arXiv subjects

Jaziel G. Coelho

Publications and source records attributed to Jaziel G. Coelho.

At least 19 recordsLinked to original sources

arXiv2026

Extending the Pulsar radio population through Quantum Vacuum Cherenkov Emission

The presence of some radio pulsars within and below the so-called ``death valley'' in the $P$--$\dot{P}$ diagram challenges the standard pair-cascade models based on curvature radiation (CR) and resonant inverse Compton scattering (RICS) in neutron star polar caps. We show that Cherenkov radiation from the quantum vacuum, a strong-field quantum electrodynamics (QED) phenomenon enabled by vacuum polarization, constitutes a distinct pair-creation channel within the vacuum gap model. Within this model, the acceleration length scale for Cherenkov photon emission and the subsequent photon mean free path are sufficiently short to ensure rapid conversion into $e^+e^-$ pairs. This process acts as a complementary pair-creation channel, remaining viable even in regimes where classical mechanisms become heavily suppressed. This leads to a new set of death lines in the $P$--$\dot{P}$ diagram that partially overlap with and extend beyond the CR and RICS death valleys, enlarging the viable parameter space for radio-emitting neutron stars. The result provides a physical basis for coherent radio emission from long-period neutron stars previously considered inactive, and offers a new window for testing non-linear QED with next-generation radio facilities such as the Square Kilometre Array.

astro-ph.HE↗

arXiv2026

Localized magnetic pressure anisotropy in strange dwarfs with crystalline crusts

We study equilibrium sequences of magnetized strange dwarfs composed of a self bound MIT bag strange matter core, with $B_{\rm bag}^{1/4}=145\,{\rm MeV}$, and an ordinary crystalline crust. The aim is to determine how nuclear composition changes and magnetic pressure anisotropy in the crust modify the mass radius relation. The strange core is kept isotropic, while the crust is described either by pure nuclei stopped at the first inverse beta threshold or by electron capture sequences extended to neutron drip for C, O, Ne, and Mg compositions; selected mixed crusts are also considered up to the first instability. The stellar structure is solved with the radial pressure as the integrated pressure and with the anisotropic contribution $2(P_t-P_r)/r$ restricted to the crust. This treatment is compared with ordinary white dwarf sequences and with scalar pressure branch calculations. We find that extending the crust to neutron drip produces more compact strange dwarf branches than stopping at the inverse beta threshold, because the evolved crust is softer and the core crust transition occurs at higher pressure. Magnetic anisotropy further shifts the selected branches toward smaller radii, with the effect visible across the compositions studied and clearer at fixed stellar mass. The comparison with compact objects from the Montreal White Dwarf Database is used only as an observational reference plane, but it indicates that compact white dwarf candidates are a relevant region for testing small core strange dwarf scenarios.

astro-ph.SR↗

arXiv2026

Ultra-High-Energy Particle Production in Binary Mergers Endowed with Magnetic Fields

We study the production of ultra-high-energy particles via the Bañados--Silk--West (BSW) mechanism in the pre-merger phase of binary systems detected by LIGO-Virgo-KAGRA. By solving the geodesic equations for charged particles in magnetized Kerr spacetime with fields of $B \sim 10^{12}$--$10^{14}$~G, we demonstrate that collisions near the horizon can achieve center-of-mass energies $E_{\mathrm{cm}} \sim 10^{18}$-- $10^{20}$~eV, placing them firmly in the ultra-high-energy cosmic-ray (UHECR) range. We systematically explore the parameter space of merger remnants, varying black hole mass ($M \sim 20$--$150\,M_\odot$, characteristic of the binary black hole population), dimensionless spin ($χ_f \sim 0.7$--$0.9$), magnetic field strength, and particle angular momenta. Our analysis reveals three distinct acceleration regimes: a gravity-dominated regime ($B < 10^{12}$~G) with negligible magnetic enhancement; a transition regime ($10^{12}~\text{G} \lesssim B \lesssim 10^{13}~\text{G}$) where gravitational and magnetic effects compete; and a magnetic-dominated regime ($B > 10^{13}$~G) where fields amplify collision energies by nearly an order of magnitude. For the 34 gravitational-wave events with high remnant spins ($χ_f > 0.7$), we compute the maximum achievable energies, finding that systems with $χ_f \gtrsim 0.85$ and $M \gtrsim 100\,M_\odot$ can reach $E_{\mathrm{max}} \sim 10^{20}$~eV. Our results establish magnetized binary mergers, particularly black hole--neutron star systems and postmerger black hole remnants formed in binary neutron star coalescences, as promising sources of UHECRs and provide quantitative predictions linking gravitational-wave observables to particle acceleration efficiency.

