Relativistic Precession Model Research Articles

Black hole spin measurements through the relativistic precession model: XTE J1550-564

Abstract We present a systematic analysis of the complete set of observations of the black hole (BH) binary XTE J1550-564 obtained by the Rossi X-ray Timing Explorer. We study the fast time variability properties of the source and determine the spin of the BH through the relativistic precession model. Similarly to what is observed in the BH binary GRO J1655-40, the frequencies of the QPOs and broad-band noise components match the general relativistic frequencies of particle motion close to the compact object predicted by the relativistic precession model. The combination of two simultaneously observed quasi-periodic oscillation (QPO) frequencies together with the dynamical BH mass from optical/infrared observations yields a spin equal to a = 0.34 ± 0.01, consistent with previous determinations from X-ray spectroscopy. Based on the derived BH parameters, the low-frequency QPO emission radii vary from ∼30 gravitational radii (Rg) to the innermost stable orbit for this spin (∼5 Rg), where they sharply disappear as observed for the case of GRO J1655-40.

Innermost stable circular orbits and epicyclic frequencies around a magnetized neutron star

A full-relativistic approach is used to compute the radius of the innermost stable circular orbit (ISCO), the Keplerian, frame-dragging, precession and oscillation frequencies of the radial and vertical motions of neutral test particles orbiting the equatorial plane of a magnetized neutron star. The space-time around the star is modelled by the six parametric solution derived by Pachon et al. It is shown that the inclusion of an intense magnetic field, such as the one of a neutron star, have non-negligible effects on the above physical quantities, and therefore, its inclusion is necessary in order to obtain a more accurate and realistic description of the physical processes occurring in the neighbourhood of this kind of objects such as the dynamics of accretion disk. The results discussed here also suggest that the consideration of strong magnetic fields may introduce non-negligible corrections in, e.g., the relativistic precession model and therefore on the predictions made on the mass of neutron stars.

Precise mass and spin measurements for a stellar-mass black hole through X-ray timing: the case of GRO J1655−40

We present a systematic analysis of the fast time variability properties of the transient black hole binary GRO J1655-40, based on the complete set of RossiXTE observations. We demonstrate that the frequencies of the quasi-periodic oscillations and of the broad band noise components and their variations match accurately the strong field general relativistic frequencies of particle motion in the close vicinity of the innermost stable circular orbit, as predicted by the relativistic precession model. We obtain high precision measurements of the black hole mass (M = (5.31 +/- 0.07) M solar masses, consistent with the value from optical/NIR observations) and spin (a/M = 0.290 +/- 0.003), through the sole use of X-ray timing.

Constraints on kilohertz quasi-periodic oscillation models and stellar equations of state from SAX J1808.4–3658, Cyg X-2 and 4U 1820–30

We test the relativistic precession model (RPM) and the magnetohydrodynamics Alfvén wave oscillation model (AWOM) for the kilohertz (kHz) quasi-periodic oscillations (QPOs) from sources with measured neutron star (NS) masses and twin kHz QPO frequencies. For the RPM, the derived NS masses of Cyg X-2, SAX J1808.4–3658 and 4U 1820–30 are 1.96 ± 0.10, 2.83 ± 0.04 and 1.85 ± 0.02 M⊙, respectively. These are, respectively, ∼30, 100 and 40 per cent higher than the measured results 1.5 ± 0.3, <1.4 and |$1.29^{+0.19}_{-0.07}$| M⊙. For the AWOM, where the free parameter of the model is the density of the star, we infer the NS radii to be around 10–20 km for the above three sources. Based on this, we can infer the matter compositions inside the NSs with the help of the equations of state. In particular, for SAX J1808.4–3658, the AWOM shows a lower mass density for its NS than those of the other known kHz QPO sources, with a radius range of 17–20 km, which excludes the strange quark matter inside its star.

