIT bag model EOS [38,
325,
98], but some quantitative differences exist [128].
Accreting strange stars in LMXBs will follow different
evolutionary paths than do accreting hadronic stars in a mass vs.
central energy density diagram [341]. When (and if) strange stars reach the mass-shedding limit, the
ISCO still exists [297
] (while it disappears for most hadronic EOSs). Stergioulas,
Kluzniak, and Bulik [297
] show that the radius and location of the ISCO for the sequence
of mass-shedding models also scales as
, while the angular velocity of particles in circular orbit at
the ISCO scales as
. Additional scalings with the constant
a
in the strange quark EOS (that were proposed in [196]) are found to hold within an accuracy of better than
for the mass-shedding sequence
In addition, it is found that models at mass-shedding can have
T
/
W
as large as 0.28 for
.
As strange quark stars are very compact, the angular velocity
at the ISCO can become very large. If the 1066 Hz upper QPO
frequency in 4U 1820-30 (see [167] and references therein) is the frequency at the ISCO, then it
rules out most models of slowly rotating strange stars in LMXBs.
However, in [297] it is shown that rapidly rotating bare strange stars are still
compatible with this observation, as they can have ISCO
frequencies
even for
models. On the other hand, if strange stars have a thin solid
crust, the ISCO frequency at the mass-shedding limit increases by
about 10% (compared to a bare strange star of the same mass), and
the above observational requirement is only satisfied for slowly
rotating models near the maximum nonrotating mass, assuming some
specific values of the parameters in the strange star EOS [342,
340
]. Moderately rotating strange stars, with spin frequencies
around 300 Hz can also be accommodated for some values of
the coupling constant
[338] (see also [131] for a detailed study of the ISCO frequency for rotating strange
stars). The 1066 Hz requirement for the ISCO frequency
depends, of course, on the adopted model of kHz QPOs in LMXBs,
and other models exist (see next section).
If strange stars can have a solid crust, then the density at
the bottom of the crust is the neutron drip density
, as neutrons are absorbed by strange quark matter. A strong
electric field separates the nuclei of the crust from the quark
plasma. In general, the mass of the crust that a strange star can
support is very small, of the order of
. Rapid rotation increases by a few times the mass of the crust
and the thickness at the equator becomes much larger than the
thickness at the poles [340
]. Zdunik, Haensel, and Gourgoulhon [340
] also find that the mass
and thickness
of the crust can be expanded in powers of the spin frequency
as
where a subscript ``0'' denotes nonrotating values. For
, the above expansion agrees well with the quadratic expansion
derived previously by Glendenning and Weber [126]. In a spinning down magnetized strange quark star with crust,
parts of the crust will gradually dissolve into strange quark
matter, in a strongly exothermic process. In [340], it is estimated that the heating due to deconfinement may
exceed the neutrino luminosity from the core of a strange star
older than
and may therefore influence the cooling of this compact object
(see also [334]).
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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 |