Quasi-periodic oscillations in low-mass X-ray binaries and constraints on the equation of state of neutron star matter

Abstract

Recently discovered quasi-periodic oscillations in the X-ray brightness of low-mass X-ray binaries are used to derive constraints on the mass of the neutron star component and the equation of state of neutron star matter. The observations are compared with models of rapidly rotating neutron stars which are calculated by means of an exact numerical method in full relativity. For the equations of state we select a broad collection of models representing different assumptions about the many-body structure and the complexity of the composition of superdense matter. The mass constraints differ from their values in the approximate treatment by ∼10 per cent. Under the assumption that the maximum frequency of the quasi-periodic oscillations originates from the innermost stable orbit, the mass of the neutron star is in the range M∼1.92ߝ2.25 M_⊙. The quasi-periodic oscillation in the Atoll-source 4U 1820-30 in particular is only consistent with equations of state that are rather stiff at high densities, which is explainable, so far, only with pure nucleonic/leptonic composition. This interpretation contradicts the hypothesis that the protoneutron star formed in SN 1987A collapsed to a black hole, since this would demand a maximum neutron star mass below 1.6 M_⊙. The recently suggested identification of quasi-periodic oscillations with frequencies of about 10 Hz with the Lense—Thirring precession of the accretion disc is found to be inconsistent with the models studied in this work, unless it is assumed that the first overtone of the precession is observed.

accretion, accretion discs, equation of state, gravitation, stars: neutron, stars: rotation, X-rays: stars

1 Introduction

Neutron stars contain matter in one of the densest forms found in the Universe. Their central density ranges from a few times the density of normal nuclear matter to about one order of magnitude higher, depending on the star's mass and the equation of state (EOS). Neutron stars therefore provide us with a powerful tool for exploring the properties of such dense matter. In the last decades, this tool has been applied to, among other topics, the determination of the EOS of dense, charge-neutral, β-equilibrated matter by means of comparing the theoretical predicted properties with observations of neutron stars. This was attempted by studying, for example, the maximum stable star mass (van Kerkwijk et al. 1995), the minimum rotation period (Friedman et al. 1986; Weber & Glendenning 1992), or the thermal behaviour (Tsuruta 1966; Schaab et al. 1996; Page 1998; see Balberg, Lichtenstadt & Cook 1999 for a recent review).

Recently, Strohmayer et al. (1996) and van der Klis et al. (1996) discovered, with the Rossi X-ray Timing Explorer (RXTE), kilohertz quasi-periodic oscillations (QPOs) in the X-ray brightness of low-mass X-ray binaries (LMXBs; see van der Klis 1998 for a recent review). In subsequent observations, two QPOs were often detected simultaneously in a given source. The frequency separation between the two QPOs is almost constant, although the frequencies of the two QPOs themselves vary by several hundred Hertz. Until now, the only exceptions are the Atoll sources 4U 1608-52 and 4U 1735-44 and the Z-source Scorpius X-1 in which the frequency separation varies with the luminosity by roughly ±15 per cent (Méndez et al. 1998c; van der Klis & Wijnands, 1997; Ford et al. 1998b). Psaltis et al. (1998) found that the QPO data of the other LMXBs are consistent both with being constant and with varying by a similar fraction as in the three sources quoted above. The frequency of oscillations during type I X-ray bursts detected in some sources is consistent with the frequency separation of the two QPOs or with its first overtone. Only in the source 4U 1636-536 does the average frequency separation, Δν=251 Hz, and half of the frequency in type I bursts, ν_burst=581 Hz, differ by approximately 15 per cent (Méndez & van Paradijs 1998). In the power spectrum of bursts of the same source Miller (1998) found a second signal at 290 Hz∼(1/2)ν_burst. These observations strongly favour the beat frequency model, where the frequency separation between the two QPOs originates from the stellar spin whereas the higher QPO is produced by accreting gas in a stable, nearly circular orbit around the neutron star. It still remains however, to clarify how the slightly varying frequency separation and the small deviation of the frequency separation from the burst frequency can be incorporated into this model.

Beside the two kilohertz QPOs and the burst oscillations, QPOs with frequencies of a few tens of Hertz were also detected in some sources. Their frequencies correlate with the high-frequency kilohertz QPOs. These low-frequency QPOs were interpreted by Stella & Vietri (1998) to originate from the Lense—Thirring precession of the accreting disc due to the frame-dragging effect of the rapidly spinning neutron star (Lense & Thirring 1918).

