Mass and radius constraints for neutron stars using X-ray timing, spectral, and polarization observations

Neutron stars (NSs) are the most dense objects in the Universe that can be directly observed. The nature of the cold ultra-dense matter inside them is still unresolved, and determining the equation of state (EoS) of that matter is a fundamental problem in nuclear physics. Measurements of sizes and masses of NSs can be used to constrain the EoS, and thus NSs can be described as astrophysical laboratories for nuclear physics. The size (or radius) and mass measurements can be done, for example, using the X-ray timing observations of millisecond pulsars (MSPs), which are very rapidly rotating NSs. In the first part of this thesis, I have presented a framework that can be used to model the observed X-ray pulse profiles from MSPs and to obtain constraints for the model parameters, including NS mass and radius. I have also estimated how the upcoming X-ray polarization measurements will improve the constraints. In addition, I have shown that there also exist problems in the current models in explaining all the features of the X-ray data. Especially, the emission pattern from an atmosphere of an accretion-powered millisecond pulsar (AMP) should be accurately solved for more robust estimates. Moreover, modelling NS atmospheres is not only important for AMPs, but also essential for pulse profile modelling of rotation-powered millisecond pulsars (RMPs), and that is considered in the second part of the thesis. I have studied the importance of the exact formulation of Compton scattering in the RMP atmospheres and created a novel model for RMP atmospheres heated by magnetospheric return-currents. This model differs from the preceding ones in that it does not assume that all the heat is released in the deepest layers of the atmosphere. The results imply that the emission pattern also from RMP surface may significantly deviate from that predicted by previous models, which could affect also the recent radius constraints obtained from the observations of Neutron star Interior Composition ExploreR (NICER) instrument. In the final part of the thesis, I have studied, in more detail, how the upcoming X-ray polarization observations of rapidly rotating NSs can be accurately modelled accounting for the flattened shape of the star and used to obtain further constraints on the EoS of ultra-dense matter. The results show that the unknown physics of NS interiors can be probed by combining X-ray timing, spectral, and polarization measurements of MSPs

Mass and radius constraints for neutron stars from pulse shape modeling

Neutron stars are the most compact directly observable objects. The matter inside a neutron star is at supranuclear densities. The equation of state (EOS) of neutron stars describes the properties of such dense matter. Separation between numerous theoretical EOSs is possible if we are able to constrain the possible masses and radii of neutron stars from observations. In this thesis we present one method that can be used to constrain masses and radii of neutron stars. The method is suitable for accreting millisecond pulsars, where a rapidly rotating neutron star, called a millisecond pulsar, accretes matter from a relatively low mass companion star, via an accretion disc, onto the magnetic poles of the neutron star. Because of the accretion, we observe radiation from two "hot spots" on the neutron star surface. This radiation is pulsating coherently at the spinning frequency of the neutron star. The exact shape of the pulses can be modeled with a theoretical model that takes into account the general and special relativistic effects via "Schwarzschild-Doppler" approximation and the oblate shape of the star caused by the fast rotation. The details of this model are discussed. The pulse profiles carry information about the mass and radius of a neutron star since e.g., the light bending and thus pulse shape depends strongly on the compactness of the star. Also many other physical parameters and observing angles affect the light curves. Therefore, we use Bayesian analysis and a novel Monte Carlo sampling method, called "ensemble sampler", to obtain probability distributions for the different parameters, especially for the mass and the radius. The ensemble sampler has shown to overcome many difficulties concerning the traditional Metropolis-Hastings sampler. We have also generated synthetic data to test our method and fitted the pulse profiles to these data. The results of our samplings, using these synthetic data, show that obtaining new constraints for radius and mass is not a very easy task. However, according to our study, prior information obtained from polarization measurements may be used the get significantly tighter constraints

Oblate Schwarzschild approximation for polarized radiation from rapidly rotating neutron stars

We have developed a complete theory for the calculation of the observed Stokes parameters for radiation emitted from the surface of a rapidly rotating neutron star (NS) using the oblate Schwarzschild approximation. We accounted for the rotation of the polarization plane due to relativistic effects along the path from the stellar surface to the observer. The results were shown to agree with those obtained by performing full numerical general relativistic ray-tracing with the \textsc{arcmancer} code. We showed that the obtained polarization angle (PA) profiles may differ substantially from those derived for a spherical star. We demonstrated that assuming incorrect shape for the star can lead to biased constraints for NS parameters when fitting the polarization data. Using a simplified model, we also made a rough estimate of how accurately the geometrical parameters of an accreting NS can be determined using the X-ray polarization measurements of upcoming polarimeters like the Imaging X-ray Polarimeter Explorer (IXPE) or the enhanced X-ray Timing and Polarimetry (eXTP) mission.Comment: 11 pages, 10 figures, accepted in A&

Polarized radiation from an accretion shock in accreting millisecond pulsars using exact Compton scattering formalism

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 202

Magnetospheric return-current-heated atmospheres of rotation-powered millisecond pulsars

