3 Oscillations and Stability2 The Equilibrium Structure of 2.4 Numerical Schemes

2.5 Properties of Equilibrium Models

2.5.1 Bulk Properties of Equilibrium Models

Neutron star models constructed with various realistic EOSs have considerably different bulk properties, due to the large uncertainties in the equation of state at high densities. Very compressible (soft) EOSs produce models with small maximum mass, small radius, and large rotation rate. On the other hand, less compressible (stiff) EOSs produce models with a large maximum mass, large radius, and low rotation rate.

The gravitational mass, equatorial radius and rotational period of the maximum mass model constructed with one of the softest EOSs (EOS B) (tex2html_wrap_inline2063, 9.3km, 0.4ms) 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) (tex2html_wrap_inline2065, 18.3km, 0.8ms). The two models differ by a factor of 5 in central energy density and a factor of 8 in the moment of inertia!

Not all properties of the maximum mass models between proposed EOSs differ considerably. For example, most realistic EOSs predict a maximum mass model with a ratio of rotational to gravitational energy T / W of tex2html_wrap_inline2069, a dimensionless angular momentum tex2html_wrap_inline2071 of tex2html_wrap_inline2073 and an eccentricity of tex2html_wrap_inline2075, [1Jump To The Next Citation Point In The Article]. Hence, between the set of realistic EOSs, some properties are directly related to the stiffness of the EOS while other properties are rather insensitive to stiffness.

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 tex2html_wrap_inline2077 higher than the mass of the maximum mass nonrotating model, for typical realistic EOSs. The corresponding increase in radius is tex2html_wrap_inline2079 .

The deformed shape of a rapidly rotating star creates a distortion, away from spherical symmetry, in its gravitational field. Far from the star, the distortion is measured by the quadrupole-moment tensor tex2html_wrap_inline2081 . For uniformly rotating, axisymmetric and equatorially symmetric configurations, one can define a scalar quadrupole moment Q, which can be extracted from the asymptotic expansion, at large r, of the metric function tex2html_wrap_inline1975 .

Laarakkers and Poisson [54], numerically compute the scalar quadrupole moment Q for several equations of state, using the rotating neutron star code rns [43]. 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 a remarkable accuracy. The quantity a varies between tex2html_wrap_inline2099 for very soft EOSs and tex2html_wrap_inline2101 for very stiff EOSs, for tex2html_wrap_inline2103 neutron stars.

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 [41Jump To The Next Citation Point In The Article]. 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 so that the star does not shed mass at the equator. Cook et al. [38Jump To The Next Citation Point In The Article, 39Jump To The Next Citation Point In The Article, 40Jump To The Next Citation Point In The Article] 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 [41Jump To The Next Citation Point In The Article] 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.

2.5.2 An Empirical Formula for the Kepler Velocity

In the Newtonian limit, the maximum angular velocity of uniformly rotating polytropic stars is, tex2html_wrap_inline2107 (see [150]). For relativistic stars, the empirical formula [55, 35, 56]


gives the maximum angular velocity in terms of the mass and radius of the maximum mass nonrotating model with an accuracy of tex2html_wrap_inline2109, without actually having to construct rotating models.

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. [57] find that, for most EOSs, the coefficient in the empirical formula is an almost linear function of the parameter


When this relation is taken into account in the empirical formula, it reproduces the exact values with a relative error of only tex2html_wrap_inline2111 .

Weber and Glendenning [30, 32] try to reproduce analytically the empirical formula in the slow rotation approximation, but the formula they obtain involves the mass and radius of the maximum mass rotating configuration, which is different from what is involved in (14Popup Equation).

2.5.3 The Upper Limit on Mass and Rotation

The maximum mass and minimum period of rotating relativistic stars computed with realistic EOSs from the Arnett and Bowers collection [8] are about tex2html_wrap_inline2113 (EOS L) and 0.4ms (EOS B), while tex2html_wrap_inline2117 neutron stars, rotating at the Kepler limit, have a rotational periods between 0.53ms (EOS B) and 1.7ms (EOS M) [40Jump To The Next Citation Point In The Article]. The maximum, accurately measured, neutron star mass is currently tex2html_wrap_inline2033, but there are also indications for tex2html_wrap_inline2123 neutron stars [58]. The minimum observed pulsar period is 1.56ms [7Jump To The Next Citation Point In The Article], which is close to the experimental sensitivity of recent pulsar searches (an ongoing experiment is designed to detect sub-millisecond pulsars, if they exist [59]).

In principle, neutron stars with maximum mass or minimum period could exist, if they are born as such in a core collapse, or if they accrete the right amount of matter and angular momentum during an accretion-induced spin-up phase. Such a phase could also follow the creation of an tex2html_wrap_inline2117 neutron star during the accretion induced collapse of a white dwarf.

In reality, only a very small fraction, if any, of neutron stars will be close to the maximum mass or minimum period limit. In addition, rapidly rotating nascent neutron stars are subject to a nonaxisymmetric instability, which lowers their initial rotation rate and neutron stars with a strong magnetic field have their rotation rate limited by the Kepler velocity at their Alfven radius, where the accretion pressure balances the magnetospheric pressure [7].

