Search

Observations show that the X-ray emission of the accreting weakly magnetized neutron stars is polarized. Here, we develop a theoretical model, where we assume the emission of the accreting neutron star coming from the spreading layer, the extension of the boundary between the disk and the neutron star surface onto the surface. We then calculate the Stokes parameters of the emission accounting for relativistic aberration and gravitational light bending in the Schwarzschild metric. We show that regardless of the geometry, for the spreading layer, we cannot expect the polarization degree to be higher than 1.5%. Our results have implications with regard to the understanding of the X-ray polarization from weakly magnetized neutron stars observed with the Imaging X-ray Polarimetry Explorer and the future enhanced X-ray Timing and Polarimetry mission., Comment: 9 pages, 7 figures, submitted to A&A

Pulse profile modelling (PPM) is a technique for inferring mass, radius and hotspot properties of millisecond pulsars. PPM is now regularly used for analysis of rotation-powered millisecond pulsars (RMPs) with data from the Neutron Star Interior Composition ExploreR (NICER). Extending PPM to accreting millisecond pulsars (AMPs) is attractive, because they are a different source class featuring bright X-ray radiation from hotspots powered by accretion. In this paper, we present a modification of one of the PPM codes, X-PSI, so that it can be used for AMPs. In particular, we implement a model of an accretion disc and atmosphere model appropriate for the hotspots of AMPs, and improve the overall computational efficiency. We then test parameter recovery with synthetic NICER data in two scenarios with reasonable parameters for AMPs. We find in the first scenario, where the hotspot is large, that we are able to tightly and accurately constrain all parameters including mass and radius. In the second scenario, which is a high inclination system with a smaller hotspot, we find degeneracy between a subset of model parameters and a slight bias in the inferred mass and radius. This analysis of synthetic data lays the ground work for future analysis of AMPs with NICER data. Such an analysis could be complemented by future (joint) analysis of polarization data from the Imaging X-ray Polarimetry Explorer (IXPE)., Comment: 14 pages, 5 figures, 3 tables. Paper submitted to MNRAS

We report on the discovery of polarized X-ray emission from an accreting millisecond pulsar. During a 10-day-long coverage of the February 2024 outburst of SRGA J144459.2-604207, the Imaging X-ray Polarimetry Explorer (IXPE) detected an average polarization degree of the 2-8 keV emission of 2.3% +/- 0.4% at an angle of 59{\deg} +/- 6{\deg} (East of North; uncertainties quoted at the 1${\sigma}$ confidence level). The polarized signal shows a significant energy dependence with a degree of 4.0% +/- 0.5% between 3 and 6 keV and < 2% (90% c.l.) in the 2-3 keV range. We used NICER, XMM-Newton, and NuSTAR observations to obtain an accurate pulse timing solution and perform a phase-resolved polarimetric analysis of IXPE data. We did not detect any significant variability of the Stokes parameters Q and U with the spin and the orbital phases. We used the relativistic rotating vector model to show that a moderately fan-beam emission from two point-like spots at a small magnetic obliquity ($\simeq$ 10{\deg}) is compatible with the observed pulse profile and polarization properties. IXPE also detected 52 type-I X-ray bursts, with a recurrence time $\Delta t_{rec}$ increasing from 2 to 8 h as a function of the observed count rate $C$ as as $\Delta t_{rec} \simeq C^{-0.8}$ We stacked the emission observed during all the bursts and obtained an upper limit on the polarization degree of 8.5% (90% c.l.)., Comment: 13 pages, 9 figures, submitted to ApJL

Cygnus X-3 is an enigmatic X-ray binary, that is both an exceptional accreting system and a cornerstone for the population synthesis studies. Prominent X-ray and radio properties follow a well-defined pattern, yet the physical reasons for the state changes observed in this system are not known. Recently, the presence of an optically thick envelope around the central source in the hard state was revealed using the X-ray polarization data obtained with Imaging X-ray Polarimetry Explorer (IXPE). In this work, we analyse IXPE data obtained in the ultrasoft (radio quenched) state of the source. The average polarization degree (PD) of $11.9\pm0.5\%$ at a polarization angle (PA) of $94^{\circ}\pm1^{\circ}$ is inconsistent with the simple geometry of the accretion disc viewed at an intermediate inclination. The high PD, the blackbody-like spectrum, and the weakness of fluorescent iron line imply that the central source is hidden behind the optically thick outflow and its beamed radiation is scattered towards our line of sight. In this picture the observed PD is directly related to the source inclination, which we conservatively determine to lie in the range $26^{\circ}

