Numerical solution of the radiative transfer equation: X-ray spectral formation from cylindrical accretion onto a magnetized neutron star

Predicting the emerging X-ray spectra in several astrophysical objects is of great importance, in particular when the observational data are compared with theoretical models. To this aim, we have developed an algorithm solving the radiative transfer equation in the Fokker-Planck approximation when both thermal and bulk Comptonization take place. The algorithm is essentially a relaxation method, where stable solutions are obtained when the system has reached its steady-state equilibrium. We obtained the solution of the radiative transfer equation in the two-dimensional domain defined by the photon energy E and optical depth of the system tau using finite-differences for the partial derivatives, and imposing specific boundary conditions for the solutions. We treated the case of cylindrical accretion onto a magnetized neutron star. We considered a blackbody seed spectrum of photons with exponential distribution across the accretion column and for an accretion where the velocity reaches its maximum at the stellar surface and at the top of the accretion column, respectively. In both cases higher values of the electron temperature and of the optical depth tau produce flatter and harder spectra. Other parameters contributing to the spectral formation are the steepness of the vertical velocity profile, the albedo at the star surface, and the radius of the accretion column. The latter parameter modifies the emerging spectra in a specular way for the two assumed accretion profiles. The algorithm has been implemented in the XSPEC package for X-ray spectral fitting and is specifically dedicated to the physical framework of accretion at the polar cap of a neutron star with a high magnetic field (> 10^{12} G), which is expected to be typical of accreting systems such as X-ray pulsars and supergiant fast X-ray transients.Comment: 13 pages, 20 figures, accepted for publication in A&

On the stability of the thermal Comptonization index in neutron star low-mass X-ray binaries in their different spectral states

Most of the spectra of neutron star low mass X-ray binaries (NS LMXBs), being them persistent or transient, are characterized by the presence of a strong thermal Comptonization bump, thought to originate in the transition layer (TL) between the accretion disk and the NS surface. The observable quantities which characterize this component dominating the emission below 30 keV, are the spectral index alpha and the rollover energy, both related to the electron temperature and optical depth of the plasma. Starting from observational results on a sample of NS LMXBs in different spectral states, we formulate the problem of X-ray spectral formation in the TL of these sources. We predict a stability of the thermal Comptonization spectral index in different spectral states if the energy release in the TL is much higher than the intercepted flux coming from the accretion disk. We use an equation for the energy balance and the radiative transfer diffusion equation for a slab geometry in the TL, to derive a formula for the thermal Comptonization index alpha. We show that in this approximation the TL electron temperature kTe and optical depth tau_0 can be written as a function of the energy flux from the disk intercepted by the corona (TL) and that in the corona itself Qdisk/Qcor, in turn leading to a relation alpha=f(Qdisk/Qcor), with alpha ~ 1 when Qdisk/Qcor <<1. We show that the observed spectral index alpha for the sample of sources here considered lies in a belt around 1 +/- 0.2 a part for the case of GX 354--0. Comparing our theoretical predictions with observations, we claim that this result, which is consistent with the condition Qdisk/Qcor <<1, can give us constraints on the accretion geometry of these systems, an issue that seems difficult to be solved using only the spectral analysis method.Comment: 7 pages, 3 figures, accepted for publication in A&

Analysis of X-ray spectral variability and black hole mass determination of the NLS1 galaxy Mrk 766

