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Patrick Weltevrede

Publications and source records attributed to Patrick Weltevrede.

At least 19 recordsLinked to original sources

Probing Neutron Star Interiors and the Properties of Cold Ultra-dense Matter with the SKAO

Matter inside neutron stars is compressed to densities several times greater than nuclear saturation density, while maintaining low temperatures and large asymmetries between neutrons and protons. Neutron stars, therefore, provide a unique laboratory for testing physics in environments that cannot be recreated on Earth. To uncover the highly uncertain nature of cold, ultra-dense matter, discovering and monitoring pulsars is essential, and SKAO will play a crucial role in this endeavour. In this chapter, we will present the current state-of-the-art in dense matter physics and dense matter superfluidity, and discuss recent advances in measuring global neutron star properties (masses, moments of inertia, and maximum rotation frequencies) as well as non-global observables (pulsar glitches and free precession). We will specifically highlight how radio observations of isolated neutron stars and those in binaries -- such as those performed with SKAO in the near future -- inform our understanding of ultra-dense physics and address in detail how SKAO's telescopes unprecedented sensitivity, large-scale survey and sub-arraying capabilities will enable novel dense matter constraints. We will also address the potential impact of dark matter and modified gravity models on these constraints and emphasise the role of synergies between SKAO and other facilities, specifically X-ray telescopes and next-generation gravitational wave observatories.

astro-ph.HE

The Thousand-Pulsar-Array programme on MeerKAT XIX: Single-pulse data analysis, nulling and pulse energy distributions

We present the Thousand Pulsar Array (TPA) single-pulse data set, obtained with the MeerKAT radio telescope and comprising time-series observations of 1192 pulsars, typically containing ~1000 consecutive pulses per source. We describe the MeerTime Single Pulse software pipeline which calibrates the data and automatically excises interference signals to produce data products suitable for typical single-pulse studies. To demonstrate the capabilities of the dataset, we carry out a population-level study of phase-averaged single-pulse energy distributions and nulling behaviour. Pulse energy distributions are modelled within a Bayesian framework choosing from a range of intrinsic energy distributions, and including an explicit nulling fraction. We find that approximately half of the pulsars require multi-component intrinsic energy distributions, while the remainder are consistent with single-component models. Nulling is detected or constrained for most pulsars in the sample, and both the occurrence and inferred nulling fraction show systematic variation across the P-$\dot{P}$ diagram. In particular, nulling fractions increase with spin period and exhibit only a weak dependence on period derivative. We also examine trends in the preferred forms of pulse energy distributions as a function of spin-down luminosity, finding modest evidence for population-level evolution. Estimates of single-pulse luminosities indicate that individual pulses can exceed the long-term average luminosity by large factors, particularly for low-$\dot{E}$ pulsars. These results characterise the statistical properties of single-pulse emission across a large pulsar sample and highlight the limitations of phase-averaged energy distributions for capturing the full complexity of pulsar emission variability.

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The MeerKAT Thousand-Pulsar Polarisation Array II: Searches for Ultralight Axion-Like Dark Matter

We construct Pulsar Polarisation Arrays (PPA), using regular pulsars monitored in MeerKAT's Thousand Pulsar Array (TPA) Programme, to search for Axion-like Dark Matter (ALDM) within Milky Way. Specifically, from a catalogue of 1237 regular pulsars, we select the 50 ones with the highest signal-to-noise ratio and set upper limits on the ALDM Chern-Simons coupling. We find no signals with statistical significance over the mass range of $[10^{-23},10^{-20}]\,{\rm eV}$ in the six-year MeerKAT's data. By combining the high-quality TPA pulsars and the accurate ionospheric subtraction of spinifex, we establish the most sensitive upper limits to the date on the ALDM Chern-Simons coupling, namely $\lesssim 10^{-14} - 3\times 10^{-13}\,{\rm GeV}^{-1}$, for the mass range of $[10^{-23},10^{-21}]\,{\rm eV}$ except at $m_a \sim 1.3 \times 10^{-22}\,$eV. This study underscores the great potential of constructing regular-pulsar PPAs for scientific tasks.

astro-ph.HE

The MeerKAT Thousand-Pulsar Polarization Array I: Properties of the Polarization and Rotation Measure Time Series Data