astro-ph.HE↗

arXiv2026

Probing Low-Luminosity Gamma-Ray Emission from SNR G296.5+10.0 and CCO 1E 1207.4-5209 with CTAO

The acceleration mechanisms of cosmic rays (CR) in supernova remnants (SNRs) and their associated compact central objects (CCOs) remain an open question in high-energy astrophysics. In this study, we perform a modeling of CR transport and gamma-ray emission from SNR G296.5+10.0 and its CCO 1E 1207.4-5209, using the latest public release of the GALPROP code (v57) and focusing, in particular, on the contribution from the CCO. Our simulations predict the contribution of CR from this source to the Galactic flux, accounting for energy losses and particle interaction processes. We find that, under time-evolving scenarios, the environment around SNR G296.5+10.0 and 1E 1207.4-5209 is suitable for CR acceleration and gamma-ray production. The analysis distinguishes between gamma rays produced by hadronic interactions in SNR G296.5+10.0 and by leptonic processes in CCO 1E 1207.4-5209, revealing that each mechanism dominates in different energy bands. We show that the Cherenkov Telescope Array Observatory (CTAO) can detect this emission with a significance of 5σ after 50 h of exposure, providing the first constraints on particle acceleration in this unique CCO-SNR system. These findings suggest that CCOs may be efficient electron accelerators, even in the absence of pulsar wind nebulae, and emphasize the critical role of next-generation observatories such as CTAO in unraveling CR acceleration processes in low-luminosity SNR-CCO systems.

astro-ph.HE↗

arXiv2026

Optical transients from non-explosive double white-dwarf mergers: the case of a central neutron star remnant

Discoveries of ultra-massive magnetic white dwarfs (WDs) and peculiar pulsars have been proposed to originate in double white dwarf (DWD) mergers. There are three possible post-merger central remnants of non-explosive mergers: 1) a stable sub-Chandrasekhar WD; 2) a rapidly rotating super-Chandrasekhar WD; 3) a neutron star (NS). In this work, we explore the thermal transient arising from non-explosive DWD mergers that leave an NS remnant from the prompt collapse of the merged core. The transient is powered by the cooling of the expanding dynamical ejecta, with energy injection from magnetic dipole radiation, which depends on the dipole factor $D = B_d^2/P_0^4$, with $B_d$ and $P_0$ being the surface magnetic field strength and initial rotation period of the newborn NS. We simulate lightcurves in the Legacy Survey of Space and Time (LSST) bands and estimate the horizon and detection rates for these transients across a range of model parameters. We find LSST detection horizons upper limits ranging $30$--$820$ Mpc and corresponding detection rates $10^2$--$10^6$ yr$^{-1}$ for $\log D = 24$--$40$. Accounting for the survey cadence, we find that only configurations with $\log D = 36$--$40$ are detectable within $240$--$760$ Mpc, with detection rates $10^4$--$10^5$ yr$^{-1}$. Combined searches across surveys can compensate for the low cadence and improve the detection rates of fast and less energetic sources. Multi-wavelength campaigns can aid in detecting the spindown radiation at higher energies observable after the optical transient. Observations of these transients will provide direct evidence of the non-explosive DWD mergers, characterise the remnants and progenitor parameters, and the fraction of explosive and non-explosive mergers.