MASS-ANGULAR-MOMENTUM RELATIONS IMPLIED BY MODELS OF TWIN PEAK QUASI-PERIODIC OSCILLATIONS

Twin peak quasi-periodic oscillations (QPOs) appear in the X-ray power-density spectra of several accreting low-mass neutron star (NS) binaries. Observations of the peculiar Z-source Circinus X-1 display unusually low QPO frequencies. Using these observations, we have previously considered the relativistic precession (RP) twin peak QPO model to estimate the mass of central NS in Circinus X-1. We have shown that such an estimate results in a specific mass-angular-momentum (M-j) relation rather than a single preferred combination of M and j. Here we confront our previous results with another binary, the atoll source 4U 1636-53 that displays the twin peak QPOs at very high frequencies, and extend the consideration to various twin peak QPO models. In analogy to the RP model, we find that these imply their own specific M-j relations. We explore these relations for both sources and note differences in the chi-square behavior that represent a dichotomy between high- and low-frequency sources. Based on the RP model, we demonstrate that this dichotomy is related to a strong variability of the model predictive power across the frequency plane. This variability naturally comes from the radial dependence of characteristic frequencies of orbital motion. As a consequence, the restrictions on the models resulting from observations of low-frequency sources are weaker than those in the case of high-frequency sources. Finally we also discuss the need for a correction to the RP model and consider the removing of M-j degeneracies, based on the twin peak QPO-independent angular momentum estimates.

Parameter constrain in neutron star by twin kHz QPOs

AbstractWe use data on twin kHz QPOs and measured neutron star (NS) masses to test the relation between the lower (ν1) and upper (ν2) peak frequencies predicted with the MHD Alfvén wave oscillation model (AWOM) and the general relativistic precession model (GRPM). In the context of AWOM, the NS radius can be constrained from kHz QPO data and a measured NS mass.

What can quasi-periodic oscillations tell us about the structure of the corresponding compact objects?

We show how one can estimate the multipole moments of the space-time, assuming that the quasi-periodic modulations of the X-ray flux (quasi-periodic oscillations), observed from accreting neutron stars or black holes, are due to orbital and precession frequencies (relativistic precession model). The precession frequencies $\Omega_{\rho}$ and $\Omega_z$ can be expressed as expansions on the orbital frequency $\Omega$, in which the moments enter the coefficients in a prescribed form. Thus, observations can be fitted to these expressions in order to evaluate the moments. If the compact object is a neutron star, constrains can be imposed on the equation of state. The same analysis can be used for black holes as a test for the validity of the no-hair theorem. Alternatively, instead of fitting for the moments, observations can be matched to frequencies calculated from analytic models that are produced so as to correspond to realistic neutron stars described by various equations of state. Observations can thus be used to constrain the equation of state and possibly other physical parameters (mass, rotation, quadrupole, etc.) Some distinctive features of the frequencies, which become evident by using the analytic models, are discussed.

Confronting the models of 3:2 quasiperiodic oscillations with the rapid spin of the microquasar GRS 1915+105

Spectral fitting of the spin a in the microquasar GRS 1915+105 estimate values higher than a=0.98. However, there are certain doubts about this (nearly) extremal number. Confirming a high value of a>0.9 would have significant concequences for the theory of high-frequency quasiperiodic oscillations (HF QPOs). Here we discuss its possible implications assuming several commonly used orbital models of 3:2 HF QPOs. We show that the estimate of a>0.9 is almost inconsistent with two hot-spot (relativistic precession and tidal disruption) models and the warped disc resonance model. In contrast, we demonstrate that the epicyclic resonance and discoseismic models assuming the c- and g- modes are favoured. We extend our discussion to another two microquasars that display the 3:2 HF QPOs. The frequencies of these QPOs scale roughly inversely to the microquasar masses, and the differences in the individual spins, such as a=0.9 compared to a=0.7, represent a generic problem for most of the discussed geodesic 3:2 QPO models. To explain the observations of all the three microquasars by one unique mechanism, the models would have to accommodate very large non-geodesic corrections.

ON MASS CONSTRAINTS IMPLIED BY THE RELATIVISTIC PRECESSION MODEL OF TWIN-PEAK QUASI-PERIODIC OSCILLATIONS IN CIRCINUS X-1

Boutloukos et al. (2006) discovered twin-peak quasi-periodic oscillations (QPOs) in 11 observations of the peculiar Z-source Circinus X-1. Among several other conjunctions the authors briefly discussed the related estimate of the compact object mass following from the geodesic relativistic precession model for kHz QPOs. Neglecting the neutron star rotation they reported the inferred mass M_0 = 2.2 +/- 0.3 M_\sun. We present a more detailed analysis of the estimate which involves the frame-dragging effects associated with rotating spacetimes. For a free mass we find acceptable fits of the model to data for (any) small dimensionless compact object angular momentum j=cJ/GM^2. Moreover, quality of the fit tends to increase very gently with rising j. Good fits are reached when M ~ M_0[1+0.55(j+j^2)]. It is therefore impossible to estimate the mass without the independent knowledge of the angular momentum and vice versa. Considering j up to 0.3 the range of the feasible values of mass extends up to 3M_\sun. We suggest that similar increase of estimated mass due to rotational effects can be relevant for several other sources.