We present the first X-ray spectropolarimetric results for Cygnus X-1 in its soft state from a campaign of five IXPE observations conducted during 2023 May-June. Companion multiwavelength data during the campaign are likewise shown. The 2-8 keV X-rays exhibit a net polarization degree PD=1.99%+/-0.13% (68% confidence). The polarization signal is found to increase with energy across IXPE's 2-8 keV bandpass. The polarized X-rays exhibit an energy-independent polarization angle of PA=-25.7+/-1.8 deg. East of North (68% confidence). This is consistent with being aligned to Cyg X-1's AU-scale compact radio jet and its pc-scale radio lobes. In comparison to earlier hard-state observations, the soft state exhibits a factor of 2 lower polarization degree, but a similar trend with energy and a similar (also energy-independent) position angle. When scaling by the natural unit of the disk temperature, we find the appearance of a consistent trendline in the polarization degree between soft and hard states. Our favored polarimetric model indicates Cyg X-1's spin is likely high (a* above ~0.96). The substantial X-ray polarization in Cyg X-1's soft state is most readily explained as resulting from a large portion of X-rays emitted from the disk returning and reflecting off the disk surface, generating a high polarization degree and a polarization direction parallel to the black hole spin axis and radio jet. In IXPE's bandpass, the polarization signal is dominated by the returning reflection emission. This constitutes polarimetric evidence for strong gravitational lensing of X-rays close to the black hole., Comment: 20 pages, accepted for publication in ApJL

Recent observations of X-ray pulsars (XRPs) performed by the Imaging X-ray Polarimetry Explorer (IXPE) have made it possible to investigate the intricate details of these objects in a new way, thanks to the added value of X-ray polarimetry. Here we present the results of the IXPE observations of SMC X-1, a member of the small group of XRPs displaying super-orbital variability. SMC X-1 was observed by IXPE three separate times during the high state of its super-orbital period. The observed luminosity in the 2-8 keV energy band of $L=2\times10^{38}$ erg/s makes SMC X-1 the brightest XRP ever observed by IXPE. We detect significant polarization in all three observations, with values of the phase-averaged polarization degree (PD) and polarization angle (PA) of $3.2\pm0.8$% and $97\deg\pm8\deg$ for Observation 1, $3.0\pm0.9$% and $90\deg\pm8\deg$ for Observation 2, and $5.5\pm1.1$% and $80\deg\pm6\deg$ for Observation 3, for the spectro-polarimetric analysis. The observed PD shows an increase over time with decreasing luminosity, while the PA decreases in decrements of 10\deg. The phase-resolved spectro-polarimetric analysis reveals significant detection of polarization in three out of seven phase bins, with the PD ranging between 2% and 10%, and a corresponding range in the PA from $\sim$70\deg\ to $\sim$100\deg. The pulse-phase resolved PD displays an apparent anti-correlation with the flux. Using the rotating vector model, we obtain constraints on the pulsar's geometrical properties for the individual observations. The position angle of the pulsar displays an evolution over time supporting the idea that we observe changes related to different super-orbital phases. Scattering in the wind of the precessing accretion disk may be responsible for the behavior of the polarimetric properties observed during the high-state of SMC X-1's super-orbital period., Comment: 11 pages, 11 figures, submitted to A&A

Radiative corrections which remove accidental degeneracy in the spectrum of the relativistic hydrogen atom and lead to the modification of the Coulomb law, are calculated within the novel approach, based on the exact solution of the Dirac equation with the Coulomb potential. The energy spectrum of the hydrogen atom is obtained with account of these corrections and the Lamb shift is calculated for the lowest energy states., Comment: 24 pages