The identification of both the high-frequency kilohertz QPOs with the orbital frequency of a stable circular orbit, and the low-frequency QPOs with the frame-dragging frequency of the same orbit allow us to constrain the mass of the neutron star and also the EOS of neutron star matter (Lamb et al. 1998; Stella & Vietri 1998). If the evidence for the detection of QPOs of the innermost stable circular orbit in the sources 4U 1608-52, 4U 1636-536 (Kaaret, Ford & Chen 1997), and 4U 1820-30 (Zhang et al. 1998b) can be confirmed by future observations, the constraints are rather severe, allowing only a few stiff EOSs.

The frequency separation of the two kilohertz QPOs shows that the neutron stars in LMXBs are rapidly rotating with periods ranging from 2.5 to 4 ms. It can therefore be expected that the geometry of the neutron star and its exterior space—time are non-spherical. Since the innermost stable orbit is located at only a few kilometres above the star's surface, the deviations from the Kerr space—time are large and should not be neglected. In order to compare theoretical neutron star models with QPO observations, a completely general relativistic calculation of the rotating neutron star structure and space—time geometry is therefore necessary.

In order to study the impact and the discrimination power of the QPO data in greater detail, we select a broad collection of modern EOSs, which were obtained utilizing numerous assumptions and approximations about the dynamics and composition of superdense matter. These include the many-body technique used to determine the EOS; the model for the nucleon—nucleon interaction; a description of electrically charged-neutral neutron star matter in terms of either only neutrons and protons in generalized chemical equilibrium (β equilibrium) with electrons and muons, or nucleons, hyperons and more massive baryon states in β equilibrium with leptons; hyperon coupling strengths in matter; inclusion of meson (π, K) condensation; treatment of the transition of confined hadronic matter into quark matter; and assumptions about the true ground state of strongly interacting matter (i.e. absolute stability of strange quark matter relative to baryon matter).

The paper is organized as follows. In Section 2 we summarize the equations that govern the space—time structure and compare the approximate values of the orbital frequencies, the radius of the innermost stable orbit, and the Lense—Thirring precession frequencies with the respective values from the exact numerical solution of Einstein's equations. The physics of the EOSs is discussed in Section 3. The high-frequency kilohertz QPOs and their interpretation in combination with their compatibility with the different EOSs are discussed in Section 4. The implications of the identification of the low-frequency QPOs with Lense—Thirring precession are presented in Section 5. We summarize our results, the constraints on the neutron star masses, and the conclusions concerning the neutron star EOS in Section 6.

2 Space—time around rapidly rotating neutron stars

2.1 Einstein equations

The stationary, axis-symmetric, and asymptotically flat metric in quasi-isotropic coordinates reads

(1)

where the metric coefficients gg(r,θ) are functions of r and θ only. The metric coefficients are determined by the Einstein equation (cG=1)

(2)

and the energy—momentum conservation

(3)

where T=(e+p)u⊗u+pg is the stress—energy tensor of an ideal fluid with the 4-velocity

(4)

the energy density e, and the pressure p. The Lorentz factor Γ is given by u·u=−1, and hence

(5)

ω=u^ϕu^t is the angular velocity of the fluid with respect to an observer at infinity. The proper velocity U of the fluid with respect to the local Eulerian Observer O₀ (Smarr & York 1978) is given by the equation

(6)

Note that if the fluid were at rest with respect to the Observer O₀, U=0, it would not necessarily be at rest for an inertial observer at infinity, since ω=N^ϕ≠0. This phenomenon of dragging of the inertial frame was first studied by Lense & Thirring (1918) and turns out to be important for the investigation of the Lense—Thirring precession of the accreting disc (see Section 5).

The four non-trivial Einstein equations together with the energy—momentum conservation are solved via a finite-difference scheme (Schaab 1999). We follow Bonazzola et al. (1993) in compacting the outer space to a finite region by using the transformation r→1/r. The boundary condition of approximate flatness can then be exactly fulfilled. The neutron star model is uniquely determined by fixing one of the parameters of central density, gravitational mass, baryon number, or Kepler orbiting frequency at the star's equator or at the marginally stable radius, as well as one of the parameters of angular velocity, angular momentum, or stability parameter tT/|W|. The models of maximum mass and/or maximum rotation velocity can also be calculated.

