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3C84 is a nearby radio source with a complex total intensity structure, showing linear polarisation and spectral patterns. A detailed investigation of the central engine region necessitates the use of VLBI above the hitherto available maximum frequency of 86GHz. Using ultrahigh resolution VLBI observations at the highest available frequency of 228GHz, we aim to directly detect compact structures and understand the physical conditions in the compact region of 3C84. We used EHT 228GHz observations and, given the limited (u,v)-coverage, applied geometric model fitting to the data. We also employed quasi-simultaneously observed, multi-frequency VLBI data for the source in order to carry out a comprehensive analysis of the core structure. We report the detection of a highly ordered, strong magnetic field around the central, SMBH of 3C84. The brightness temperature analysis suggests that the system is in equipartition. We determined a turnover frequency of $\nu_m=(113\pm4)$GHz, a corresponding synchrotron self-absorbed magnetic field of $B_{SSA}=(2.9\pm1.6)$G, and an equipartition magnetic field of $B_{eq}=(5.2\pm0.6)$G. Three components are resolved with the highest fractional polarisation detected for this object ($m_\textrm{net}=(17.0\pm3.9)$%). The positions of the components are compatible with those seen in low-frequency VLBI observations since 2017-2018. We report a steeply negative slope of the spectrum at 228GHz. We used these findings to test models of jet formation, propagation, and Faraday rotation in 3C84. The findings of our investigation into different flow geometries and black hole spins support an advection-dominated accretion flow in a magnetically arrested state around a rapidly rotating supermassive black hole as a model of the jet-launching system in the core of 3C84. However, systematic uncertainties due to the limited (u,v)-coverage, however, cannot be ignored., Comment: 15 pages, 6 figures, published in A&A

Tracking the motions of transient jets launched by low-mass X-ray binaries (LMXBs) is critical for determining the moment of jet ejection, and identifying any corresponding signatures in the accretion flow. However, these jets are often highly variable and can travel across the resolution element of an image within a single observation, violating a fundamental assumption of aperture synthesis. We present a novel approach in which we directly fit a single time-dependent model to the full set of interferometer visibilities, where we explicitly parameterise the motion and flux density variability of the emission components, to minimise the number of free parameters in the fit, while leveraging information from the full observation. This technique allows us to detect and characterize faint, fast-moving sources, for which the standard time binning technique is inadequate. We validate our technique with synthetic observations, before applying it to three Very Long Baseline Array (VLBA) observations of the black hole candidate LMXB MAXI J1803-298 during its 2021 outburst. We measured the proper motion of a discrete jet component to be $1.37\pm0.14$ mas/hr, and thus we infer an ejection date of MJD $59348.08_{-0.06}^{+0.05}$, which occurs just after the peak of a radio flare observed by the Australia Telescope Compact Array (ATCA) and the Atacama Large Millimeter/Sub-Millimeter Array (ALMA), while MAXI J1803-298 was in the intermediate state. Further development of these new VLBI analysis techniques will lead to more precise measurements of jet ejection dates, which, combined with dense, simultaneous multi-wavelength monitoring, will allow for clearer identification of jet ejection signatures in the accretion flow., Comment: 15 pages, 9 figures, 4 tables; Accepted for publication in MNRAS

Since the discovery of cosmic rays (CRs) over a century ago, their origin remains an open question. Galactic CRs with energy up to the knee ($10^{15}$ eV) are considered to originate from supernova remnants, but this scenario has recently been questioned due to lack of TeV $\gamma$-ray counterparts in many cases. Extragalactic CRs on the other hand, are thought to be associated with accelerated particles in the relativistic jets launched by supermassive accreting black holes at the center of galaxies. Scaled down versions of such jets have been detected in X-ray binaries hosting a stellar black hole (BHXBs). In this work, we investigate the possibility that the smaller-scale jets in transient outbursts of low-mass BHXBs could be sources of Galactic CRs. To better test this scenario, we model the entire electromagnetic spectrum of such sources focusing on the potential TeV regime, using the `canonical' low-mass BHXB GX 339-4 as a benchmark. Taking into account both the leptonic radiative processes and the $\gamma$-rays produced via neutral pion decay from inelastic hadronic interactions, we predict the GeV and TeV $\gamma$-ray spectrum of GX 339-4 using lower-frequency emission as constraints. Based on this test-case of GX 339-4 we investigate whether other, nearby low-mass BHXBs could be detected by the next-generation very-high-energy $\gamma$-ray facility the Cherenkov Telescope Array, which would establish them as additional and numerous potential sources of CRs in the Galaxy., Comment: 13 pages, 12 figures, accepted to MNRAS