astro-ph.HE↗

arXiv2026

Observational bounds on Dark Matter Admixed Neutron Stars from Gravitational Wave Data

Recent gravitational-wave (GW) observations offer a unique opportunity to probe the fundamental nature of compact objects. A growing body of research has focused on exploring the role of dark matter (DM) through the concept of DM-admixed neutron stars (NSs), where the presence of DM can significantly alter key physical properties of NSs, such as their mass, radius, and tidal deformability, ultimately affecting the predicted GW waveform emitted during binary coalescences. In this work, we present a novel observational test that, for the first time, places constraints on the influence of DM inside NSs using real GW data. By reanalyzing signals from events such as GW230529, GW200115, and GW200105, we derive new upper bounds on the DM fraction, $F_χ$, and particle mass, $m_χ$, under the assumption that DM is described by a scalar field with a self-interaction potential. We find that the upper bound on $F_χ$ depends on the specific binary system under analysis, indicating that different DM configurations can be consistent with observations in different ways. In particular, the event GW190814 may be compatible with a DM halo configuration. In contrast, the other events analyzed (GW230529, GW200105 and GW200115) are consistent with DM forming a core inside the NS, yielding strong upper bounds on $F_χ$. The corresponding values for the mass scale $m_χ$ are also discussed in the text. This work offers a new approach to probing DM in the context of compact NS objects through GW observations.

gr-qc↗

arXiv2026

Double White Dwarf Mergers as Progenitors of Long-Period Transients

There is an ongoing discussion in the literature on the nature of long-period transients (LPTs), radio-emitting sources with periods ranging from hundreds to tens of thousands of seconds. Although some of these objects have been identified as white dwarf (WD) + M-dwarf binaries, this description currently does not fit the entire class. An example is GLEAM-X J162759.5-523504.3 (hereafter GLEAM-X J1627-5235), with a period of 1091 s, for which the lack of an optical counterpart disfavors the presence of such a binary system. In this case, GLEAM-X J1627-5235 could be interpreted as an isolated, massive, fast-rotating, and highly magnetized (~ 1e+9 G) WD pulsar. Its properties are consistent with a carbon-oxygen WD of mass ~1.3 Msun and radius ~2500 km, possibly supported by small-scale multipolar magnetosphere structures that keep it above the death line for WD-pulsars. We assess a double WD merger origin, modeling the post-merger rotational evolution under accretion, propeller, and magnetic braking torques. We find rotational age of ~572 Myr for GLEAM-X J1627-5235, i.e., the post-merger time required to reach its observed period. This result is consistent with current optical upper limits for GLEAM-X J1627-5235 and support the WD pulsar interpretation for this source. We also discuss how the same model can apply to other LPTs.

astro-ph.HE↗

arXiv2025

Exploring Temperature Influences on Gravitational Wave Production in Binary White Dwarfs

This study investigates the conditions under which gravitational waves (GWs) are emitted during the merger of hot white dwarfs (WDs) in a binary system. Traditionally, these systems consist of two low-mass stars or a more massive WD paired with a less massive companion. In addition, recent work has investigated the possibility that double white dwarf (DWD) mergers are possibly the leading formation channel of massive, rapidly rotating, high-field magnetic WDs , particularly SDSS J221141.80 + 113604.4 (hereafter J2211+1136) and ZTF J190132.9 + 145808.7 (hereafter J1901 + 14588). Motivated by these findings and the Laser Interferometer Space Antenna (LISA) prospects, this study aims to calculate the tidal Love number, the dimensionless tidal deformability, as well as the frequency and amplitude of GWs of hot WDs. The results indicate that the tidal deformability is more pronounced in stars with higher central temperatures and lower masses, which would lead to reduced emission of GWs. In contrast, more massive stars exhibit less deformability, making them prime candidates for generating stronger GWs. Additionally, the analysis of frequency and amplitude reveals that the frequencies of high-mass binaries are smaller and evolve more rapidly, reaching a limit that aligns with the operational detection capabilities of LISA during its initial phase.