The polarimetry of recent pulsar observations has provided a wealth of observational data with which to test physical theories of emission mechanisms, radiative transfer and even theories that extend beyond the Standard Model. In this work, we have outlined the data analysis of the polarisation time series data of a population of 513 pulsars from the Thousand Pulsar Array observing programme, laying the foundation for building the MeerKAT Thousand-Pulsar Polarization Array as a probe for ultralight Axion-Like Dark Matter (ALDM). From this large dataset we have focused on the temporal trends in the observed polarisation angle (PA) through a measure we call the PA offset, and characterised the trends due to the effects of Faraday Rotation within the interstellar medium and the Earth's ionosphere, as well as generic white and red noise models that are estimated within a Bayesian MCMC analysis. Then, motivated by potential extra contributions to the rotation of the PA that may not be Faraday-like, arising from the proposed ALDM field, we have investigated a derived time dependence for the rotation measure (RM) required to explain the observed PA offset. Comparison of these estimates to RM values that are measured in typical pulsar studies, through a technique known as RM Synthesis, provides a probe of any wavelength-independent contribution to the rotation of the PA. Although we find no evidence for oscillatory behaviour within our dataset's observation timespan, we do find cases of deviation from the usual RM values in several `pulsars of interest', as well as long-term linear trends in the time evolution of Faraday rotation that have been presented in the literature before.

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Temporal Variations in Pulsar Spectro-Polarimetry: Findings from millisecond pulsar J2144-5237 using Parkes UWL receiver

While the temporal variations of the spectro-polarimetric nature of pulsars remains unexplored, this investigation offers significant potential for uncovering key insights into pulsar emission mechanisms, magnetic field geometry, and propagation effects within the magnetosphere. We developed a package for investigating time-varying spectral behavior for full Stokes parameters and demonstrate it on a millisecond pulsar (MSP) J2144-5237 in a binary system (orbital period ~10 days) using the Parkes UWL receiver. In this study we report rotation measure (RM) variation with orbital phase. We find that the temporal variations in the spectra of Stokes I, Q, and V are generally correlated throughout the orbit, while Stokes U exhibits intervals of both correlation and anticorrelation with Stokes I, depending on the orbital phase. We also provide a Poincare sphere representation of the polarization properties of J2144-5237, demonstrating a systematic temporal change of Poincare sphere location for the main component with orbital phase. To our knowledge, this is the first investigation of time-varying properties of the spectro-polarimetric nature of any pulsars or MSPs. Extending this study to probe the spectro-temporal nature of full Stokes data on a larger sample of MSPs or pulsars has the potential to provide vital information on emission mechanisms inside the magnetosphere, interstellar propagation effects, and binary interactions.

astro-ph.HE

Detection of giant pulses from the transitional millisecond pulsar J1227$-$4853

We report the discovery of giant pulse (GP) emission from the transitional millisecond pulsar (tMSP) PSR J1227$-$4853, using 174 hours of single-pulse data from the upgraded Giant Metrewave Radio Telescope (uGMRT). This marks the first detection of GPs from a transitional MSP and adds to the small number of millisecond pulsars known to exhibit such extreme variability. A total of 235 GPs were detected across observations at 550-750 MHz, with widths as narrow as 1.28 $\mu$s and flux densities up to $\sim 10^4$ times the pulsar's mean flux density. The GPs are strongly localized in pulse phase, originating predominantly from the second and third main-pulse components, and are absent in the inter-pulse region. Their cumulative fluence distribution follows a power law above the completeness threshold, consistent with a defining characteristic of GP emission. The arrival times of the GPs deviate significantly from Poisson statistics, with the waiting-time distribution well described by a Weibull model having a shape parameter of $k = 0.30$, indicative of strong temporal clustering. During an epoch of enhanced activity, the GP rate increased by nearly two orders of magnitude to 124 $\mathrm{hr}^{-1}$, with a corresponding shape parameter of $k = 0.47$. This value is similar to that reported for a burst storm from the repeating fast radio burst FRB 20200120E, suggesting possible phenomenological parallels between GPs from compact binary systems and repeating FRBs.

astro-ph.HE

Probing neutron star interiors and the properties of cold ultra-dense matter with the SKAO