Discovery of pulsations from a number of ultra-luminous X-ray (ULX) sources proved that accretion onto neutron stars can produce luminosities exceeding the Eddington limit by a couple of orders of magnitude. The conditions necessary to achieve such high luminosities as well as the exact geometry of the accretion flow in the neutron star vicinity are, however, a matter of debate. The pulse phase-resolved polarization measurements that became possible with the launch of the IXPE can be used to determine the pulsar geometry and its orientation relative to the orbital plane. They provide an avenue to test different theoretical models of ULX pulsars. In this paper we present the results of three IXPE observations of the first Galactic ULX pulsar Swift J0243.6+6124 during its 2023 outburst. We find strong variations of the polarization characteristics with the pulsar phase. The average polarization degree increases from about 5% to 15% as the flux dropped by a factor of three in the course of the outburst. The polarization angle (PA) as function of the pulsar phase shows two peaks in the first two observations, but changes to a characteristic sawtooth pattern in the remaining data set. This is not consistent with a simple rotating vector model. Assuming the existence of an additional constant polarized component, we were able to fit the three observations with a common rotating vector model and obtain constraints on the pulsar geometry. In particular, we find the pulsar angular momentum inclination with respect to the line-of-sight of 15-40 deg, the magnetic obliquity of 60-80 deg, and the pulsar spin position angle of -50 deg, which differs from the constant component PA of about 10 deg. Combining these X-ray measurements with the optical PA, we find evidence for a 30 deg misalignment between the pulsar spin and the binary orbital axis., Comment: 13 pages, 10 figures, submitted to A&A

Weakly magnetized neutron stars (WMNS) are complicated sources with challenging phenomenology. For decades, they have been studied via spectrometry and timing. It has been established that the spectrum of WMNSs consists of several components traditionally associated with the accretion disk, the boundary or spreading layer, and the wind and their interactions with each other. Since 2022, WMNSs have been actively observed with the Imaging X-ray Polarimetry Explorer (IXPE). Polarimetric studies provided new information about the behavior and geometry of these sources. One of the most enigmatic sources of the class, galactic X-ray burster GX 13+1 was first observed with IXPE in October 2023. A strongly variable polarization at the level 2-5$\%$ was detected with the source showing a rotation of the polarization angle (PA) that hinted towards the misalignment within the system. The second observation was performed in February 2024 with a complementary observation by Swift/XRT. IXPE measured an overall polarization degree (PD) of 2.5$\%$ and the PA of 24 degrees, and the Swift/XRT data helped us evaluate the galactic absorption and fit the continuum. Here we study the similarities and differences between the polarimetric properties of the source during the two observations. We confirm the expectation of the misalignment in the system and the assignment of the harder component to the boundary layer. We emphasize the importance of the wind in the system. We note the difference in the variation of polarimetric properties with energy and with time., Comment: 8 pages, 7 figures

Weakly magnetized neutron stars in X-ray binaries show complex phenomenology with several spectral components that can be associated with the accretion disk, boundary and/or spreading layer, a corona, and a wind. Spectroscopic information alone is, however, not enough to disentangle these components. Additional information about the nature of the spectral components and in particular the geometry of the emission region can be provided by X-ray polarimetry. One of the objects of the class, a bright, persistent, and rather peculiar galactic Type I X-ray burster was observed with the Imaging X-ray Polarimetry Explorer (IXPE) and the X-ray Multi-Mirror Mission Newton (XMM-Newton). Using the XMM-Newton data we estimated the current state of the source as well as detected strong absorption lines associated with the accretion disk wind. IXPE data showed the source to be significantly polarized in the 2-8 keV energy band with the overall polarization degree (PD) of 1.4% at a polarization angle (PA) of -2 degrees (errors at 68% confidence level). During the two-day long observation, we detected rotation of the PA by about 70 degrees with the corresponding changes in the PD from 2% to non-detectable and then up to 5%. These variations in polarization properties are not accompanied by visible changes in spectroscopic characteristics. The energy-resolved polarimetric analysis showed a significant change in polarization, from being strongly dependent on energy at the beginning of the observation to being almost constant with energy in the later parts of the observation. As a possible interpretation, we suggest the presence of a constant component of polarization, strong wind scattering, or different polarization of the two main spectral components with individually peculiar behavior. The rotation of the PA suggests a 30-degree misalignment of the neutron star spin from the orbital axis., Comment: 12 pages, 12 figures