Accreting black holes are sources of major interest in astronomy, particular those launching jets because of their ability to accelerate particles, and dramatically affect their surrounding environment up to very large distances. The spatial, energy and time scales at which a central active black hole radiates and impacts its environment depend on its mass. The implied scale-invariance of accretion/ejection physics between black hole systems of different central masses has been confirmed by several studies. Therefore, designing a self-consistent theoretical model that can describe such systems, regardless of their mass, is of crucial importance to tackle a variety of astrophysical sources. We present here a new and significantly improved version of a scale invariant, steady-state, multi-zone jet model, which we rename bhjet, resulting from the efforts of our group to advance the modelling of black hole systems. We summarise the model assumptions and basic equations, how they have evolved over time, and the additional features that we have recently introduced. These include additional input electron populations, the extension to cyclotron emission in near-relativistic regime, an improved multiple inverse Compton scattering method, external photon seeds typical of AGN and a magnetically-dominated jet dynamical model as opposed to the pressure-driven jet configuration present in older versions. In this paper, we publicly release the code on Github and, in order to facilitate the user's approach to its many possibilities, showcase a few applications as a tutorial., Comment: 28 pages, 16 figures, accepted for publication on MNRAS, code available at https://github.com/matteolucchini1/BHJet/, updated to the latest version of the manuscript

During outbursts, the observational properties of black hole X-ray binaries (BHXBs) vary on timescales of days to months. These relatively short timescales make these systems ideal laboratories to probe the coupling between accreting material and outflowing jets as a the accretion rate varies. In particular, the origin of the hard X-ray emission is poorly understood and highly debated. This spectral component, which has a power-law shape, is due to Comptonisation of photons near the black hole, but it is unclear whether it originates in the accretion flow itself, or at the base of the jet, or possibly the interface region between them. In this paper we explore the disk-jet connection by modelling the multi-wavelength emission of MAXI J1836-194 during its 2011 outburst. We combine radio through X-ray spectra, X-ray timing information, and a robust joint-fitting method to better isolate the jet's physical properties. Our results demonstrate that the jet base can produce power-law hard X-ray emission in this system/outburst, provided that its base is fairly compact and that the temperatures of the emitting electrons are sub-relativistic. Because of energetic considerations, our model favours mildly pair-loaded jets carrying at least 20 pairs per proton. Finally, we find that the properties of the X-ray power spectrum are correlated with the jet properties, suggesting that an underlying physical process regulates both., Comment: 17 pages, 10 figures, accepted for publication on MNRAS

Outflows, spanning a wide range of dynamical properties and spatial extensions, have now been associated with a variety of accreting astrophysical objects, from supermassive black holes at the core of active galaxies to young stellar objects. The role of such outflows is key to the evolution of the system that generates them, for they extract a fraction of the orbiting material and angular momentum from the region close to the central object and release them in the surroundings. The details of the launching mechanism and their impact on the environment are fundamental to understand the evolution of individual sources and the similarities between different types of outflow-launching systems. We solve semi-analytically the non-relativistic, ideal, magnetohydrodynamics (MHD) equations describing outflows launched from a rotating disk threaded with magnetic fields using our new numerical scheme. We present here a parameter study of a large sample of new solutions. We study the different combinations of forces that lead to a successfully launched jet and discuss their global properties. We show how these solutions can be applied to the outflow of the water fountain W43A for which we have observational constraints on magnetic field, density and velocity of the flow at the location of two symmetrical water maser emitting regions., Comment: 22 pages, 20 figures, the version includes changes based on referee comments