On magnetic-field-induced non-geodesic corrections to relativistic orbital and epicyclic frequencies

We discuss non-geodesic corrections to orbital and epicyclic frequencies of charged test particles orbiting a non-rotating neutron star with a dipole magnetic field. Using a fully relativistic approach we consider the influence of both the magnetic attraction and repulsion on the orbital and epicyclic motion. The magnetic repulsion introduces a rather complex and unusual behaviour of the circular orbital motion that is well defined down to the radius where the vertical epicyclic frequency loses its meaning. We demonstrate that for the intensity of the magnetic interaction appropriately restricted, the stable circular orbits extend down to the magnetic innermost stable circular orbit (MISCO) that is located well under the geodetic innermost stable circular orbit (GISCO) and even can reach the region under the photon circular orbit. The lowest stable circular orbit at rMISCOmin = 2.73M, associated with the highest possible orbital frequency , corresponds to the critical value of the particle-specific charge and the neutron star magnetic dipole moment product . For the magnetic attraction acting above the GISCO, the situation is much more simple and we demonstrate that the most significant correction arises for the radial epicyclic frequency and consequently for the location of the MISCO when the strong magnetic attraction pushes its location far behind the location of GISCO. We show that the Lorentz force also naturally violates the equality of the orbital and vertical epicyclic frequencies implied by the spherical symmetry of the background Schwarzschild geometry giving rise to the new effect of nodal precession of the orbital motion plane. Finally, we apply the magnetic attraction corrections on the relativistic precession model of the twin-peak high-frequency quasiperiodic oscillations observed in the galactic low mass x-ray binaries, showing possible high relevance of the modified radial epicyclic frequency.

The emission positions of kHz QPOs and Kerr spacetime influence

Based the Alfven wave oscillation model (AWOM) and relativistic precession model (RPM) for twin kHz QPOs, we estimate the emission positions of most detected kHz QPOs to be at r=18+-3 km (R/15km) except Cir X-1 at r = 30\+-5 km (R/15km). For the proposed Keplerian frequency as an upper limit to kHz QPO, the spin effects in Kerr Spacetime are discussed, which have about a 5% (2%) modification for that of the Schwarzchild case for the spin frequency of 1000 (400) Hz.The application to the four typical QPO sources, Cir X-1, Sco X-1, SAX J1808.4-3658 and XTE 1807-294, is mentioned.

The estimations of neutron star mass and radius by the kHz QPOs

AbstractThe kHz quasi‐periodic oscillations (QPOs) have been detected by the RXTE satellite in about thirty neutron stars (NSs) in low mass X‐ray binaries (LMXBs), which are usually interpreted to be related to the Keplerian motions in the orbit close to NS surface where the accreted matter is sucked onto the star. Based on the MHD Alfvén wave oscillation model and the relativistic precession model for the neutron star (NS) kHz QPOs, estimations of mass M and radius R of some NSs are given, which can give clues to evaluate the models. Furthermore, comparisons with theoretical M ‐R relations by stellar equations of state (EOSs) are presented (© 2009 WILEY‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)

Orbital resonances in discs around braneworld Kerr black holes

Rotating black holes in the brany universe of the Randall-Sundrum type are described by the Kerr geometry with a tidal charge b representing the interaction of the brany black hole and the bulk spacetime. For b<0 rotating black holes with dimensionless spin a>1 are allowed. We investigate the role of the tidal charge b in the orbital resonance model of QPOs in black hole systems. The orbital Keplerian, the radial and vertical epicyclic frequencies of the equatorial, quasicircular geodetical motion are given and their radial profiles are discussed. The resonant conditions are given in three astrophysically relevant situations: for direct (parametric) resonances, for the relativistic precession model, and for some trapped oscillations of the warped discs, with resonant combinational frequencies. It is shown, how b could influence matching of the observational data indicating the 3:2 frequency ratio observed in GRS 1915+105 microquasar with prediction of the orbital resonance model; limits on allowed range of the black hole parameters a and b are established. The "magic" dimensionless black hole spin enabling presence of strong resonant phenomena at the radius where \nu_K:\nu_{\theta}:\nu_r=3:2:1 is determined in dependence on b. Such strong resonances could be relevant even in sources with highly scattered resonant frequencies, as those expected in Sgr A*. The specific values of a and b are given also for existence of specific radius where \nu_K:\nu_{\theta}:\nu_r=s:t:u with 5>=s>t>u being small natural numbers. It is shown that for some ratios such situation is impossible in the field of black holes. We can conclude that analysing the microquasars high-frequency QPOs in the framework of orbital resonance models, we can put relevant limits on the tidal charge of brany Kerr black holes.