2.5.4 The Upper Limit on Mass and Rotation Set by Causality

Current proposed EOSs are reliable only to about twice nuclear density and result in very different values for the maximum mass and minimum period of neutron stars. If one is interested in obtaining upper limits on the mass and rotation rate, independent of the proposed EOSs, one has to rely on fundamental physical principles.

Instead of using realistic EOSs, one constructs a set of artificial EOSs that satisfy only a minimal set of physical constraints, which represent what we know about the equation of state of matter with high confidence. One then searches among all these EOSs to obtain the one that gives the maximum mass or minimum period. The minimal set of constraints that have been used in such searches is:

  1. the high density EOS matches to the known low density EOS at some matching energy density tex2html_wrap_inline2127,
  2. the matter at high densities satisfies the causality constraint (the speed of sound is less than the speed of light).
In relativistic perfect fluids, the speed of sound is the characteristic velocity of the fluid evolution equations, and the causality constraint translates into the requirement


(see e.g. Geroch and Lindblom [61]). 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, tex2html_wrap_inline2129), that matches directly to the known low density EOS. For a chosen matching density tex2html_wrap_inline2127, they computed a maximum mass of tex2html_wrap_inline2133 . However, this is not the theoretically maximum mass of nonrotating neutron stars, as is often quoted in the literature. Hartle and Sabbadini [62] point out that tex2html_wrap_inline2135 is sensitive to the matching energy density, and Hartle [63] computes tex2html_wrap_inline2135 as a function of tex2html_wrap_inline2127 .


In the case of rotating stars, Friedman and Ipser [64] assume that the absolute maximum mass is obtained by the same EOS as in the nonrotating case and compute tex2html_wrap_inline2135 as a function of matching density, assuming the BPS EOS holds at low densities. Stergioulas and Friedman [41] recompute tex2html_wrap_inline2143 for rotating stars using the more recent FPS EOS at low densities, obtaining very nearly the same result


where, tex2html_wrap_inline2145 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 [65] by constructing nonrotating models and using the empirical formula (14Popup Equation) to estimate the minimum period. Koranda, Stergioulas and Friedman [66Jump To The Next Citation Point In The Article] improve on Glendenning's results by constructing accurate rapidly rotating models and 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, 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 tex2html_wrap_inline2127 (the above result was computed for a matching number density of tex2html_wrap_inline2149).

In [66], 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 (This is not true for the limit on the maximum mass.). Thus, using causality as the only constraint on the EOS, tex2html_wrap_inline2151 is lowered by only tex2html_wrap_inline2153, 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.

2.5.5 Spin-Up Prior to Collapse

Since rotation increases the mass that a neutron star of given central density can support, there exist sequences of neutron stars with constant baryon number that have no nonrotating member. Such sequences are called supramassive as opposed to normal sequences that do have a nonrotating member. A nonrotating star can become supramassive by accreting matter and spinning-up to large rotation rates; in another scenario, neutron stars could be born supramassive after a core collapse. A supramassive star evolves along a sequence of constant baryon mass, slowly loosing angular momentum. Eventually, the star reaches a point where it becomes unstable to axisymmetric perturbations and collapses to a black hole. The instability grows on a secular timescale, in the sense that it is limited by the time required for viscosity to redistribute the star's angular momentum. This timescale is comparable with the spin-up time following a glitch [1].

Cook et al. [38Jump To The Next Citation Point In The Article, 39, 40] have discovered that a supramassive 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 equations of state, and it is more pronounced for rapidly rotating massive stars. In a similar phenomenon, normal stars can spin-up by loss of angular momentum near the Kepler limit, if the equation of state is extremely stiff or extremely soft.

2.5.6 Rotating Magnetized Neutron Stars

The presence of a magnetic field was ignored in the models of rapidly rotating relativistic stars that were considered in the previous sections. The reason is that the observed surface dipole magnetic field strength of pulsars ranges between tex2html_wrap_inline2155 G and tex2html_wrap_inline2157 G. These values of B imply a magnetic field energy density that is too small compared to the energy density of the fluid to significantly affect the structure of a neutron star. However, one cannot exclude the existence of neutron stars with higher magnetic field strengths or the possibility that neutron stars are born with much stronger magnetic fields, which then decay to the observed values. (Of course there are also many arguments against magnetic field decay in neutron stars [6].) In addition, even though moderate magnetic field strengths do not alter the bulk properties of neutron stars, they may have an effect on the damping or growth rate of various perturbations of an equilibrium star, affecting its stability. For these reasons, a fully relativistic description of magnetized neutron stars is desirable; and, in fact, Bocquet et al. [67] achieved the first numerical computation of such configurations. Here we give a brief summary of their work:

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 and the pressure gradient, but the Lorentz force due to the electric currents also enters the balance. For simplicity, Bocquet et al. consider only poloidal magnetic fields, which preserve the circularity of the space-time. 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) [25]. Under these assumptions, the electromagnetic field tensor tex2html_wrap_inline2161 is derived from a potential 1-form tex2html_wrap_inline2163 with only two non-vanishing components, tex2html_wrap_inline2165 and tex2html_wrap_inline2167, which are given by 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.