astro-ph.SR↗

arXiv2024

X-ray pulsed light curves of highly compact neutron stars as probes of scalar-tensor theories of gravity

The strong gravitational potential of neutron stars (NSs) makes them ideal astrophysical objects for testing extreme gravity phenomena. We explore the potential of NS X-ray pulsed lightcurve observations to probe deviations from general relativity (GR) within the scalar-tensor theory (STT) of gravity framework. We compute the flux from a single, circular, finite-size hot spot, accounting for light bending, Shapiro time delay, and Doppler effect. We focus on the high-compactness regime, i.e., close to the critical GR value GM/(Rc^2) = 0.284, over which multiple images of the spot appear and impact crucially the lightcurve. Our investigation is motivated by the increased sensitivity of the pulse to the scalar charge of the spacetime in such high compactness regimes, making these systems exceptionally suitable for scrutinizing deviations from GR, notably phenomena such as spontaneous scalarization, as predicted by STT. We find significant differences in NS observables, e.g., the flux of a single spot can differ up to 80% with respect to GR. Additionally, reasonable choices for the STT parameters that satisfy astrophysical constraints lead to changes in the NS radius relative to GR of up to approximately 10%. Consequently, scalar parameters might be better constrained when uncertainties in NS radii decrease, where this could occur with the advent of next-generation gravitational wave detectors, such as the Einstein Telescope and LISA, as well as future electromagnetic missions like eXTP and ATHENA. Thus, our findings suggest that accurate X-ray data of the NS surface emission, jointly with refined theoretical models, could constrain STTs.

astro-ph.HE↗

arXiv2024

Prospects for gamma-ray emission from magnetar regions in CTAO observations

Recent multi-wavelength observations have highlighted magnetars as significant sources of cosmic rays, particularly through their gamma-ray emissions. This study examines three magnetar regions - CXOU J171405.7-31031, Swift J1834-0846, and SGR 1806-20 - known for emitting detectable electromagnetic signals. We assess the detectability of these regions using the upcoming Cherenkov Telescope Array Observatory (CTAO) by conducting an ON/OFF spectral analysis and compare the expected results with existing observations. Our findings indicate that CTAO will detect gamma-ray emissions from these three magnetar regions with significantly reduced emission flux errors compared to current instruments. In special, the study shows that the CXOUJ1714-3810 and SwiftJ1834-0846 magnetar regions can be observed by the full southern and northern CTAO arrays in just five hours of observation, with mean significances above $10 \,σ$ and $30 \,σ$, respectively. This paper discusses the regions analyzed, presents key results, and concludes with insights drawn from the study.

astro-ph.HE↗

arXiv2024

Impact of stratified rotation on the moment of inertia of neutron stars

Rigid (Uniform) rotation is usually assumed when investigating the properties of mature neutron stars (NSs). Although it simplifies their description, it is an assumption because we cannot observe the NS's innermost parts. Here, we analyze the structure of NSs in the simple case of ''almost rigidity,'' where the innermost and outermost parts rotate with different angular velocities. This is motivated by the possibility of NSs having superfluid interiors, phase transitions, and angular momentum transfer during accretion processes. We show that, in general relativity, the relative difference in angular velocity between different parts of an NS induces a change in the moment of inertia compared to that of rigid rotation. The relative change depends nonlinearly on where the angular velocity jump occurs inside the NS. For the same observed angular velocity in both configurations, if the jump location is close to the star's surface-which is possible in central compact objects (CCOs) and accreting stars-the relative change in the moment of inertia is close to that of the angular velocity (which is expected due to total angular momentum aspects). If the jump occurs deep within the NS, for instance, due to phase transitions or superfluidity, smaller relative changes in the moment of inertia are observed; we found that if it is at a radial distance smaller than approximately $40\%$ of the star's radius, the relative changes are negligible. Additionally, we outline the relevance of systematic uncertainties that nonrigidity could have on some NS observables, such as radius, ellipticity, and the rotational energy budget of pulsars, which could explain the X-ray luminosity of some sources. Finally, we also show that non-rigidity weakens the universal $I$-Love-$Q$ relations.