Matter inside neutron stars is compressed to densities several times greater than nuclear saturation density, while maintaining low temperatures and large asymmetries between neutrons and protons. Neutron stars, therefore, provide a unique laboratory for testing physics in environments that cannot be recreated on Earth. To uncover the highly uncertain nature of cold, ultra-dense matter, discovering and monitoring pulsars is essential, and the SKA will play a crucial role in this endeavour. In this paper, we will present the current state-of-the-art in dense matter physics and dense matter superfluidity, and discuss recent advances in measuring global neutron star properties (masses, moments of inertia, and maximum rotation frequencies) as well as non-global observables (pulsar glitches and free precession). We will specifically highlight how radio observations of isolated neutron stars and those in binaries -- such as those performed with the SKA in the near future -- inform our understanding of ultra-dense physics and address in detail how SKAO's telescopes unprecedented sensitivity, large-scale survey and sub-arraying capabilities will enable novel dense matter constraints. We will also address the potential impact of dark matter and modified gravity models on these constraints and emphasise the role of synergies between the SKA and other facilities, specifically X-ray telescopes and next-generation gravitational wave observatories.

astro-ph.HE

Hierarchical Bayesian estimation of population-level torque law parameters from $68$ young radio pulsars observed with the Murriyang telescope

Abridged. The measured braking index, $n=\nu \ddot{\nu}/\dot{\nu}^2$, of a rotation-powered pulsar with spin frequency $\nu$ and braking torque $K \nu^{n_{\rm pl}}$, features secular and stochastic anomalies arising from $\dot{K} \neq 0$ and random torque noise respectively. Previous studies quantified the variance $\langle n^{2} \rangle = (n_{\rm pl}+\dot{K}_{\rm dim})^{2}+\sigma_{\rm dim}^{2}$, where the secular anomaly, $\dot{K}_{\rm dim}$, is inversely proportional to the characteristic time-scale $\tau_{K}$ over which $K$ varies; the stochastic anomaly, $\sigma_{\rm dim}^{2} = \sigma_{\ddot{\nu}}^{2}\nu^{2}\gamma_{\ddot{\nu}}^{-2}\dot{\nu}^{-4}T_{\rm obs}^{-1}$, is a function of the timing noise amplitude $\sigma_{\ddot{\nu}}$, a damping time-scale $\gamma_{\ddot{\nu}}^{-1}$ and the total observing time $T_{\rm obs}$; and the average is taken over an ensemble of random realizations of the noise process. Here, we use a hierarchical Bayesian scheme, based on the formula for $\langle n^{2} \rangle$, to infer the population-level distribution of $n_{\rm pl}+\dot{K}_{\rm dim}$ for a sample of $68$ young radio pulsars, observed for $\gtrsim 10~{\rm years}$ with Murriyang, the 64-m Parkes radio telescope. Upon assuming that the $n_{\rm pl}+\dot{K}_{\rm dim}$ values are drawn from a population-level Gaussian, $N(\mu_{\rm pl}, \sigma_{\rm pl})$, the Bayesian scheme returns the mean $\mu_{\rm pl} = 9.95^{+5.58}_{-5.26}$ and standard deviation $\sigma_{\rm pl}=10.89^{+5.14}_{-3.69}$. At a per-pulsar level it returns posterior medians satisfying $-13.86 \leq n_{\rm pl}+\dot{K}_{\rm dim} \leq 30.38$. The secular anomaly dominates the stochastic anomaly, with posterior medians satisfying $|n_{\rm pl} + \dot{K}_{\rm dim}| \geq \sigma_{\rm dim}$ in 10 out of 68 objects.

astro-ph.HE

Pulsar B1237+25 Aberration/Retardation Analysis from Decimeter to Decameter Wavelength: Challenge to "Radius-to-Frequency Mapping"

PSR B1237+25 is perhaps the canonical example of a pulsar with a core/double cone profile. Moreover, it is bright with little spectral turnover, and its profile perhaps uniquely remains undistorted by scattering far into the decametric band. Here we assemble more than a dozen of the highest quality profiles (30 MHz to 5 GHz) from half a dozen observatories, where possible polarimetric. The pulsar's 2.6$^{\circ}$ core component marks the magnetic axis longitude, and we confirm that this point coincides both with the linear polarization angle inflection point and the zero-crossing of its antisymmetric circular signature -- thus providing the possibility to estimate emission heights over a very broad band using aberration/retardation (A/R). We then carefully fit the profile components with Gaussians to identify and study the subtle asymmetries produced by A/R. We find a consistent A/R in the pulsar's profiles of some 0.5$^{\circ}$ longitude or 2 ms -- corresponding to a putative conal emission height of 200-400 km -- with a formal error of about 100 km. Our analysis finds no evidence whatsoever for an emission height increase with wavelength, the so-called ``radius-to-frequency mapping''. Nor do we find any significant difference in A/R effect between the outer and inner cones.