2.2 Stable circular orbits

Since e_t and e are Killing vectors, the components p_t and p of the 4-momentum of a freely falling particle are constants of motion and can be identified with the negative of the specific energy E (in units of the rest mass m₀) and the specific angular momentum L, respectively. For a particle motion confined to the equatorial plane, the geodesic equation yields

(7)

where V is the effective potential for the particle motion (Misner, Thorne & Wheeler 1973, p. 655). A circular orbit, p_r=0, is given by expression (4) for the 4-velocity of the fluid inside the star. The specific energy E and angular momentum L can then be expressed in terms of the Lorentz factor Γ (equation 5) and the proper velocity U (equation 6):

(8)

(9)

The circular orbit exists if V_,r=0, thus¹

(10)

The proper velocity U of a particle corotating with the star is given by

(11)

The circular orbit is stable if V_,rr<0, with (see also Cook, Shapiro & Teukolsky 1992 and Datta, Thampan & Bombacci 1998)

(12)

The zero of V_,rr determines the radius r^ms of the innermost stable or marginally stable circular orbit. The Kepler frequency ν_K of a circular orbit as measured by a distant observer is given by the proper velocity U through the expression

(13)

and the Lense—Thirring precession frequency is given by

(14)

The respective values at the innermost stable orbit with radius r^ms are denoted by and , respectively. The circumferential radius r_circ of the object is linked to the equatorial coordinate radius r by the expression

(15)

2.3 Approximate expressions

Although we shall use the exact expressions for ν_K, ν_LT, , and in our investigation, we also give, for the purpose of comparison, the approximate expressions valid to first order of the dimensionless angular momentum jJM², where J and M denote the angular momentum and gravitational mass, respectively. The approximate expressions have the advantage of constraining the mass and the angular momentum independently of the EOS. To first order in j the corresponding expressions are (Miller et al. 1998b):

(16)

(17)

(18)

(19)

(20)

3 Equations of state

3.1 Neutron star matter

The EOS of neutron star matter is the basic input quantity to the structure equations discussed in Section 2. Knowledge of it over a wide range of densities is necessary. Whereas the density at the star's surface corresponds to the density of iron, the density in the centre of a very massive star can reach 15 times the density of normal nuclear matter. Since neutron star matter in chemical equilibrium (i.e. generalized β-equilibrium) is highly isospin-asymmetric and may carry net-strangeness its properties cannot be explored in laboratory experiments. Therefore, one is left with models for the EOS that depend on theoretically motivated assumptions and/or speculations about the behaviour of superdense matter. One main source of uncertainty is the competition between non-relativistic and relativistic models. Although both treatments give satisfactory results for normal nuclear densities, they provide quite different results if one extrapolates to higher densities (see, for example Huber et al. 1998). Moreover, one encounters substantial differences in the high-density regime depending on the underlying dynamics, the many-body approximation, and assumptions about the composition. The simplest approach describes neutron star matter by pure neutron matter. Since pure neutron matter is certainly not the true ground state of neutron star matter, it will quickly decay by means of the weak interaction into chemically equilibrated neutron star matter, whose fundamental constituents — besides neutrons — are protons, hyperons and possibly more massive baryons. Even a transition to quark matter (so-called hybrid stars) and the occurrence of pion- or kaon-condensates is possible.

The cross-section of a neutron star can be split roughly into three distinct regimes (Börner 1973; Weber & Glendenning 1993). The first one is the star's outer crust, which consists of a lattice of atomic nuclei and a Fermi liquid of relativistic, degenerate electrons. The inner crust extends from neutron drip density, ρ=4.3×10¹¹ g cm⁻³, to a transition density of about ρ_tr=1.7×10¹⁴ g cm⁻³ (Pethick, Ravenhall & Lorenz 1995). Beyond this transition density ρ_tr one enters the star's third regime; that is, its core where all atomic nuclei have dissolved into their constituent protons and neutrons. Furthermore, as outlined above, due to the high Fermi pressure the core will contain hyperons, eventually more massive baryon resonances, and possibly a gas of free up, down and strange quarks.

The EOS of the outer and inner crust has been studied in several investigations and is well known. We shall adopt for these regions the models derived by Haensel & Pichon (1994) and by Negele & Vautherin (1973), respectively. The models for the EOS of the star's core are discussed in detail in Schaab et al. (1996).

An overview of the collection of EOSs used in this paper for the core region is given in Table 1. We have chosen a representative collection of models in order to check which EOSs are compatible with the QPOs and their theoretical interpretation.