Cygnus X--1 is the first Galactic source confirmed to host an accreting black hole. It has been detected across the entire electromagnetic spectrum from radio to GeV $\gamma$-rays. The source's radio through mid-infrared radiation is thought to originate from the relativistic jets. The observed high degree of linear polarisation in the MeV X-rays suggests that the relativistic jets dominate in this regime as well, whereas a hot accretion flow dominates the soft X-ray band. The origin of the GeV non-thermal emission is still debated, with both leptonic and hadronic scenarios deemed to be viable. In this work, we present results from a new semi-analytical, multi-zone jet model applied to the broad-band spectral energy distribution of Cygnus X--1 for both leptonic and hadronic scenarios. We try to break this degeneracy by fitting the first-ever high-quality, simultaneous multiwavelength data set obtained from the CHOCBOX campaign (Cygnus X--1 Hard state Observations of a Complete Binary Orbit in X-rays). Our model parameterises dynamical properties, such as the jet velocity profile, the magnetic field, and the energy density. Moreover, the model combines these dynamical properties with a self-consistent radiative transfer calculation including secondary cascades, both of leptonic and hadronic origin. We conclude that sensitive TeV $\gamma$-ray telescopes like Cherenkov Telescope Array (CTA) will definitively answer the question of whether hadronic processes occur inside the relativistic jets of Cygnus X--1., Comment: 16 pages, 4 figures, accepted for publication in MNRAS

We present results from six epochs of quasi-simultaneous radio, (sub-)millimetre, infrared, optical, and X-ray observations of the black hole X-ray binary MAXI~J1535$-$571. These observations show that as the source transitioned through the hard-intermediate X-ray state towards the soft intermediate X-ray state, the jet underwent dramatic and rapid changes. We observed the frequency of the jet spectral break, which corresponds to the most compact region in the jet where particle acceleration begins (higher frequencies indicate closer to the black hole), evolve from the IR band into the radio band (decreasing by $\approx$3 orders of magnitude) in less than a day. During one observational epoch, we found evidence of the jet spectral break evolving in frequency through the radio band. Estimating the magnetic field and size of the particle acceleration region shows that the rapid fading of the high-energy jet emission was not consistent with radiative cooling; instead the particle acceleration region seems to be moving away from the black hole on approximately dynamical timescales. This result suggests that the compact jet quenching is not caused by local changes to the particle acceleration, rather we are observing the acceleration region of the jet travelling away from the black hole with the jet flow. Spectral analysis of the X-ray emission show a gradual softening in the few days before the dramatic jet changes, followed by a more rapid softening $\sim$1--2\,days after the onset of the jet quenching., Comment: 17 pages, 6 figures, data provided in the appendices. Accepted for publication in MNRAS

Submillimetre and millimetre observations are important in probing the properties of the molecular gas and dust around obscured active galactic nuclei (AGNs) and their feedback. With very high-resolution (0."02x0."03 (2x3 pc)) ALMA 345 GHz observations of CO 3-2, HCO$^+$ 4-3, HCN 4-3 $\nu_2$=1$f$, and continuum we have studied the molecular outflow and nucleus of the extremely radio-quiet lenticular galaxy NGC1377. The outflow is resolved, revealing a 150 pc long, clumpy, high-velocity, collimated molecular jet. The molecular emission is emerging from the spine of the jet with an average diameter of 3-7 pc. A narrow-angle, rotating molecular wind surrounds the jet and is enveloped by a larger-scale, slower CO-emitting structure. The jet and narrow wind are turbulent ($\sigma>$40 kms$^{-1}$) and have steep radial gas excitation gradients. The jet shows velocity reversals that we propose are caused by precession, or episodic directional changes. We suggest that an important process powering the outflow is magneto-centrifugal driving. In contrast, the large-scale CO-envelope may be a slow wind, or cocoon that stems from jet-wind interactions. An asymmetric, nuclear r$\sim$2 pc and hot (>180 K) dust structure with a high molecular column density, N(H$_2$)$\sim1.8 \times 10^{24}$ cm$^{-2}$, is detected in continuum and vibrationally excited HCN. Its luminosity is likely powered by a buried AGN. The mass of the supermassive black hole (SMBH) is estimated to $\sim9\times10^6$ M$_\odot$ and the SMBH of NGC1377 appears to be at the end of an intense phase of accretion. The nuclear growth may be fuelled by low-angular momentum gas inflowing from gas ejected in the molecular jet and wind. Such a feedback-loop of cyclic accretion and outflows would be an effective process in growing the nuclear SMBH. This result invites new questions as to SMBH growth processes in obscured, dusty galaxies., Comment: 16 pages, 12 figures, accepted for publication in Astronomy and Astrophysics. Updated affiliations, added reference