We present an XMM-Newton time-resolved spectral analysis of the NLS1 galaxy Mrk 766. We analyse eight available observations of the EPIC-pn camera taken between May 2000 and June 2005 to investigate the X-ray spectral variability as produced by changes in the mass accretion rate. The 0.2-10 keV spectra are extracted in time bins longer than 3 ks to accurately trace the variations of the best fit parameters of our adopted Comptonisation spectral model. We test a bulk-motion Comptonisation (BMC) model which is in general applicable to any physical system powered by accretion onto a compact object, and assumes that soft seed photons are efficiently up-scattered via inverse Compton scattering in a hot and dense electron corona. The Comptonised spectrum has a characteristic power-law shape, whose slope was found to increase for large values of the normalisation of the seed component, that is proportional to the mass accretion rate (in Eddington units). Our baseline spectral model also includes a warm absorber lying on the line of sight and radiation reprocessing from the accretion disk or from outflowing matter in proximity of the central compact object. Our study reveals that the normalisation-slope correlation, observed in Galactic Black Hole sources (GBHs), also holds for Mrk 766: variations of the photon index in the range Gamma~1.9-2.4 are indeed likely to be related to the variations of m-dot, as observed in X-ray binary systems. We finally applied a scaling technique based on the observed correlation to estimate the BH mass in Mrk 766. This technique is commonly and successfully applied to measure masses of GBHs, and this is the first time it is applied in detail to estimate the BH mass in an AGN. We obtain a value of M_{BH}=1.26^{+1.00}_{-0.77}x10^6 M_{sun} that is in very good agreement with that estimated by the reverberation mappingComment: 26 pages, 7 figures, 4 tables to be published in Astronomy and Astrophysic

Kilohertz QPOs in Neutron Star Binaries modeled as Keplerian Oscillations in a Rotating Frame of Reference

Since the discovery of kHz quasi-periodic oscillations (QPO) in neutron star binaries, the difference between peak frequencies of two modes in the upper part of the spectrum, i.e. Delta (omega)=omega_h-omega_K has been studied extensively. The idea that the difference Delta(omega) is constant and (as a beat frequency) is related to the rotational frequency of the neutron star has been tested previously. The observed decrease of Delta(omega) when omega_h and omega_k increase has weakened the beat frequency interpretation. We put forward a different paradigm: a Keplerian oscillator under the influence of the Coriolis force. For such an oscillator, omega_h and the assumed Keplerian frequency omega_k hold an upper hybrid frequency relation: omega^2_h-omega^2_K=4*Omega^2, where Omega is the rotational frequency of the star's magnetosphere near the equatorial plane. For three sources (Sco X-1, 4U 1608-52 and 4U 1702-429), we demonstrate that the solid body rotation Omega=Omega_0=const. is a good first order approximation. Within the second order approximation, the slow variation of Omega as a function of omega_K reveals the structure of the magnetospheric differential rotation. For Sco X-1, the QPO have frequencies approximately 45 and 90 Hz which we interpret as the 1st and 2nd harmonics of the lower branch of the Keplerian oscillations for the rotator with vector Omega not aligned with the normal of the disk: omega_L/2 pi=(Omega/pi)(omega_K/omega_h)sin(delta) where delta is the angle between vector Omega and the vector normal to the disk.Comment: 13 pages, 3 figures, accepted for publications in ApJ Letter

Wide band observations of the X-ray burster GS 1826-238

GS 1826-238 is a well-studied X-ray bursting neutron star in a low mass binary system. Thermal Comptonisation by a hot electron cloud is a widely accepted mechanism accounting for its high energy emission, while the nature of most of its soft X-ray output is not completely understood. A further low energy component is typically needed to model the observed spectra: pure blackbody and Comptonisation-modified blackbody radiation by a lower temperature (a few keV) electron plasma were suggested to explain the low energy data. We studied the steady emission of GS 1826-238 by means of broad band (X to soft Gamma-rays) measurements obtained by the INTEGRAL observatory in 2003 and 2006. The newly developed, up-to-date Comptonisation model CompTB is applied for the first time to study effectively the low-hard state variability of a low-luminosity neutron star in a low-mass X-ray binary system. We confirm that the 3-200 keV emission of \GS is characterised by Comptonisation of soft seed photons by a hot electron plasma. A single spectral component is sufficient to model the observed spectra. At lower energies, no direct blackbody emission is observed and there is no need to postulate a low temperature Compton region. Compared to the 2003 measurements, the plasma temperature decreased from 20 to 14 keV in 2006, together with the seed photons temperature. The source intensity was also found to be 30% lower in 2006, whilst the average recurrence frequency of the X-ray bursts significantly increased. Possible explanations for this apparent deviation from the typical limit-cycle behaviour of this burster are discussed.Comment: 6 pages, 2 figures. Accepted for publication in A&