Table 1.

Dynamics and approximation schemes for EOSs derived for the cores of neutron stars.

An important characteristic of relativistic models is the appearance of a new saturation mechanism. In relativistic theories the repulsive force is caused by the exchange of vector mesons coupled to the baryon densities, whereas the attraction is coupled to the scalar densities by means of the scalar mesons. Hence, the repulsion increases with increasing density with respect to the attraction. Since non-relativistic treatments do not distinguish between the two kinds of densities, one has to introduce ad hoc forces depending explicitly on density in order to reproduce the properties of saturated nuclear matter.

The potential parameters and coupling constants in the pure nucleonic sector are adjusted to nucleon—nucleon scattering data and properties of the deuteron in the case of the non-relativistic microscopic models and the relativistic Brückner—Hartree—Fock models. In this sense, these models are called parameterfree. For the Hartree and the Hartree—Fock approximations as well as for the Thomas—Fermi model, such an adjustment is not possible, since the free force parameters would not yield saturation of nuclear matter. In such more phenomenological approximations, the coupling constants are adjusted to properties of saturated nuclear matter or atomic nuclei (Weber & Weigel 1989; Schaffner & Mishustin 1996; Myers & Swiatecki 1996). In a more recent and more elaborate investigation (Huber et al. 1998), the adjustment was performed with respect to the properties of neutron star matter at 2–3 times nuclear density known from relativistic Brückner—Hartree—Fock calculations. Relativistic Brückner—Hartree—Fock calculations cannot be performed over the whole density range at present, since an inclusion of hyperons leads to complex equation systems. Even for pure nucleonic matter, the self-consistent method is numerically stable only up to 2–3 times nuclear density (Huber et al. 1998).

Since the coupling constants of hyperons are not obtainable from properties of normal nuclei, and data of hypernuclei are relatively scarce, there is greater freedom in the selection of parameter sets in the hyperonic sector. From a theoretical standpoint, one may first utilize the SU(6) symmetry of the quark model for the vector coupling constants and adjust the Σ-coupling from the binding energy of the Λ hyperon in nuclear matter. It turns out, however, that this procedure gives different Σ-couplings depending on the many-body approximation chosen. Therefore, it seems reasonable to vary both couplings with the constraint of the Λ binding energy in nuclear matter in such phenomenological many-body theories. A further constraint in this procedure is the compatibility with the allowed neutron star masses (for more details, see Huber et al. 1998 and Huber 1998). The coupling constants for the strange mesons can be obtained to a certain extent from the data of double Λ hypernuclei (Schaffner & Mishustin 1996). Batty, Friedman & Gal (1994) claim some doubts on the existence of Σ hyperons in neutron star matter. However, the disappearance of Σ hyperons would only slightly change the EOS (Balberg & Gal 1997). A further open question is the existence of Δ isobars in the interior of neutron stars. In relativistic Hartree treatments, the isospin unfavoured Δ⁻ isobar does not appear because of the large ρ-coupling, which is necessary to reproduce the empirical symmetry coefficient (Huber et al. 1998). Δ isobars are therefore often neglected a priori in relativistic Hartree treatments (Schaffner & Mishustin 1996). However, in relativistic Hartree—Fock approximations, the ρ-coupling becomes smaller due to the exchange contributions. For this reason, the charge-favoured Δ⁻ may now enter the composition depending on the behaviour of the effective Δ mass in neutron star matter, which is quite uncertain (Huber et al. 1998).

The possibility of a transition of confined hadronic matter into quark matter at high densities is included in the EOS labelled G^K240_B180 (Glendenning 1997) (so-called hybrid stars). The transition was determined for a bag constant of B^1/4=180 MeV, which places the energy per baryon of strange quark matter at 1100 MeV, well above the energy per nucleon in saturated nuclear matter as well as in the most stable nucleus, ⁵⁶Fe (EA≈930 MeV). This model predicts a transition to a mixed phase consisting of quark matter and hadronic matter at a density as low as 1.6 n₀. The pure quark-matter phase is reached at a density of about 10 n₀, which is close to the central density of the heaviest and thus most compact star constructed from such an EOS. A phase transition to pion condensation is accounted for in . This EOS predicts a phase transition at n≈1.3 n₀. The possibility of absolutely stable strange quark matter will be considered in the following section.