We consider electronic spectra of graphene nanotubes and their perturbation by impurity atoms absorbed at different positions on nanotube surfaces, within the framework of Anderson hybrid model. A special attention is given to the cases when Dirac-like 1D modes appear in the nanotube spectrum and their hybridization with localized impurity states produces, at growing impurity concentration $c$, onset of a mobility gap near the impurity level and even opening, at yet higher $c$, of some narrow delocalized range within this mobility gap. Such behaviors are compared with the similar effects in the previously studied 2D graphene and armchair type graphene nanoribbons. Some possible practical applications are discussed., Comment: 14 pages, 17 figures

We report on an observational campaign on the bright black hole X-ray binary Swift J1727.8$-$1613 centered around five observations by the Imaging X-ray Polarimetry Explorer (IXPE). These observations track for the first time the evolution of the X-ray polarization of a black hole X-ray binary across a hard to soft state transition. The 2--8 keV polarization degree decreased from $\sim$4\% to $\sim$3\% across the five observations, but the polarization angle remained oriented in the North-South direction throughout. Based on observations with the Australia Telescope Compact Array (ATCA), we find that the intrinsic 7.25 GHz radio polarization aligns with the X-ray polarization. Assuming the radio polarization aligns with the jet direction (which can be tested in the future with higher spatial resolution images of the jet), our results imply that the X-ray corona is extended in the disk plane, rather than along the jet axis, for the entire hard intermediate state. This in turn implies that the long ($\gtrsim$10 ms) soft lags that we measure with the Neutron star Interior Composition ExploreR (NICER) are dominated by processes other than pure light-crossing delays. Moreover, we find that the evolution of the soft lag amplitude with spectral state does not follow the trend seen for other sources, implying that Swift J1727.8$-$1613 is a member of a hitherto under-sampled sub-population., Comment: Accepted for publication in ApJ. 21 pages, 8 figures

Pulse profiles of accreting millisecond pulsars can be used to determine neutron star (NS) parameters, such as their masses and radii, and therefore provide constraints on the equation of state of cold dense matter. Information obtained by the Imaging X-ray Polarimetry Explorer (IXPE) can be used to decipher pulsar inclination and magnetic obliquity, providing ever tighter constraints on other parameters. In this paper, we develop a new emission model for accretion-powered millisecond pulsars based on thermal Comptonization in an accretion shock above the NS surface. The shock structure was approximated by an isothermal plane-parallel slab and the Stokes parameters of the emergent radiation were computed as a function of the zenith angle and energy for different values of the electron temperature, the Thomson optical depth of the slab, and the temperature of the seed blackbody photons. We show that our Compton scattering model leads to a significantly lower polarization degree of the emitted radiation compared to the previously used Thomson scattering model. We computed a large grid of shock models, which can be combined with pulse profile modeling techniques both with and without polarization included. In this work, we used the relativistic rotating vector model for the oblate NS in order to produce the observed Stokes parameters as a function of the pulsar phase. Furthermore, we simulated the data to be produced by IXPE and obtained constraints on model parameters using nested sampling. The developed methods can also be used in the analysis of the data from future satellites, such as the enhanced X-ray Timing and Polarimetry mission., Comment: Accepted to A&A on 11 August 2023

Spectro-polarimetric signatures of accretion disks in X-ray binaries and active galactic nuclei contain information about the masses and spins of their central black holes, as well as the geometry of matter close to the compact objects. This information can be extracted using the means of X-ray polarimetry. In this work, we present a fast analytical ray-tracing technique for polarized light \textsc{artpol} that helps obtain the spinning black hole parameters from the observed properties. This technique can replace the otherwise time-consuming numerical ray-tracing calculations. We show that \textsc{artpol} proves accurate for Kerr black holes with dimensionless spin parameter $a\leq0.94$ while being over four orders of magnitude faster than direct ray-tracing calculations. This approach opens broad prospects for directly fitting the spectro-polarimetric data from the \textit{Imaging X-ray Polarimetry Explorer}., Comment: Submitted to Astronomy & Astrophysics