astro-ph.HE

The Thousand-Pulsar-Array programme on MeerKAT -- XVIII. Complex sub-pulse modulation patterns, bi-drifting and mode changing of nine radio pulsars

We present a detailed analysis of sub-pulse modulations in nine pulsars which show evidence of changes in sub-pulse drift direction as a function of pulse longitude in the Thousand Pulsar Array single pulse survey with MeerKAT. We confirm that all of these are consistent with persistent drift direction changes. These 'bi-drifting' pulsars present a challenge to the classical carousel model for sub-pulse drifting. In general, bi-drifting in this expanded sample is less clear than some of the previously published cases, which we attribute to narrower profile widths ($<20$ degrees) or smaller $P_3$ values (close to 2$P$). However, given the broad variety of pulse shapes and drift behaviours across the pulsar population, it is unsurprising that the phenomenon is not limited to only those where it can most easily be detected. Four of our samples show at least two emission modes with different profile shapes and drift properties, which seems to be a relatively common feature of bi-drifting pulsars. We also find jumps in sub-pulse phase between adjacent components in two pulsars. In addition to our MeerKAT L-band data, we used GMRT observations for four, and MeerKAT UHF observations for two of these pulsars to investigate the frequency dependence of sub-pulse drift. We find subtle changes in the drift as a function of frequency, but no clear overall pattern. Looking at the distribution of bi-drifting pulsars over $P$, $\dot{P}$ and $P_3$ suggests they are consistent with the underlying population of all drifting pulsars.

astro-ph.HE

Deciphering Profile Stability in Millisecond Pulsars: Timescales, Frequency Evolution, and Implications on Emission Mechanisms

Pulse profile stability in millisecond pulsars (MSPs) is a key factor in achieving high-precision timing essential for detecting nanohertz gravitational waves with Pulsar Timing Arrays (PTAs). In this work, we present a systematic analysis of profile stabilization timescales in MSPs using a direct method based on pulse stacking, applied to long-term multi-epoch observations. Our study utilizes data from the upgraded GMRT (uGMRT) between 300--750 MHz for nine MSPs over 3--5 years and Parkes Ultra-Wideband low-frequency receiver observations (Parkes UWL; covering 704--4032 MHz) for three of them. We find that stable profiles typically require averaging over $10^{5}$--$10^{6}$ pulses. This is the first time such a quantitative approach has been applied to MSPs across a wide frequency range, providing an indirect but practical estimate of jitter noise, a dominant noise source in PTA datasets. We observe that stabilization timescales depend on signal-to-noise ratio, pulse morphology, and surface magnetic field strength, with a moderate correlation indicating a possible role of the magnetic field in emission stability. A complementary single-epoch analysis of nine bright MSPs with uGMRT Band-3 (300--500 MHz) reinforces these results and demonstrates the method's applicability to broader MSP populations. We show that a strong correlation exists between profile-stability slope and the jitter parameter, implying that for faint MSPs, profile-stability analysis can act as an effective proxy for intrinsic pulse-shape variability. Our work provides a novel and scalable framework to assess intrinsic profile variability, helping to guide integration time choices and reduce timing noise in PTA experiments.

astro-ph.HE

Measuring pulsar profile variations with 2-D Gaussian process regression

Time-correlated variations in the pulse profiles of radio pulsars provide insights into changes in their magnetospheres. For a small number of pulsars (~20), these variations have been shown to correlate with spin-down rate. Many of these profile changes involve small (few percent) variations in the relative intensity of different profile components, and hence tools such as Gaussian process regression have been employed to separate the time-correlated profile variation from intrinsic noise. In this paper, we present a computationally efficient approximation of a 2-D Gaussian process model that enhances sensitivity by simultaneously tracking time- and phase-correlated signals. Applying this model to 26 pulsars observed at the Jodrell Bank Observatory, we detect significant profile shape variations in 21 pulsars. Using principal component analysis, we confirm spin-down correlated shape variations in 11 pulsars where this had been previously observed. Additionally, we find evidence of spin-down correlated shape changes in 7 pulsars for the first time (PSRs B0105+65, B0611+22, B0626+24, B1740-03, B1826-17, B1917+00, and B2148+63). We look in greater detail at PSR B0740-28, where the correlation between profile shape and spin-down itself seems to switch between quasi-stable states. Notably the profile shape associated with greater spin-down seems to invert at times, presenting a challenge to our understanding of the physical processes at work.