The stiffness of the EOS depends strongly on the internal degrees of freedom. Generally, the more degrees of freedom that are taken into account, the softer the EOS becomes (Fig. 1). A softer EOS, in turn, leads to lower maximum neutron star masses and, for fixed mass, to higher central densities.

Figure 1.

Pressure—density relation for several EOSs.

3.2 Strange stars

The hypothesis that strange quark matter may be the absolute ground state of the strong interaction (not ⁵⁶Fe) has been raised independently by Bodmer (1971) and Witten (1984). If the hypothesis is true, then a separate class of compact stars could exist, ranging from dense strange stars to strange dwarfs to strange planets (Weber et al. 1995; Glendenning, Kettner & Weber 1995a,b). In principle both strange and neutron stars could exist. If strange stars exist, however, the galaxy is likely to be contaminated by strange quark nuggets which would convert via impact ‘conventional’ neutron stars into strange stars (Glendenning 1990; Madsen & Olesen 1991; Caldwell & Friedman 1991). This would imply that the objects known to astronomers as pulsars are probably rotating strange matter stars and not neutron matter stars, as is usually assumed. At present there is no sound scientific basis on which to either confirm or reject the hypothesis, so it remains a serious possibility of fundamental significance for compact stars (Weber et al. 1997a,b; Madsen 1998) plus various other physical phenomena (Madsen & Haensel, 1991). Below we will explore the implications of the existence of strange stars with respect to the QPO phenomenon. This enables one to test the possible existence of strange stars and thus draw definitive conclusions about the true ground state of strongly interacting matter.

As pointed out by Alcock, Farhi & Olinto (1986), a strange star can carry a solid nuclear crust whose density is strictly limited by neutron drip. This possibility is caused by the displacement of electrons at the surface of the strange matter core, which generates an electric dipole layer there. It can be sufficiently strong to produce a gap between ordinary atomic matter (crust) and the quark-matter surface, which prevents a conversion of ordinary atomic matter into the assumed lower-lying ground state of strange matter. Obviously, free neutrons, being electrically charge neutral, cannot exist in the crust, because they do not feel the Coulomb barrier and thus would gravitate towards the strange-quark matter core, where they would be converted by hypothesis into strange matter. Consequently, the density at the base of the crust (inner crust density) must always be smaller than the neutron drip density. The situation is illustrated in Fig. 2, where the EOS of a strange star with crust is shown.

Figure 2.

Pressure—density relation for strange stars with crust. The bag constant is B^1/4=145 MeV or 158 MeV; the mass of the strange quark is m_s=100 MeV.

The strange-star models presented in the following are constructed for an EOS of strange matter derived by Farhi & Jaffe (1984). We shall study models with a fixed strange quark mass, m_s=100 MeV, and the two bag constants B^1/4=145 MeV and B^1/4=158 MeV.

4 Kilohertz quasi-periodic oscillations

In Table 2 the observations of kilohertz QPOs from 16 sources are summarized. The sources can be classified into two classes, the Atoll- and the Z-sources, depending on the form of their colour—colour diagram. Almost all sources show simultaneously two kilohertz QPOs. Although the frequencies of the two QPOs vary by several hundred hertz with the accretion rate, the variation of the frequency separation is only small.

Table 2.

Observational data of kilohertz QPOs.

Therefore the observations strongly favour some kind of beat-frequency model. In such models the higher kilohertz QPO corresponds to the Kepler rotation at the inner edge of the accretion disc. The lower QPO corresponds to the beat frequency between the upper QPO and the spin of the neutron star. This interpretation is supported by the fact that the frequency separation is equal to the QPO frequency (or half of it) in the type I X-ray bursts observed in some of the Atoll sources (see Table 2). The question remains, however, where the higher QPO is generated. Strohmayer et al. (1996) suggested that this radius is identical to the magnetospheric radius, whereas Miller et al. (1998c) proposed that it is identical to the sonic radius (see van der Klis 1998 for a critical discussion of these models). In both cases, the orbital radius has to be larger than the radius of the innermost stable orbit (see Section 2.2), or equivalently, the frequency ν_QPO1 of the higher QPO has to be smaller than the Kepler frequency at the innermost stable orbit:

(21)

where the approximate expression (19) was used. The exact expression depends on the EOS and on the spin period of the neutron star. This inequality sets an upper limit to the mass of the star. Obviously, the orbital radius has to be larger than the star's radius, which is larger than the radius of the innermost stable orbit for low star masses. This constraint sets a lower limit to the star mass which again depends on the spin period and the EOS.