MAXI J1535-571 is a Galactic black hole candidate X-ray binary that was discovered going into outburst in 2017 September. In this paper, we present comprehensive radio monitoring of this system using the Australia Telescope Compact Array (ATCA), as well as the MeerKAT radio observatory, showing the evolution of the radio jet during its outburst. Our radio observations show the early rise and subsequent quenching of the compact jet as the outburst brightened and then evolved towards the soft state. We constrain the compact jet quenching factor to be more than 3.5 orders of magnitude. We also detected and tracked (for 303 days) a discrete, relativistically-moving jet knot that was launched from the system. From the motion of the apparently superluminal knot, we constrain the jet inclination (at the time of ejection) and speed to $\leq 45^{\circ}$ and $\geq0.69$c, respectively. Extrapolating its motion back in time, our results suggest that the jet knot was ejected close in time to the transition from the hard intermediate state to soft intermediate state. The launching event also occurred contemporaneously with a short increase in X-ray count rate, a rapid drop in the strength of the X-ray variability, and a change in the type-C quasi-periodic oscillation (QPO) frequency that occurs $>$2.5 days before the first appearance of a possible type-B QPO., Comment: 19 pages, 9 figures, supplementary tables and figures included as appendices. Accepted for publication in ApJ

Despite their importance in the star formation process, measurements of magnetic field strength in proto-planetary discs remain rare. While linear polarisation of dust and molecular lines can give insight into the magnetic field structure, only observations of the circular polarisation produced by Zeeman splitting provide a direct measurement of magnetic field strenghts. One of the most promising probes of magnetic field strengths is the paramagnetic radical CN. Here we present the first Atacama Large Millimeter/submillimeter Array (ALMA) observations of the Zeeman splitting of CN in the disc of TW~Hya. The observations indicate an excellent polarisation performance of ALMA, but fail to detect significant polarisation. An analysis of eight individual CN hyperfine components as well as a stacking analysis of the strongest (non-blended) hyperfine components yields the most stringent limits obtained so far on the magnetic field strength in a proto-planetary disc. We find that the vertical component of the magnetic field $|B_z|<0.8$~mG ($1\sigma$ limit). We also provide a $1\sigma$ toroidal field strength limit of $<30$~mG. These limits rule out some of the earlier accretion disc models, but remain consistent with the most recent detailed models with efficient advection. We detect marginal linear polarisation from the dust continuum, but the almost purely toroidal geometry of the polarisation vectors implies that his is due to radiatively aligned grains., Comment: 10 pages, 5 figures, accepted for publication in A&A letters

We report on the results of optical, near-infrared (NIR) and mid-infrared observations of the black hole X-ray binary candidate (BHB) MAXI J1535-571 during its 2017/2018 outburst. During the first part of the outburst (MJD 58004-58012), the source shows an optical-NIR spectrum that is consistent with an optically thin synchrotron power-law from a jet. After MJD 58015, however, the source faded considerably, the drop in flux being much more evident at lower frequencies. Before the fading, we measure a de-reddened flux density of $\gtrsim$100 mJy in the mid-infrared, making MAXI J1535-571 one of the brightest mid-infrared BHBs known so far. A significant softening of the X-ray spectrum is evident contemporaneous with the infrared fade. We interpret it as due to the suppression of the jet emission, similar to the accretion-ejection coupling seen in other BHBs. However, MAXI J1535-571 did not transition smoothly to the soft state, instead showing X-ray hardness deviations, associated with infrared flaring. We also present the first mid-IR variability study of a BHB on minute timescales, with a fractional rms variability of the light curves of $\sim 15-22 \%$, which is similar to that expected from the internal shock jet model, and much higher than the optical fractional rms ($\lesssim 7 \%$). These results represent an excellent case of multi-wavelength jet spectral-timing and demonstrate how rich, multi-wavelength time-resolved data of X-ray binaries over accretion state transitions can help refining models of the disk-jet connection and jet launching in these systems., Comment: 20 pages, 8 figures. Accepted for publication in the ApJ