astro-ph.HE

Exploring Unusual High-frequency Eclipses in MSP J1908+2105

This paper presents a comprehensive study of the eclipse properties of the spider millisecond pulsar (MSP) J1908$+$2105, using wide-band observations from the uGMRT and Parkes UWL. For the first time, we observed that this pulsar exhibits extended eclipses up to 4 GHz, the highest frequency band of the UWL, making it one of only three MSPs known to have such high-frequency eclipses. This study reveals synchrotron absorption as the primary eclipse mechanism for J1908$+$2105. We present modeling of synchrotron optical depth with various possible combinations of the parameters to explain the observed eclipsing in this as well as other spider MSPs. Observed eclipses at unusually high frequencies for J1908$+$2105 significantly aided in constraining the magnetic field and electron column density in the eclipse medium while modeling the synchrotron optical depth. Combining our findings with data from other MSPs in the literature, for the first time we note that a higher cutoff frequency of eclipsing, particularly above 1 GHz, is consistently associated with a higher electron column density ($>$ 10$^{17}$ cm$^{-2}$) in the eclipse medium. Additionally, we present the first evidence of lensing effects near eclipse boundaries in this MSP, leading to significant magnification of radio emissions. The orbital phase resolved polarization analysis presented in this paper further indicates variation in rotation measure and consequently stronger magnetic fields in the eclipse region.

astro-ph.HE

Unveiling frequency-dependent eclipsing in spider millisecond pulsars using broadband polarization observations with the Parkes

This study presents an orbital phase-dependent analysis of three black widow spider millisecond pulsars (BW MSPs), aiming to investigate the magnetic field within the eclipse environment. The ultra-wide-bandwidth low-frequency receiver (UWL) of the Parkes 'Murriyang' radio telescope is utilised for full polarisation observations covering frequencies from 704-4032 MHz. Depolarisation of pulsed emission is observed during the eclipse phase of three BW MSPs namely, J0024-7204J, J1431-4715 and PSR J1959+2048, consistent with previous studies of other BW MSPs. We estimated orbital phase dependent RM values for these MSPs. The wide bandwidth observations also provided the constraints on eclipse cutoff frequency for these BW MSPs. For PSR J0024-7204J, we report temporal variation of the eclipse cutoff frequency coupled with changes in the electron column density within the eclipse medium across six observed eclipses. Moreover, the eclipse cutoff frequency for PSR J1431-4715 is determined to be 1251 $\pm$ 80 MHz, leading to the conclusion that synchrotron absorption is the primary mechanism responsible for the eclipsing. Additionally, for PSR J1959+2048, the estimated cutoff frequency exceeded 1400 MHz, consistent with previous studies. With this investigation, we have doubled the sample size of BW MSPs with orbital phase-resolved studies allowing a better probe to the eclipse environment.

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Measuring glitch recoveries and braking indices with Bayesian model selection

For a selection of 35 pulsars with large spin-up glitches ($Δν/ν\geq10^{-6}$), which are monitored by the Jodrell Bank Observatory, we analyse 157 glitches and their recoveries. All parameters are measured consistently and we choose the best model to describe the post-glitch recovery based on Bayesian evidence. We present updated glitch epochs, sizes, changes of spin down rate, exponentially recovering components (amplitude and corresponding timescale) when present, as well as pulsars' second frequency derivatives and their glitch associated changes if detected. We discuss the different observed styles of post-glitch recovery as well as some particularly interesting sources. Several correlations are revealed between glitch parameters and pulsar spin parameters, including a very strong correlation between a pulsar's interglitch $|\ddotν|$ and $\dotν$, as well as between the glitch-induced spin-down rate change $Δ\dotν_{\rm p}$ that does not relax exponentially and $\dotν$. We find that the ratio $\left|Δ\dotν_{\mathrm{p}}/\ddotν\right|$ can be used as an estimate of glitch recurrence times, especially for those pulsars for which there are indications of a characteristic glitch size and interglitch waiting time. We calculate the interglitch braking index $n$ and find that pulsars with large glitches typically have $n$ greater than $3$, suggesting that internal torques dominate the rotational evolution between glitches. The external torque, e.g. from electromagnetic dipole radiation, could dominate the observed $\ddotν$ for the youngest pulsars ($\lesssim10^{4}\;\mathrm{yr}$), which may be expected to display $n\sim3$.