Fig. 3 shows the maximal allowed Kepler frequency as function of the gravitational mass for two EOSs, RHF_pn and RHF8. If the radius of the innermost stable orbit is smaller than the star's radius, the maximal allowed Kepler frequency is given by the Kepler frequency at the star's equator. This kind of solution is represented by the rising branch of ν_K(M), since ν_K increases with increasing gravitational mass. If the innermost stable orbit is located outside the star, the maximal allowed Kepler frequency decreases again with increasing mass. The two EOSs differ for M≳1.0 M_⊙ in composition. In RHF8 hyperons are included, whereas RHF_pn assumes pure nucleonic matter. RHF8 is therefore softer than RHF_pn at high densities and can support only smaller masses than RHF_pn. The maximal allowed Kepler frequency depends only slightly on the neutron star's spin frequency, as long as the spin frequency is not larger than about 10 per cent of the star's Kepler frequency. If one compares the ν_K curves for example with the frequency of the highest kilohertz QPO observed in 4U 1820-30 (ν_s=275 Hz), one obtains that the mass of the neutron star has to be within the range between M=1.0 M_⊙ and M_max=2.25 M_⊙ (for RHF_pn) or 1.65 M_⊙ (for RHF8).

Figure 3.

Maximal allowed Kepler frequency ν_K as function of the neutron star's mass M for the two EOSs RHF_pn and RHF8 and for two spin frequencies ν_s=275 and 355 Hz.

If the interpretation of the sonic-point model is adopted, it is expected that the inner boundary of the accretion disc will be close to the innermost stable orbit. Indeed, the observations of the sources 4U 1820-30, 4U 1608-52, and 4U 1636-536 seem to confirm this interpretation, since the QPO frequency remains constant for high mass accretion rates (Kaaret et al. 1997; Zhang et al. 1998b). Therefore, only the right intersection point of the ν_K(M)-curve with the line is allowed by the observations.² The mass of the neutron star in the binary system 4U 1820-30 can now be determined. For RHF_pn the mass is equal to M=2.25 M_⊙, whereas RHF8 would be excluded.

Figs 4–6 show the respective ranges of masses for which the Kepler frequency at the equator or at the innermost stable orbit is larger than the highest observed QPO frequency for all EOSs considered here (see Table 1). The vertical line refers to the approximate mass, which is obtained by setting j=0 in equation (19). This expression underestimates the upper limit of the mass. The dashed bars refer to the EOSs for which models with do not exist; that is, those whose maximal stable mass is to small. If the interpretation of the highest observed QPO frequency in the three sources 4U 1820-30, 4U 1608-52, and 4U 1636-536 is confirmed, the mass of the neutron star is constrained to the respective right ends of the solid bars. The EOSs for which models with do not exist (dashed bars) are excluded.

Figure 4.

Range of masses for which the Kepler frequency at the equator or at the innermost stable orbit is larger than the highest observed QPO frequency in the source 4U 1636-536 (ν_s=290 Hz). The dashed bars refer to the EOSs for which models with do not exist. The approximate value M_appr=1.80 M_⊙ is also shown.

Figure 5.

As Fig. 4, but for the source 4U 1728-34 ν_s=355 Hz, M_appr=2.01 M_⊙).

Figure 6.

As Fig. 4, but for the source 4U 1820-30 ν_s=275 Hz, M_appr=2.07 M_⊙).

The observation of 4U 1820-30 agrees only with the two nucleonic EOSs UV₁₄+UVII and RHF_pn. If the interpretation of the highest observed QPO frequency in this source is confirmed by further observations, the existence of hyperons, meson condensates, quark condensates, and strange stars can be rejected. Neutron star matter thus consists only of nucleons and leptons. At the very least, the hyperon fraction (i.e. strangeness) is much smaller than predicted by actual relativistic calculations.

Figs 7 and 8 show the allowed ranges of mass of the neutron stars in the considered binaries for the EOSs RHF_pn and RHF8, respectively. For RHF8, the masses are limited by the maximal stable star mass M_max∼1.65 M_⊙, which depends on the spin period. All sources are consistent with a canonical star mass of M=1.4 M_⊙. Only if one assumes that the highest QPO frequency is equal to the Kepler frequency at the innermost stable orbit do the masses lie between M=2.0 M_⊙ and 2.5 M_⊙ (for RHF_pn).

Figure 7.