We report on Imaging X-ray polarimetry explorer (IXPE) observations of the Be-transient X-ray pulsar LS V +44 17/RX J0440.9+4431 made at two luminosity levels during the giant outburst in January--February 2023. Considering the observed spectral variability and changes in the pulse profiles, the source was likely caught in supercritical and subcritical states with significantly different emission-region geometry, associated with the presence of accretion columns and hot spots, respectively. We focus here on the pulse-phase-resolved polarimetric analysis and find that the observed dependencies of the polarization degree and polarization angle (PA) on the pulse phase are indeed drastically different for the two observations. The observed differences, if interpreted within the framework of the rotating vector model (RVM), imply dramatic variations in the spin axis inclination, the position angle, and the magnetic colatitude by tens of degrees within the space of just a few days. We suggest that the apparent changes in the observed PA phase dependence are predominantly related to the presence of an unpulsed polarized component in addition to the polarized radiation associated with the pulsar itself. We then show that the observed PA phase dependence in both observations can be explained with a single set of RVM parameters defining the pulsar's geometry. We also suggest that the additional polarized component is likely produced by scattering of the pulsar radiation in the equatorial disk wind., Comment: 10 pages, 7 figures, accepted in A&A

We present an X-ray spectro-polarimetric analysis of the bright Seyfert galaxy IC 4329A. The Imaging X-ray Polarimetry Explorer (IXPE) observed the source for ~500 ks, supported by XMM-Newton (~60 ks) and NuSTAR (~80 ks) exposures. We detect polarisation in the 2-8 keV band with 2.97 sigma confidence. We report a polarisation degree of $3.3\pm1.1$ per cent and a polarisation angle of $78\pm10$ degrees (errors are 1 sigma confidence). The X-ray polarisation is consistent with being aligned with the radio jet, albeit partially due to large uncertainties on the radio position angle. We jointly fit the spectra from the three observatories to constrain the presence of a relativistic reflection component. From this, we obtain constraints on the inclination angle to the inner disc (< 39 degrees at 99 per cent confidence) and the disc inner radius (< 11 gravitational radii at 99 per cent confidence), although we note that modelling systematics in practice add to the quoted statistical error. Our spectro-polarimetric modelling indicates that the 2-8 keV polarisation is consistent with being dominated by emission directly observed from the X-ray corona, but the polarisation of the reflection component is completely unconstrained. Our constraints on viewer inclination and polarisation degree tentatively favour more asymmetric, possibly out-flowing, coronal geometries that produce more highly polarised emission, but the coronal geometry is unconstrained at the 3 sigma level., Comment: Accepted for publication in MNRAS, 13 pages, 9 figures

Large energy-dependent X-ray polarization degree is detected by the Imaging X-ray Polarimetry Explorer ({IXPE}) in the high-soft emission state of the black hole X-ray binary 4U 1630--47. The highly significant detection (at $\approx50\sigma$ confidence level) of an unexpectedly high polarization, rising from $\sim6\%$ at $2$ keV to $\sim10\%$ at $8$ keV, cannot be easily reconciled with standard models of thin accretion discs. In this work we compare the predictions of different theoretical models with the {IXPE} data and conclude that the observed polarization properties are compatible with a scenario in which matter accretes onto the black hole through a thin disc, covered by a partially-ionized atmosphere flowing away at mildly relativistic velocities., Comment: Published in ApJ (https://doi.org/10.3847/1538-4357/ad226e)