We computed accurate atmosphere models of rotation-powered millisecond pulsars in which the polar caps of a neutron star (NS) are externally heated by magnetospheric return currents. The external ram pressure, energy losses, and stopping depth of the penetrating charged particles were computed self-consistently with the atmosphere model, instead of assuming a simplified deep-heated atmosphere in radiative equilibrium. We used exact Compton scattering formalism to model the properties of the emergent X-ray radiation. The deep-heating approximation was found to be valid only if most of the heat originates from ultra-relativistic bombarding particles with Lorentz factors of $\gamma \gtrsim 100$. In the opposite regime, the atmosphere attains a distinct two-layer structure with an overheated optically thin skin on top of an optically thick cool plasma. The overheated skin strongly modifies the emergent radiation: it produces a Compton-upscattered high-energy tail in the spectrum and alters the radiation beaming pattern from limb darkening to limb brightening for emitted hard X-rays. This kind of drastic change in the emission properties can have a significant impact on the inferred NS pulse profile parameters as performed, for example, by Neutron star Interior Composition ExploreR. Finally, the connection between the energy distribution of the return current particles and the atmosphere emission properties offers a new tool to probe the exact physics of pulsar magnetospheres.Comment: 13 pages, 10 figures, published in A&

Neutron star parameter constraints for accretion-powered millisecond pulsars from the simulated IXPE data

We have simulated the X-ray polarization data that can be obtained with the Imaging X-ray Polarimetry Explorer, when observing accretion-powered millisecond pulsars. We estimated the necessary exposure times for SAX J1808.4$-$3658 in order to obtain different accuracies in the measured time-dependent Stokes profiles integrated over all energy channels. We found that the measured relative errors depend strongly on the relative configuration of the observer and the emitting hotspot. The improvement in the minimum relative error in Stokes $Q$ and $U$ parameters as a function of observing time $t$ scales as $1/\sqrt{t}$, and spans the range from 30-90% with 200 ks exposure time to 20-60% with 500 ks exposure time (in case of data binned in 19 phase bins). The simulated data were also used to predict how accurate measurements of the geometrical parameters of the neutron star can be made when modelling only $Q$ and $U$ parameters, but not the flux. We found that the observer inclination and the hotspot co-latitude could be determined with better than 10 deg accuracy for most of the cases we considered. These measurements can be used to further constrain neutron star mass and radius when combined with modelling of the X-ray pulse profile.Comment: 12 pages, 11 figures, published in A&

X-PSI Parameter Recovery for Temperature Map Configurations Inspired by PSR J0030+0451

In the last few years, the NICER collaboration has provided mass and radius inferences, via pulse profile modeling, for two pulsars: PSR J0030+0451 and PSR J0740+6620. Given the importance of these results for constraining the equation of state of dense nuclear matter, it is crucial to validate them and test their robustness. We therefore explore the reliability of these results and their sensitivity to analysis settings and random processes, including noise, focusing on the specific case of PSR J0030+0451. We use X-PSI, one of the two main analysis pipelines currently employed by the NICER collaboration for mass and radius inferences. With synthetic data that mimic the PSR J0030+0451 NICER data set, we evaluate the recovery performances of X-PSI under conditions never tested before, including complex modeling of the thermally emitting neutron star surface. For the test cases explored, our results suggest that X-PSI is capable of recovering the true mass and radius within reasonable credible intervals. This work also reveals the main vulnerabilities of the analysis: a significant dependence on noise and the presence of multi-modal structure in the posterior surface. Noise particularly impacts our sensitivity to the analysis settings and widths of the posterior distributions. The multi-modal structure in the posterior suggests that biases could be present if the analysis is unable to exhaustively explore the parameter space. Convergence testing, to ensure an adequate coverage of the parameter space and a suitable representation of the posterior distribution, is one possible solution to these challenges.Comment: 27 pages, 13 figure

Oblate Schwarzschild approximation for polarized radiation from rapidly rotating neutron stars

We have developed a complete theory for the calculation of the observed Stokes parameters for radiation emitted from the surface of a rapidly rotating neutron star (NS) using the oblate Schwarzschild approximation. We accounted for the rotation of the polarization plane due to relativistic effects along the path from the stellar surface to the observer. The results were shown to agree with those obtained by performing full numerical general relativistic ray-tracing with the ARCMANCER code. We showed that the obtained polarization angle profiles may differ substantially from those derived for a spherical star. We demonstrated that assuming incorrect shape for the star can lead to biased constraints for NS parameters when fitting the polarization data. Using a simplified model, we also made a rough estimate of how accurately the geometrical parameters of an accreting NS can be determined using the X-ray polarization measurements of upcoming polarimeters like the Imaging X-ray Polarimeter Explorer or the enhanced X-ray Timing and Polarimetry mission

Magnetospheric return-current-heated atmospheres of rotation-powered millisecond pulsars

We computed accurate atmosphere models of rotation-powered millisecond pulsars in which the polar caps of a neutron star (NS) are externally heated by magnetospheric return currents. The external ram pressure, energy losses, and stopping depth of the penetrating charged particles were computed self-consistently with the atmosphere model, instead of assuming a simplified deep-heated atmosphere in radiative equilibrium. We used exact Compton scattering formalism to model the properties of the emergent X-ray radiation. The deep-heating approximation was found to be valid only if most of the heat originates from ultra-relativistic bombarding particles with Lorentz factors of gamma greater than or similar to 100. In the opposite regime, the atmosphere attains a distinct two-layer structure with an overheated optically thin skin on top of an optically thick cool plasma. The overheated skin strongly modifies the emergent radiation: It produces a Compton-upscattered high-energy tail in the spectrum and alters the radiation beaming pattern from limb darkening to limb brightening for emitted hard X-rays. This kind of drastic change in the emission properties can have a significant impact on the inferred NS pulse profile parameters as performed, for example, by Neutron star Interior Composition ExploreR. Finally, the connection between the energy distribution of the return current particles and the atmosphere emission properties offers a new tool to probe the exact physics of pulsar magnetospheres