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Linear to circular conversion in the polarized radio emission of a magnetar

Radio emission from magnetars provides a unique probe of the relativistic, magnetized plasma within the near-field environment of these ultra-magnetic neutron stars. The transmitted waves can undergo birefringent and dispersive propagation effects that result in frequency-dependent conversions of linear to circularly polarized radiation and vice-versa, thus necessitating classification when relating the measured polarization to the intrinsic properties of neutron star and fast radio burst (FRB) emission sites. We report the detection of such behavior in 0.7-4 GHz observations of the P = 5.54 s radio magnetar XTE J1810$-$197 following its 2018 outburst. The phenomenon is restricted to a narrow range of pulse phase centered around the magnetic meridian. Its temporal evolution is closely coupled to large-scale variations in magnetic topology that originate from either plastic motion of an active region on the magnetar surface or free precession of the neutron star crust. Our model of the effect deviates from simple theoretical expectations for radio waves propagating through a magnetized plasma. Birefringent self-coupling between the transmitted wave modes, line-of-sight variations in the magnetic field direction and differences in particle charge or energy distributions above the magnetic pole are explored as possible explanations. We discuss potential links between the immediate magneto-ionic environments of magnetars and those of FRB progenitors.

astro-ph.HE

Radio Pulse Profile Evolution of Magnetar Swift J1818.0-1607

The shape and polarisation properties of the radio pulse profiles of radio-loud magnetars provide a unique opportunity to investigate their magnetospheric properties. Gaussian Process Regression analysis was used to investigate the variation in the total intensity shape of the radio pulse profiles of the magnetar Swift J1818.0-1607. The observed profile shape was found to evolve through three modes between MJDs 59104 and 59365. The times at which these transitions occurred coincided with changes in the amplitude of modulations in the spin-down rate. The amount of linear and circular polarisation was also found to vary significantly with time. Lomb-Scargle periodogram analysis of the spin-down rate revealed three possibly harmonically related frequencies. This could point to the magnetar experiencing seismic activity. However, no profile features exhibited significant periodicity, suggesting no simple correlations between the profile variability and fluctuations of the spin-down on shorter timescales within the modes. Overall, this implies the mode changes seen are a result of local magnetospheric changes, with other theories, such as precession, less able to explain these observations.

astro-ph.HE

Multiwavelength pulsations and surface temperature distribution in the middle-aged pulsar B1055-52

We present a detailed study of the X-ray emission from PSR B1055-52 using XMM-Newton observations from 2019 and 2000. The phase-integrated X-ray emission from this pulsar is poorly described by existing neutron star atmosphere models. Instead, we confirm that, similar to other middle-aged pulsars, the best-fitting spectral model consists of two blackbody components, with substantially different temperatures and emitting areas, and a nonthermal component characterized by a power law. Our phase-resolved X-ray spectral analysis using this three-component model reveals variations in the thermal emission parameters with the pulsar's rotational phase. These variations suggest a nonuniform temperature distribution across the neutron star's surface, including the cold thermal component and probable hot spot(s). Such a temperature distribution can be caused by external and internal heating processes, likely a combination thereof. We observe very high pulse fractions, 60\%--80\% in the 0.7-1.5, keV range, dominated by the hot blackbody component. This could be related to temperature non-uniformity and potential beaming effects in an atmosphere. We find indication of a second hot spot that appears at lower energies (0.15-0.3, keV) than the first hot spot (0.5-1.5, keV) in the X-ray light curves, and is offset by about half a rotation period. This finding aligns with the nearly orthogonal rotator geometry suggested by radio observations of this interpulse pulsar. If the hot spots are associated with polar caps, a possible explanation for their temperature asymmetry could be an offset magnetic dipole and/or an additional toroidal magnetic field component in the neutron star crust

astro-ph.HE