Quasiperiodic oscillations in a strong gravitational field around neutron stars testing braneworld models

The strong gravitational field of neutron stars in the brany universe could be described by spherically symmetric solutions with a metric in the exterior to the brany stars being of the Reissner–Nordström type containing a brany tidal charge representing the tidal effect of the bulk spacetime onto the star structure. We investigate the role of the tidal charge in orbital models of high-frequency quasiperiodic oscillations (QPOs) observed in neutron star binary systems. We focus on the relativistic precession model. We give the radial profiles of frequencies of the Keplerian (vertical) and radial epicyclic oscillations. We show how the standard relativistic precession model modified by the tidal charge fits the observational data, giving estimates of the allowed values of the tidal charge and the brane tension based on the processes going in the vicinity of neutron stars. We compare the strong field regime restrictions with those given in the weak-field limit of solar system experiments.

Discovery of Twin kHz QPOs in the Peculiar X‐Ray Binary Circinus X‐1

We report the discovery with the RXTE PCA of twin kHz QPOs in the peculiar X-ray binary Circinus X-1. Eleven cases of simultaneous double QPOs occurred, with significances of up to 6.3 and 5.5 σ and centroid frequencies ranging between approximately 56 and 225 and 230 and 500 Hz for the two QPO peaks, respectively, i.e., for the most part at frequencies well below those of other sources. The QPO properties clearly indicate that these double peaks are the kHz QPOs known from low magnetic field neutron stars, and not black hole high-frequency QPOs, confirming that Cir X-1 is a neutron star, as suspected since the detection of type I X-ray bursts from the field of the source 20 years ago. The kHz QPO peak separation varies over a wide range, ~175-340 Hz, and increases as a function of kHz QPO frequency. This is contrary to what has been observed in other sources but in good agreement with predictions of the relativistic precession model and Alfven wave models; beat-frequency models require modification to accommodate this. In other observations, single kHz QPOs can be seen down to frequencies as low as ~12 Hz, as well as a strong low-frequency quasi-periodic oscillation (LF QPO) between 1 and 30 Hz. The relations between the kHz QPOs and the LF QPO are in good agreement with those found previously in Z sources, confirming that Cir X-1 may be a peculiar Z source. We suggest that the low-frequency range over which the kHz QPOs occur in Cir X-1 and to a lesser extent in (other) Z sources, might be due to a stronger radial accretion flow relative to the disk flow than in other kHz QPO sources, possibly related to the nature of the companion star.

Radial Epicyclic Frequency in a Perfect Fluid Disk Around Compact Objects

The widely used expression of the radial epicyclic frequency, which is important in the relativistic precession model to interpret the observed quasi-periodic oscillation(QPO) in X-ray binaries, is derived by considering the perturbation of a circular-orbiting test particle around a compact object. However, this is not consistent with the real configuration of an accretion disk. Here we obtain the expressions of the radial epicyclic frequency of a perfect fluid disk in both Schwarzschild and Kerr metrics and find that they are different from the previous expressions if the pressure of fluid is considered. The effects of these new expressions on the theoretical QPO models are discussed.

Evolution of the 0.01–25 Hz power spectral components in Cygnus X-1

Analyzing the archival data from the Rossi X-ray Timing Explorer (RXTE), we study the power density spectra (PDS) of Cygnus X-1 from 1996 to 2003 in the frequency range of 0.01–25 Hz. Using a model consisting of one or two Lorentzians and/or an exponentially cut-off power-law, we are able to achieve a good fit to the PDS during the observations. With our model we are also able to track the evolution of the Lorentzian components through all spectral states of the source. We confirm the relation between characteristic frequencies seen both in black hole candidate and neutron star sources, and show the changes in this relation during the transitional and soft states of the source. The connection between the Lorentzian components is investigated by analyzing similarities and differences in their behavior. We find that the spectral state of the source can be uniquely determined from the parameters of these components. The parameter correlations can all be described by continuous functions, which differ between components. We discuss our results in the context of relativistic precession model for the accretion disk, and show a remarkable agreement between the model prediction and the data in the hard state. We estimate a value for the specific angular momentum of (-0.57) in the case of prograde (retrograde) rotation and an estimate for the inner radius of 22 to 50 (25 to 55) gravitational radii. Additional assumptions are required to explain the soft state data, and attempting to invoke rotational reversal for state transitions shows that it is insufficient to explain the differences between the hard and soft state data.