The construction of magnetized models with tex2html_wrap_inline2169 G confirms that magnetic fields of this strength have a negligible effect on the structure of the star. However, if one increases the strength of the magnetic field above tex2html_wrap_inline2171 G, one observes significant effects, such as a flattening of the star. The magnetic field cannot be increased indefinitely, but there exists a maximum value of the magnetic field strength, of the order of tex2html_wrap_inline2173 G, for which the magnetic field pressure at the center of the star equals the fluid pressure. Above this value, the fluid pressure decreases more rapidly away from the center along the symmetry axis than the magnetic pressure. Instead of pressure, there is tension along the symmetry axis and no stationary configuration can exist.

The shape of a strongly magnetized star is flattened because the Lorentz forces exerted by the E/M field on the fluid act as centrifugal forces. A star with a magnetic field near the maximum value for stationary configurations displays a pinch along the symmetry axis because there, the magnetic pressure exceeds the fluid pressure. The maximum fluid density inside the star is not attained at the center, but away from it. The presence of a strong magnetic field also allows a maximum mass configuration with larger tex2html_wrap_inline2135 than for the same EOS with no magnetic field; this is in analogy with the increase of tex2html_wrap_inline2135 induced by rotation. For nonrotating stars, the increase in tex2html_wrap_inline2135, due to a strong magnetic field, is tex2html_wrap_inline2181, depending on the EOS. Following the increase in mass, the maximum allowed angular velocity for a given EOS also increases in the presence of a magnetic field.

Bocquet et al. are planning to use their code in the study of two types of possible instabilities in magnetized neutron stars: i) a pure E/M instability towards another electric current/magnetic field distribution of lower energy, and ii) a nonaxisymmetric instability for rapidly rotating models, which would be the analog of a Jacobi-type transition in non-magnetized stars. In perfect fluid models with a magnetic field, one would also expect a CFS-instability driven by electromagnetic waves.

2.5.7 Rapidly Rotating Proto-Neutron Stars

Following the gravitational collapse of a massive stellar core, a proto-neutron star (PNS) is born. Initially it has a large radius of about 100km and a temperature of 50-100MeV. The PNS may be born with a large rotational kinetic energy, and initially it will be differentially rotating. Due to the violent nature of the gravitational collapse, the PNS pulsates heavily, emitting significant amounts of gravitational radiation. After a few hundred pulsational periods, bulk viscosity will damp the pulsations significantly. Rapid cooling due to deleptonization transforms the PNS to a hot neutron star of tex2html_wrap_inline2183 MeV shortly after its formation. In addition, viscosity reduces the differential rotation to a nearly uniform rotation on a timescale of seconds [2Jump To The Next Citation Point In The Article], and the neutron star becomes quasi-stationary. Since the details of the PNS evolution determine the exact properties of the resulting cold NSs, proto-neutron stars must be modeled realistically in order to understand the structure of cold neutron stars.

Hashimoto et al. [68Jump To The Next Citation Point In The Article] and Goussard et al. [69Jump To The Next Citation Point In The Article] recently constructed fully relativistic models of rapidly rotating, hot proto-neutron stars. The authors use finite-temperature EOSs [70Jump To The Next Citation Point In The Article, 71] to model the interior of PNSs. Important parameters, which determine the local state of matter but are largely unknown, are the lepton fraction tex2html_wrap_inline2185 and the temperature profile. Hashimoto et al. consider only the limiting case of zero lepton fraction tex2html_wrap_inline2187 and classical isothermality, while Goussard et al. consider several non-zero values for tex2html_wrap_inline2185 and two different limiting temperature profiles - a constant entropy profile and a relativistic isothermal profile. In both [68Jump To The Next Citation Point In The Article] and [70], 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 tex2html_wrap_inline2191 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 25MeV 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 a stable PNS transforms to a stable neutron star. If, however, one considers the hypothetical case of a large amplitude phase transition which softens the cold EOS (such as a Kaon condensate), then tex2html_wrap_inline2135 of cold neutron stars is lower than tex2html_wrap_inline2135 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 [12] and investigated by Baumgarte et al. [72].

An analysis of radial stability of PNSs [73Jump To The Next Citation Point In The Article] shows that, for hot PNSs, the maximum angular velocity star almost coincides with the maximum mass star, 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 [68Jump To The Next Citation Point In The Article] and [69Jump To The Next Citation Point In The Article] PNSs have a maximum angular velocity that is considerably less than the maximum angular velocity allowed by the cold EOS. 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 loosing 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 [68] and confirmed in [69Jump To The Next Citation Point In The Article]. 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, a more recent study by Goussard et al. [2] suggests that proto-neutron stars created in a gravitational collapse cannot spin faster than 1.7 ms.

3 Oscillations and Stability2 The Equilibrium Structure of 2.4 Numerical Schemes

image Rotating Stars in Relativity
Nikolaos Stergioulas
© Max-Planck-Gesellschaft. ISSN 1433-8351
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