The `fundamental plane of black hole accretion' (FP), a relation between the radio luminosities ($L_R$), X-ray luminosities ($L_X$), and masses ($M_{BH}$) of hard/quiescent state black hole binaries and low-luminosity active galactic nuclei, suggests some aspects of black hole accretion may be scale invariant. However, key questions still exist concerning the relationship between the inflow/outflow behaviour in the `classic' hard state and quiescence, which may impact this scaling. We show that the broadband spectra of A0620-00 and~\sgra~(the least luminous stellar mass/supermassive black holes on the FP) can be modelled simultaneously with a physically-motivated outflow-dominated model where the jet power and all distances are scaled by the black hole mass. We find we can explain the data of both A0620-00 and~\sgra~(in its non-thermal flaring state) in the context of two outflow-model scenarios: (1) a synchrotron-self-Compton dominated state in which the jet plasma reaches highly sub-equipartition conditions (for the magnetic field with respect to that of the radiating particles), and (2) a synchrotron dominated state in the fast-cooling regime in which particle acceleration occurs within the inner few gravitational radii of the black hole and plasma is close to equipartition. We show that it may be possible to further discriminate between models (1) and (2) through future monitoring of its submm/IR/X-ray emission, in particular via time lags between the variable emission in these bands., Comment: 19 pages (including appendix), 6 figures, 5 tables. Accepted for publication in MNRAS

We consider the radiative transfer problem in a plane-parallel slab of thermal electrons in the presence of an overcritical magnetic field. Under such conditions, the magnetic field behaves as a birefringent medium for the propagating photons, and the electromagnetic radiation is split into two polarization modes, ordinary and extraordinary, having different cross-sections. When the optical depth of the slab is large, the ordinary-mode photons are strongly Comptonized and the photon field is dominated by an isotropic component. The radiative transfer problem in strong magnetic fields presents many mathematical issues and analytical or numerical solutions can be obtained only under some given approximations. We investigate this problem both from the analytical and numerical point of view, providing a test of the previous analytical estimates and extending these results introducing numerical techniques. We consider the case of low temperature blackbody photons propagating in a sub-relativistic temperature plasma, which allows us to deal with a semi Fokker-Planck approximation of the radiative transfer equation. We report the numerical solution obtained for eigenvalues and eigenfunctions of the space operator, and the emerging Comptonization spectrum of the ordinary-mode photons for any eigenvalue of the space equation and for energies significantly less than the cyclotron energy, which is of the order of MeV for the intensity of the magnetic field here considered. We derived the specific intensity of the ordinary photons, under the approximation of large angle and large optical depth. These assumptions allow the equation to be treated using a diffusion-like approximation.

We present a review of the Supergiant Fast X-ray Transients (SFXT) Project, a systematic investigation of the properties of SFXTs with a strategy that combines Swift monitoring programs with outburst follow-up observations. This strategy has quickly tripled the available sets of broad-band data of SFXT outbursts, and gathered a wealth of out-of-outburst data, which have led us to a broad-band spectral characterization, an assessment of the fraction of the time these sources spend in each phase, and their duty cycle of inactivity. We present some new observational results obtained through our outburst follow-ups, as fitting examples of the exceptional capabilities of Swift in catching bright flares and monitor them panchromatically., Comment: Accepted for publication in Advances in Space Research. Proceedings of the 39th COSPAR Scientific Assembly, E1.4, "Unveiling Properties of Supergiant Fast X-ray Transients". 9 pages, 6 figures, 2 tables