We present an X-ray spectro-polarimetric analysis of the bright Seyfert galaxy NGC4151. The source has been observed with the Imaging X-ray Polarimetry Explorer (IXPE) for 700 ks, complemented with simultaneous XMM-Newton (50 ks) and NuSTAR (100 ks) pointings. A polarization degree ${\Pi} = 4.9 {\pm} 1.1 \%$ and angle ${\Psi}= 86{\deg} {\pm} 7{\deg}$ east of north ($68\%$ confidence level) are measured in the 2-8 keV energy range. The spectro-polarimetric analysis shows that the polarization could be entirely due to reflection. Given the low reflection flux in the IXPE band, this requires however a reflection with a very large ($> 38 \%$) polarization degree. Assuming more reasonable values, a polarization degree of the hot corona ranging from ${\sim}4$ to ${\sim}8\%$ is found. The observed polarization degree excludes a spherical lamppost geometry for the corona, suggesting instead a slab-like geometry, possibly a wedge, as determined via Monte Carlo simulations. This is further confirmed by the X-ray polarization angle, which coincides with the direction of the extended radio emission in this source, supposed to match the disc axis. NGC4151 is the first AGN with an X-ray polarization measure for the corona, illustrating the capabilities of X-ray polarimetry and IXPE in unveiling its geometry., Comment: 9 pages, 4 figures

The accretion of matter by compact objects can be inhibited by radiation pressure if the luminosity exceeds the critical value, known as the Eddington limit. Discovery of ultraluminous X-ray sources has shown that accretion can proceed even when the apparent luminosity significantly exceeds this limit. High apparent luminosity might be produced thanks to geometric beaming of the radiation by an outflow. The outflow half-opening angle, which determines the amplification due to beaming, has never been robustly constrained. Using the Imaging X-ray Polarimetry Explorer, we made the measurement of X-ray polarization in the Galactic X-ray binary Cyg X-3. We find high, over 20%, nearly energy-independent linear polarization, orthogonal to the direction of the radio ejections. These properties unambiguously indicate the presence of a collimating outflow in the X-ray binary Cyg~X-3 and constrain its half-opening angle, <15 degrees. Thus, the source can be used as a laboratory for studying the super-critical accretion regime. This finding underscores the importance of X-ray polarimetry in advancing our understanding of accreting sources., Comment: 52 pages, 14 figures, 3 tables. Author version of the article published in Nature Astronomy

We present a pulse timing analysis of NICER observations of the accreting millisecond X-ray pulsar SAX J1808.4$-$3658 during the outburst that started on 2022 August 19. Similar to previous outbursts, after decaying from a peak luminosity of $\simeq 1\times10^{36} \, \mathrm{erg \, s^{-1}}$ in about a week, the pulsar entered in a $\sim 1$ month-long reflaring stage. Comparison of the average pulsar spin frequency during the outburst with those previously measured confirmed the long-term spin derivative of $\dot{\nu}_{\textrm{SD}}=-(1.15\pm0.06)\times 10^{-15} \, \mathrm{Hz\,s^{-1}}$, compatible with the spin-down torque of a $\approx 10^{26} \, \mathrm{G \, cm^3}$ rotating magnetic dipole. For the first time in the last twenty years, the orbital phase evolution shows evidence for a decrease of the orbital period. The long-term behaviour of the orbit is dominated by a $\sim 11 \, \mathrm{s}$ modulation of the orbital phase epoch consistent with a $\sim 21 \, \mathrm{yr}$ period. We discuss the observed evolution in terms of a coupling between the orbit and variations in the mass quadrupole of the companion star., Comment: 11 pages, 3 figures, 1 table. Accepted for publication in ApJ Letters

It is shown that the Dirac equation with the Coulomb potential can be solved using the algebra of the three spinor invariants of the Dirac equation without the involvement of the methods of supersymmetric quantum mechanics. The Dirac Hamiltonian is invariant with respect to the rotation transformation, which indicates the dynamical (hidden) symmetry $ SU(2) $ of the Dirac equation. The total symmetry of the Dirac equation is the symmetry $ SO(3) \otimes SU(2) $. The generator of the $ SO(3) $ symmetry group is given by the total momentum operator, and the generator of $ SU(2) $ group is given by the rotation of the vector-states in the spinor space, determined by the Dirac, Johnson-Lippmann, and the new spinor invariants. It is shown that using algebraic approach to the Dirac problem allows one to calculate the eigenstates and eigenenergies of the relativistic hydrogen atom and reveals the fundamental role of the principal quantum number as an independent number, even though it is represented as the combination of other quantum numbers., Comment: 16 pages, no figures

