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) (, 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 of 0.11 ± 0.02, a dimensionless angular momentum of 0.64 ± 0.06, and an eccentricity of 0.66 ± 0.04 . 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 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 , which can be extracted from the asymptotic expansion of the metric function at large , as in Equation (10).
Laarakkers and Poisson  numerically compute the scalar quadrupole moment for several equations of state, using the rotating neutron star code rns . They find that for fixed gravitational mass , the quadrupole moment is given as a simple quadratic fit,.
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 , as shown in Figure 1 for EOS L. The surface is limited by the nonrotating models () 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  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 .
Mass-shedding occurs when the angular velocity of the star reaches the angular velocity of a particle in a circular Keplerian orbit at the equator, i.e., when
In the Newtonian limit the maximum angular velocity of uniformly rotating polytropic stars is approximately (this is derived using the Roche model, see ). For relativistic stars, the empirical formula [142, 114, 109]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 .
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.  find that, for most EOSs, the coefficient in the empirical formula is an almost linear function of the parameteret al. empirical formula
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).
The maximum mass and minimum period of rotating relativistic stars computed with realistic hadronic EOSs from the Arnett and Bowers collection  are about (EOS L) and 0.4 ms (EOS B), while neutron stars, rotating at the Kepler limit, have rotational periods between 0.53 ms (EOS B) and 1.7 ms (EOS M) . The maximum, accurately measured, neutron star mass is currently still (see, e.g., ), but there are also indications for neutron stars . Core collapse simulations have yielded a bi-modal mass distribution of the remnant, with peaks at about and  (the second peak depends on the assumption for the high-density EOS – if a soft EOS is assumed, then black hole formation of this mass is implied). Compact stars of much higher mass, created in a neutron star binary merger, could be temporarily supported against collapse by strong differential rotation .
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 ). The existence of a magnetic field may complicate this picture. Spruit and Phinney  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  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 . A new investigation by Heger et al., yielding intermediate initial rotation rates, is presented in . Clearly, more detailed computations are needed to resolve this important question.
The minimum observed pulsar period is still 1.56 ms , 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) . Other models are based on the standard magnetospheric model for accretion-induced spin-up  or on the idea that gravitational radiation (produced by accretion-induced quadrupole deformations of the deep crust) balances the spin-up torque [35, 312]. It has also been suggested  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 .
If one is interested in obtaining upper limits on the mass and rotation rate, independently of the proposed EOSs, one has to rely on fundamental physical principles. Instead of using realistic EOSs, one constructs a set of schematic 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 are that
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). 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  point out that is sensitive to the matching energy density and Hartle  computes as a function of :
In the case of rotating stars, Friedman and Ipser  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  uses the FPS EOS at low densities, arriving at a similar result as in :× 1014 g cm–3 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  by constructing nonrotating models and using the empirical formula (32) to estimate the minimum period. Koranda, Stergioulas, and Friedman  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,–3). In , 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 , where it is shown that the CL EOS has , independent of , and the empirical formula (34) reproduces the numerical result (38) to within 2%.
Since rotation increases the mass that a neutron star of given central density can support, there exist sequences of neutron stars with constant baryon mass 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 losing angular momentum. Eventually, the star reaches a point where it becomes unstable to axisymmetric perturbations and collapses to a black hole.
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 .
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 ; 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.
The presence of a magnetic field has been 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 108 G and 2 × 1013 G. These values of the magnetic field strength 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 ). 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.  achieved the first numerical computation of such configurations. Following 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, 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) . 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 1014 G, 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 1018 G, 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  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 .
Going further An slow rotation approach for the construction of rotating magnetized relativistic stars is presented in .
Following the gravitational collapse of a massive stellar core, a proto-neutron star (PNS) is born. Initially it has a large radius of about 100 km and a temperature of 50 – 100 MeV. 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, shortly after its formation, into a hot neutron star of . In addition, viscosity or other mechanisms (see Section 2.1) enforce uniform rotation and the neutron star becomes quasi-stationary. Since the details of the PNS evolution determine the properties of the resulting cold NSs, proto-neutron stars need to be modeled realistically in order to understand the structure of cold neutron stars.
Hashimoto et al.  and Goussard et al.  construct fully relativistic models of rapidly rotating, hot proto-neutron stars. The authors use finite-temperature EOSs [238, 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  and , 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 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 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  and investigated by Baumgarte et al. .
An analysis of radial stability of PNSs  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  and , 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  and confirmed in .
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.  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. . 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 .
Most rotational properties of strange quark stars differ considerably from the properties of rotating stars constructed with hadronic EOSs. First models of rapidly rotating strange quark stars were computed by Friedman (, quoted in [123, 122]) and by Lattimer et al. . Colpi and Miller  use the approximation and find that the spin of strange stars (newly-born, or spun-up by accretion) may be limited by the CFS instability to the f -mode, since rapidly rotating strange stars tend to have . Rapidly rotating strange stars at the mass-shedding limit have been computed first by Gourgoulhon et al. , and the structure of a representative model is displayed in Figure 3.
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.  also obtain scaling relations for the model with maximum rotation rate. The maximum angular velocity scales as, 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 , the ratioset al. EOS  are similar to those of the MIT bag model EOS [38, 325, 98], but some quantitative differences exist .
Accreting strange stars in LMXBs will follow different evolutionary paths than do accreting hadronic stars in a mass vs. central energy density diagram . When (and if) strange stars reach the mass-shedding limit, the ISCO still exists  (while it disappears for most hadronic EOSs). Stergioulas, Kluźniak, and Bulik  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 in the strange quark EOS (that were proposed in ) are found to hold within an accuracy of better than 9% for the mass-shedding sequence
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  and references therein) is the frequency at the ISCO, then it rules out most models of slowly rotating strange stars in LMXBs. However, in  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 [340, 341]. Moderately rotating strange stars, with spin frequencies around 300 Hz can also be accommodated for some values of the coupling constant  (see also  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 . Zdunik, Haensel, and Gourgoulhon  also find that the mass and thickness of the crust can be expanded in powers of the spin frequency as. 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 , it is estimated that the heating due to deconfinement may exceed the neutrino luminosity from the core of a strange star older than 1000 yr and may therefore influence the cooling of this compact object (see also ).
© Max Planck Society and the author(s)