As Fig. 4, but for selected sources and the fixed EOS RHF_pn.

Figure 8.

As Fig. 7, but for RHF8.

5 Constraints on the equation of state by Lense—Thirring precession

Stella & Vietri (1998) suggested that a third oscillation with a frequency ν_QPO3 at about ∼10 Hz is also produced at the inner border of the accretion disc by Lense—Thirring precession. The observed frequencies ν_QPO3 (Table 3) can again be compared with theoretical neutron star models. The neutron star models are constructed for a given spin frequency ν_s and a fixed, unfortunately unknown, star mass M. The obtained monotonic relations ν_K(r,θ=π/2) and ν_LT(r,θ=π/2) are transformed into the relation ν_LT(ν_K). Since the Lense—Thirring precession frequency ν_LT is, to a first approximation, proportional to ν_s the relation ν_LTν_s as a function of ν_K is shown in Fig. 9 for all EOSs. The observations are shown as circles. The curves depend only weakly on the neutron star mass. One can therefore draw conclusions from a comparison of the theoretical curves with the observations, although the star masses are unknown. Since both ν_LTν_s and ν_K do not depend on ν_s to a first approximation, the calculation of the theoretical curves for one specific spin frequency, ν_s=363 Hz (this corresponds to the spin frequency of the neutron star in 4U 1728-34), is sufficient.

Table 3.

Observational data of QPOs in the 10 Hz range.

Figure 9.

Ratio ν_LTν_s of the Lense—Thirring precession frequency ν_LT and the spin frequency ν_s=363 Hz as function of the Kepler frequency ν_K. The used mass is M=1.4 M_⊙. The first group of curves contains the EOSs NLSH1, NLSH2, TM1-m2, TM1-m1, RHF_pn, G₃₀₀, HVpn, and HV (from left to right; the bold typed label corresponds to the bold curve). Group 2: , RHF8, RH1, and. Group 3: SM_B145, TF, UV₁₄+UVII, UV₁₄+TNI, and . Group 4: RHF1 and SM_B160. The full (open) circles correspond to (half of) the observed frequencies of the Atoll sources in Table 3.

As can be seen in Fig. 9, the observations of KS 1731-260 and 4U 1735-44 agree with the theoretical curves of group 1, which contains the stiffest EOSs of our sample. This is in agreement with the results of Stella & Vietri (1998). However, the other observations, including the observations of the Z-sources which are not shown, lie above all theoretical curves. The observed frequency ν_QPO3 is thus higher than the Lense—Thirring precession frequency even for the stiffest EOSs. If one assume that the observed frequencies correspond to the first overtone of the Lense—Thirring precession, the Atoll-source observations are in the range of the theoretical curves.

In the case of the Z-sources, the detected frequencies ν_QPO3 are not only too large in most sources, but additionally the slope of the relation ν_QPO3(ν_QPO1) of the Z-source observations is much higher than the slope of ν_LT(ν_K) for the theoretically determined models. Even the assumption that the first overtone of the Lense—Thirring precession frequency ν_LT is detected (Stella & Vietri 1998) cannot explain these discrepancies.

6 Conclusions

In this paper, we derived models of rapidly rotating neutron stars and strange stars by solving the general relativistic structure equations for a broad collection of modern EOSs. We compared the space—time geometry of these models with recently discovered QPOs in the X-ray brightness of LMXBs.

If one follows the general beat-frequency interpretation of the kilohertz QPOs, i.e. that the higher-frequency QPO originates at a stable circular orbit, one can constrain the mass of the neutron star to a range that depends on the EOS. This mass range is for all sources and for all EOSs consistent with a canonical mass, M=1.4 M_⊙. As stated by Miller, Lamb & Cook (1998a) and Thampan, Bhattacharya & Datta (1999), the exact lower and upper limits of the neutron star's mass can only be determined by using fully relativistic models of rapidly rotating neutron stars. The exact limits differ from the approximations with j=0 by roughly 10 per cent.

As shown by Kaaret et al. (1997) and Zhang et al. (1998b), the observation of a maximum frequency of the high-frequency QPO in the sources 4U 1820-30, 4U 1608-52, and 4U 1636-536 favours the interpretation that this QPO originates at the innermost stable orbit. If this interpretation is correct, the mass of the neutron star can be exactly (within the observational errors) determined for a given EOS. The approximately obtained mass, M=2.07 M_⊙, of the source 4U 1820-30 is larger than the maximum mass of most of the considered EOSs. This conclusion is strengthened if the observed maximum frequency is compared with the exact neutron star models. The only EOSs allowed out of our broad collection are then UV₁₄+UVII and RHF_pn, which both describe neutron star matter as consisting of nucleons and leptons only. The derived masses of the three sources lie in the narrow range between M=1.92 M_⊙ and 2.25 M_⊙.