We consider finite ribbons of graphene with armchair orientation of their edges to study in detail impurity effects on specific Dirac-like modes. In the framework of Anderson hybrid model of impurity perturbation, a possibility for Mott localization and for opening of a mobility gap under local impurity perturbations is found and analyzed in function of this model parameters: the impurity energy level, its hybridization with the host Dirac modes, and the impurity concentration. Possible electronic phase states in such disordered system and subsequent phase transitions between them are discussed., Comment: 11 pages, 15 figures

A black hole x-ray binary (XRB) system forms when gas is stripped from a normal star and accretes onto a black hole, which heats the gas sufficiently to emit x-rays. We report a polarimetric observation of the XRB Cygnus X-1 using the Imaging x-ray Polarimetry Explorer. The electric field position angle aligns with the outflowing jet, indicating that the jet is launched from the inner x-ray emitting region. The polarization degree is (4.01+-0.20)% at 2 to 8 kiloelectronvolts, implying that the accretion disk is viewed closer to edge-on than the binary orbit. The observations reveal that hot x-ray emitting plasma is spatially extended in a plane perpendicular to the jet axis, not parallel to the jet., Comment: 51 pages, 15 figures, 5 tables

Solutions of the Dirac equation for an electron in the Coulomb potential are obtained using operator invariants of the equation, namely the Dirac, Johnson-Lippmann and recently found new invariant. It is demonstrated that these operators are the spin invariants. The generalized invariant is constructed and the exact general solution of the Dirac equation are found. In particular, the explicit expressions of the bispinors corresponding to the three complete sets of the invariants, their eigenvalues and quantum numbers are calculated. It is shown that the general solution of one center Coulomb Dirac equation contains free parameters. Changing one or more of these parameters, one can transform one solution of the Dirac equation into any other. It is shown for the first time that these invariants determine electron spatial probability amplitude and spin polarization in each quantum state. Electron probability densities and spin polarizations are explicitly calculated in the general form for several electron states in the hydrogen-like energy spectrum. Spatial distributions of these characteristics are shown to depend essentially on the invariant set, demonstrating, in spite of the accidental degeneracy of energy levels, physical difference of the states corresponding to different spin invariants., Comment: 24 pages, 6 figures

Emission from an accretion disc around compact objects, such as neutron stars and black holes, is expected to be significantly polarized. The polarization can be used to put constraints on geometrical and physical parameters of the compact sources -- their radii, masses and spins -- as well as to determine the orbital parameters. The radiation escaping from the innermost parts of the disc is strongly affected by the gravitational field of the compact object and relativistic velocities of the matter. The straightforward calculation of the observed polarization signatures involves computationally expensive ray-tracing technique. At the same time, having fast computational routines for direct data fitting becomes increasingly important in light of the currently observed images of the accretion flow around supermassive black hole in M87 by the Event Horizon Telescope, infrared polarization signatures coming from Sgr A*, as well as for the upcoming X-ray polarization measurements by the Imaging X-ray Polarimetry Explorer and enhanced X-ray Timing and Polarimetry mission. In this work, we obtain an exact analytical expression for the rotation angle of polarization plane in Schwarzschild metric accounting for the effects of light bending and relativistic aberration. We show that the calculation of the observed flux, polarization degree and polarization angle as a function of energy can be performed analytically with high accuracy using approximate light-bending formula, lifting the need for the pre-computed tabular models in fitting routines., Comment: 14 pages, 11 figures

The ideal Bose-gas with finite number $N$ of particles is investigated. The exact expressions for the partition functions and occupation numbers in the grand canonical, canonical and microcanonical ensembles are found. The asymp\-totic expressions (in the case $N\gg1$) for the partition functions and occupation numbers in the canonical and microcanonical ensembles are evaluated. It is shown that the chemical potential $\mu$ of the ideal Bose-gas can lie in the range $-\infty<\mu<\infty$ oppositely to the widely adopted opinion that the value of this potential is negative., Comment: 7 pages, no figures