Supergiant fast X-ray transients (SFXTs) are a class of high-mass X-ray binaries with possible counterparts in the high energy gamma rays. The Swift SFXT Project has conducted a systematic investigation of the properties of SFTXs on timescales ranging from minutes to years and in several intensity states (from bright flares, to intermediate intensity states, and down to almost quiescence). We also performed broad-band spectroscopy of outbursts, and intensity-selected spectroscopy outside of outbursts. We demonstrated that while the brightest phase of the outburst only lasts a few hours, further activity is observed at lower fluxes for a remarkably longer time, up to weeks. Furthermore, we assessed the fraction of the time these sources spend in each phase, and their duty cycle of inactivity. We present the most recent results from our investigation. The spectroscopic and, most importantly, timing properties of SFXTs we have uncovered with Swift will serve as a guide in search for the high energy emission from these enigmatic objects., Comment: Proceedings, 5th International Symposium on High-Energy Gamma-Ray Astronomy, (Gamma2012) Heidelberg, Germany, July 9-13th, 2012

We report on the results from observations of the most recent outbursts of XTE J1739-302 and IGR J17544-2619, which are considered to be the prototypes of the supergiant fast X-ray transient (SFXT) class. They triggered the Swift/BAT on 2011 February 22 and March 24, respectively, and each time a prompt Swift slew allowed us to obtain the rich broad-band data we present. The XRT light curves show the descending portion of very bright flares that reached luminosities of ~2x10^{36} and ~5x10^{36} erg/s, respectively. The broad-band spectra, when fit with the usual phenomenological models adopted for accreting neutron stars, yield values of both high energy cut-off and e-folding energy consistent with those obtained from previously reported outbursts from these sources. In the context of more physical models, the spectra of both sources can be well fitted either with a two-blackbody model, or with a single unsaturated Comptonization model. In the latter case, the model can be either a classical static Comptonization model, such as COMPTT, or the recently developed COMPMAG model, which includes thermal and bulk Comptonization for cylindrical accretion onto a magnetized neutron star. We discuss the possible accretion scenarios derived by the different models, and we also emphasize the fact that the electron density derived from the Comptonization models, in the regions where the X-ray spectrum presumably forms, is lower than that estimated using the continuity equation at the magnetospheric radius and the source X-ray luminosity, and we give some possible explanations., Comment: 11 pages, 7 figures, accepted for publication in MNRAS

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&A

We consider the radiative transfer problem in a plane-parallel slab of thermal electrons in the presence of an ultra-strong magnetic field (B approximately greater than B(sub c) approx. = 4.4 x 10(exp 13) G). Under these conditions, the magnetic field behaves like a birefringent medium for the propagating photons, and the electromagnetic radiation is split into two polarization modes, ordinary and extraordinary, that have different cross-sections. When the optical depth of the slab is large, the ordinary-mode photons are strongly Comptonized and the photon field is dominated by an isotropic component. Aims. The radiative transfer problem in strong magnetic fields presents many mathematical issues and analytical or numerical solutions can be obtained only under some given approximations. We investigate this problem both from the analytical and numerical point of view, provide a test of the previous analytical estimates, and extend these results with numerical techniques. Methods. We consider here the case of low temperature black-body photons propagating in a sub-relativistic temperature plasma, which allows us to deal with a semi-Fokker-Planck approximation of the radiative transfer equation. The problem can then be treated with the variable separation method, and we use a numerical technique to find solutions to the eigenvalue problem in the case of a singular kernel of the space operator. The singularity of the space kernel is the result of the strong angular dependence of the electron cross-section in the presence of a strong magnetic field. Results. We provide the numerical solution obtained for eigenvalues and eigenfunctions of the space operator, and the emerging Comptonization spectrum of the ordinary-mode photons for any eigenvalue of the space equation and for energies significantly lesser than the cyclotron energy, which is on the order of MeV for the intensity of the magnetic field here considered. Conclusions. We derived the specific intensity of the ordinary photons, under the approximation of large angle and large optical depth. These assumptions allow the equation to be treated using a diffusion-like approximation.

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. This requires developing numerical routines for the solution of the radiative transfer equation according to the expected physical conditions of the systems under study. Aims. 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. Methods. 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 pi 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. Results. 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 pi 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. Conclusions. The algorithm has been implemented in the XPEC package for X-ray fitting and is specifically dedicated to the physical framework of accretion at the polar cap of a neutron star with a high magnetic field (approx > 10(exp 12) G). This latter case is expected to be of typical accreting systems such as X-ray pulsars and supergiant fast X ray transients.