2.10 Rotating relativistic stars in LMXBs

2.10.1 Particle orbits and kHz quasi-periodic oscillations

In the last few years, X-ray observations of accreting sources in LMXBs have revealed a rich phenomenology that is waiting to be interpreted correctly and could lead to significant advances in our understanding of compact objects (see [191313Jump To The Next Citation Point248Jump To The Next Citation Point]). The most important feature of these sources is the observation of (in most cases) twin kHz quasi-periodic oscillations (QPOs). The high frequency of these variabilities and their quasi-periodic nature are evidence that they are produced in high-velocity flows near the surface of the compact star. To date, there exist a large number of different theoretical models that attempt to explain the origin of these oscillations. No consensus has been reached, yet, but once a credible explanation is found, it will lead to important constraints on the properties of the compact object that is the source of the gravitational field in which the kHz oscillations take place. The compact stars in LMXBs are spun up by accretion, so that many of them may be rotating rapidly; therefore, the correct inclusion of rotational effects in the theoretical models for kHz QPOs is important. Under simplifying assumptions for the angular momentum and mass evolution during accretion, one can use accurate rapidly rotating relativistic models to follow the possible evolutionary tracks of compact stars in LMXBs [72342].

In most theoretical models, one or both kHz QPO frequencies are associated with the orbital motion of inhomogeneities or blobs in a thin accretion disk. In the actual calculations, the frequencies are computed in the approximation of an orbiting test particle, neglecting pressure terms. For most equations of state, stars that are massive enough possess an ISCO, and the orbital frequency at the ISCO has been proposed to be one of the two observed frequencies. To first order in the rotation rate, the orbital frequency at the prograde ISCO is given by (see Kluźniak, Michelson, and Wagoner [170])

(1-M-⊙) fISCO ≃ 2210(1 + 0.75j) M Hz, (44 )
where 2 j = J∕M. At larger rotation rates, higher order contributions of j as well as contributions from the quadrupole moment Q become important and an approximate expression has been derived by Shibata and Sasaki [276], which, when written as above and truncated to the lowest order contribution of Q and to 𝒪 (j2), becomes
( ) fISCO ≃ 2210(1 + 0.75j + 0.78j2 − 0.23Q2 ) 1-M-⊙- Hz, (45 ) M
where Q2 = − Q ∕M 3.

Notice that, while rotation increases the orbital frequency at the ISCO, the quadrupole moment has the opposite effect, which can become important for rapidly rotating models. Numerical evaluations of fISCO for rapidly rotating stars have been used in [228] to arrive at constraints on the properties of the accreting compact object.

In other models, orbits of particles that are eccentric and slightly tilted with respect to the equatorial plane are involved. For eccentric orbits, the periastron advances with a frequency νpa that is the difference between the Keplerian frequency of azimuthal motion νK and the radial epicyclic frequency νr. On the other hand, particles in slightly tilted orbits fail to return to the initial displacement ψ from the equatorial plane, after a full revolution around the star. This introduces a nodal precession frequency ν pa, which is the difference between ν K and the frequency of the motion out of the orbital plane (meridional frequency) νψ. Explicit expressions for the above frequencies, in the gravitational field of a rapidly rotating neutron star, have been derived recently by Marković [221], while in [222] highly eccentric orbits are considered. Morsink and Stella [229] compute the nodal precession frequency for a wide range of neutron star masses and equations of state and (in a post-Newtonian analysis) separate the precession caused by the Lense–Thirring (frame-dragging) effect from the precession caused by the quadrupole moment of the star. The nodal and periastron precession of inclined orbits have also been studied using an approximate analytic solution for the exterior gravitational field of rapidly rotating stars [277]. These precession frequencies are relativistic effects and have been used in several models to explain the kHz QPO frequencies [29024921685Jump To The Next Citation Point].

It is worth mentioning that it has recently been found that an ISCO also exists in Newtonian gravity, for models of rapidly rotating low-mass strange stars. The instability in the circular orbits is produced by the large oblateness of the star [1693395].

2.10.2 Angular momentum conservation during burst oscillations

Some sources in LMXBs show signatures of type I X-ray bursts, which are thermonuclear flashes on the surface of the compact star [197]. Such bursts show nearly-coherent oscillations in the range 270 – 620 Hz (see [313298] for recent reviews). One interpretation of the burst oscillations is that they are the result of rotational modulation of surface asymmetries during the burst. In such a case, the oscillation frequency should be nearly equal to the spin frequency of the star. This model currently has difficulties in explaining some observed properties, such as the oscillations seen in the tail of the burst, the frequency increase during the burst, and the need for two anti-podal hot spots in some sources that ignite at the same time. Alternative models also exist (see, e.g., [248]).

In the spin-frequency interpretation, the increase in the oscillation frequency by a few Hz during the burst is explained as follows: The burning shell is supposed to first decouple from the neutron star and then gradually settle down onto the surface. By angular momentum conservation, the shell spins up, giving rise to the observed frequency increase. Cumming et al. [76] compute the expected spin-up in full general relativity and taking into account rapid rotation. Assuming that the angular momentum per unit mass is conserved, the change in angular velocity with radius is given by

[( ) ] d lnΩ v2 R ∂ν ( ω ) R ∂ ω ------ = − 2 1 − ---− -- --- 1 − -- − ------ , (46 ) dln r 2 2 ∂r Ω 2Ω ∂r
where R is the equatorial radius of the star and all quantities are evaluated at the equator. The slow rotation limit of the above result was derived previously by Abramowicz et al. [3]. The fractional change in angular velocity during spin-up can then be estimated as
Δ Ω dln Ω (Δr ) ----= ------ --- , (47 ) Ω d ln r R
where Δr is the coordinate expansion of the burning shell, a quantity that depends on the shell’s composition. Cumming et al. find that the spin down expected if the atmosphere rotates rigidly is a factor of two to three times smaller than observed values. More detailed modeling is needed to fully explain the origin and properties of burst oscillations.

Going further   A very interesting topic is the modeling of the expected X-ray spectrum of an accretion disk in the gravitational field of a rapidly rotating neutron star as it could lead to observational constraints on the source of the gravitational field. See, e.g., [3022782793433], where work initiated by Kluzniak and Wilson [171] in the slow rotation limit is extended to rapidly rotating relativistic stars.

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