The Atoll Source States of 4U 1608−52

We have studied the atoll source 4U 1608-52 using a large data set obtained with the Rossi X-ray Timing Explorer. We find that the timing properties of 4U 1608-52 are almost exactly identical to those of the atoll sources 4U 0614+09 and 4U 1728-34 despite the fact that contrary to these sources 4U 1608-52 is a transient covering two orders of magnitude in luminosity. The frequencies of the variability components of these three sources follow a universal scheme when plotted versus the frequency of the upper kilohertz QPO, suggesting a very similar accretion flow configuration. If we plot the Z sources on this scheme only the lower kilohertz QPO and HBO follow identical relations. Using the mutual relations between the frequencies of the variability components we tested several models; the transition layer model, the sonic point beat frequency model, and the relativistic precession model. None of these models described the data satisfactory. Recently, it has been suggested that the atoll sources (among them 4U 1608-52) trace out similar three-branch patterns as the Z sources in the color-color diagram. We have studied the relation between the power spectral properties and the position of 4U 1608-52 in the color-color diagram and conclude that the timing behavior is not consistent with the idea that 4U 1608-52 traces out a three-branched Z shape in the color-color diagram along which the timing properties vary gradually, as Z sources do.

Correlation of the Quasi‐Periodic Oscillation Frequencies of White Dwarf, Neutron Star, and Black Hole Binaries

Using data obtained in 1994 June/July with the Extreme Ultraviolet Explorer deep survey photometer and in 2001 January with the Chandra X-ray Observatory Low Energy Transmission Grating Spectrograph, we investigate the extreme-ultraviolet (EUV) and soft X-ray oscillations of the dwarf nova SS Cyg in outburst. We find quasi-periodic oscillations (QPOs) at nu_0 ~ 0.012 Hz and nu_1 ~ 0.13 Hz in the EUV flux and at nu_0 ~ 0.0090 Hz, nu_1 ~ 0.11 Hz, and possibly nu_2 ~ nu_0 + nu_1 ~ 0.12 Hz in the soft X-ray flux. These data, combined with the optical data of Woudt & Warner for VW Hyi, extend the Psaltis, Belloni, & van der Klis nu_high-nu_low correlation for neutron star and black hole low-mass X-ray binaries (LMXBs) nearly two orders of magnitude in frequency, with nu_low ~ 0.08 nu_high. This correlation identifies the high-frequency quasi-coherent oscillations (so-called ``dwarf nova oscillations'') of cataclysmic variables (CVs) with the kilohertz QPOs of LMXBs, and the low-frequency QPOs of CVs with the horizontal branch oscillations (or the broad noise component identified as such) of LMXBs. Assuming that the same mechanisms produce the QPOs in white dwarf, neutron star, and black hole binaries, we find that the data exclude the relativistic precession model and the magnetospheric and sonic-point beat-frequency models (as well as any model requiring the presence or absence of a stellar surface or magnetic field); more promising are models that interpret QPOs as manifestations of disk accretion onto any low-magnetic field compact object.

Can Any “Invariants” Be Revealed in Quasi-periodic Phenomena Observed from Scorpius X-1?

Using a large number of Rossi X-Ray Timing Explorer observations of Scorpius X-1, we present a detailed investigation of the transition layer and relativistic precession models (TLM and RPM, respectively). These models predict the existence of the invariant quantities: an inclination angle δ of the magnetospheric axis with the normal to the disk for the TLM and a neutron star (NS) mass MNS for the RPM. Theoretical predictions of both models are tested, and their self-consistency is checked. We establish the following: (1) The inferred δ-value is 556 ± 009. Correlation of the δ-values with the horizontal-branch oscillation (HBO) frequency is rather weak. (2) There is a strong correlation between an inferred MNS and the HBO frequency in the RPM frameworks. (3) We infer MNS for different assumptions regarding the relations between the HBO frequency νHBO and the nodal frequency νnod. We find that the inferred MNS = 2.7 ± 0.1 M☉ cannot be consistent with any equation of state of NS matter. We conclude that the RPM fails to describe the data while the TLM seems to be compatible.