This result is also of some importance for the nature of the object left in the supernova 1987A. During the first 10 s after the supernova, neutrinos were detected (Chevalier 1997). This means that a protoneutron star was formed in the supernova. The fact that up to now no pulsar emission has been detected was interpreted by Bethe & Brown (1995) to mean that the protoneutron star collapsed to a black hole when the star became transparent to neutrinos after roughly 10 s. The estimated value of the baryonic mass M_B∼1.63–1.76 M_⊙ (Bethe & Brown 1995; Thielemann, Nomoto & Hashimoto 1996) thus gives an upper limit to the maximum gravitational mass of a neutron star: M_max ≲ 1.6 M_⊙. This limit is in clear contradiction to the derived mass of, for example, 4U 1820-30.

It is generally believed that neutron stars are born with a mass of about 1.4–1.5 M_⊙. Neutron stars in LMXBs would therefore accrete 0.4–0.8 M_⊙ during their lifetime. It seems reasonable to assume that some of the neutron stars in LMXBs accrete enough matter to reach the maximally stable mass (M_max=2.20 M_⊙ for UV₁₄+UVII; M_max=2.44 M_⊙ for RHF_pn). The neutron star would then collapse to a black hole.

The interpretation of the QPO with frequency ν_QPO3 of about 10 Hz as the Lense—Thirring precession frequency of the accretion disc seems not to be consistent with the theoretical star models, unless one assumes that the first overtone of the precession is observed. In the case of Z-sources, however, the necessary ratio of the frame-dragging frequency and spin frequency, and thus the moment of inertia, of the models is too small compared with the observed frequency ν_QPO3 or half of it.

Our general conclusion is that the observations of kilohertz QPOs in LMXBs provide another powerful tool for probing the interior of neutron stars. Compared with the other tools such as observations of the maximum mass (van Kerkwijk et al. 1995), the limiting spin period (Friedman et al. 1986; Weber & Glendenning 1992), and cooling simulations (Tsuruta 1966; Schaab et al. 1996; Page 1998), the derived constraints, especially in the sonic-point interpretation, are fairly strong. The lower limit M_max≳2.15 M_⊙ is only consistent with two EOSs, UV₁₄+UVII and RHF_pn, which are relatively stiff at high densities. Their stiff behaviour at high densities seems to be possible only if the neutron star matter consists of neutrons, protons, and leptons only. At the most, a small admixture of hyperons may be allowed. However, one has to admit that such a composition somehow contradicts our conception of neutron star matter, since field-theoretical models lead more or less inevitably to a more complex composition at high density. If the given interpretation is correct, one has to reinvestigate the problem of superdense neutron star matter by dropping, for instance, the standard assumptions about the coupling of the hyperons in such regimes.

This specific conclusion depends, however, on the interpretation of the maximum frequency of the kilohertz QPO within the sonic-point model. As a kind of beat-frequency model, this model predicts a constant frequency separation Δνν_QPO1-ν_QPO2. Recently, two further examples where this constancy is not observed have been discovered: 4U 1608-52 and 4U 1735-44 (Méndez et al. 1998c; Ford et al. 1998b). Moreover, the frequency separation in the Atoll source 4U 1636-536 seems not to be consistent with half of the frequency of the QPO in type I bursts (Méndez & van Paradijs 1998). Although the beat-frequency models are the most hopeful candidates in explaining the QPO phenomenology it remains to be seen how the variation of the frequency separation and deviation from the QPO frequency in bursts can be incorporated to these models.

Acknowledgments

One of us (CS) gratefully acknowledges the Bavarian State for financial support. We would like to thank Norman Glendenning and Jürgen Schaffner-Bielich for providing us with tables of their EOSs.

1

We use the usual convention, that ·_,μ represents the partial differential ∂/∂x^μ.

2

The left intersection point is unlikely, since it is expected that the lower and the upper frequency QPOs could not be observed at the same time if the accretion disc extended to the neutron star surface (Zhang et al. 1997). This would be contrary to the simultanous observation of both QPOs in several sources.