astro-ph.HE↗

arXiv2024

Evidence for 3XMM J185246.6+003317 as a massive magnetar with a low magnetic field

3XMM J185246.6+003317 is a transient magnetar located in the vicinity of the supernova remnant Kes\,79. So far, observations have only set upper limits to its surface magnetic field and spindown, and there is no estimate for its mass and radius. Using ray-tracing modelling and Bayesian inference for the analysis of several light curves spanning a period of around three weeks, we have found that it may be one of the most massive neutron stars to date. In addition, our analysis suggests a multipolar magnetic field structure with a subcritical field strength and a carbon atmosphere composition. Due to the time-resolution limitation of the available light curves, we estimate the surface magnetic field and the mass to be $\log_{10} (B/{\rm G}) = 11.89^{+0.19}_{-0.93}$ and $M=2.09^{+0.16}_{-0.09}$~$M_{\odot}$ at $1σ$ confidence level, while the radius is estimated to be $R=12.02^{+1.44}_{-1.42}$ km at $2σ$ confidence level. They were verified by simulations, i.e., data injections with known model parameters, and their subsequent recovery. The best-fitting model has three small hot spots, two of them in the southern hemisphere. These are, however, just first estimates and conclusions, based on a simple ray-tracing model with anisotropic emission; we also estimate the impact of modelling on the parameter uncertainties and the relevant phenomena on which to focus in more precise analyses. We interpret the above best-fitting results as due to accretion of supernova layers/interstellar medium onto 3XMM J185246.6+003317 leading to burying and a subsequent re-emergence of the magnetic field, and a carbon atmosphere being formed possibly due to hydrogen/helium diffusive nuclear burning. Finally, we briefly discuss some consequences of our findings for superdense matter constraints.

astro-ph.HE↗

arXiv2024

Binary coalescences as sources of Ultra-High Energy Cosmic Rays

Binary coalescences are known sources of gravitational waves (GWs) and they encompass combinations of black holes (BHs) and neutron stars (NSs). Here we show that when BHs are embedded in magnetic fields ($B$s) larger than approximately $10^{10}$ G, charged particles colliding around their event horizons can easily have center-of-mass energies in the range of ultra-high energies ($\gtrsim 10^{18}$ eV) and escape. Such B-embedding and high-energy particles can take place in BH-NS binaries, or even in BH-BH binaries with one of the BHs being charged (with charge-to-mass ratios as small as $10^{-5}$, which do not change GW waveforms) and having a residual accretion disk. Ultra-high center-of-mass energies for particle collisions arise for basically any rotation parameter of the BH when $B \gtrsim 10^{10}$ G, meaning that it should be a common aspect in binaries, especially in BH-NS ones given the natural presence of a $B$ onto the BH and charged particles due to the NS's magnetosphere. We estimate that up to millions of ultra-high center-of-mass collisions may happen before the merger of BH-BH and BH-NS binaries. Thus, binary coalescences can also be efficient sources of ultra-high energy cosmic rays (UHECRs) and constraints to NS/BH parameters would be possible if UHECRs are detected along with GWs.

astro-ph.HE↗

arXiv2022

An updated view and perspectives on high-energy gamma-ray emission from SGR J1935+2154 and its environment

SGR J1935+2154 was discovered in 2016 and is currently one of the most burst-active Soft Gamma-ray Repeaters (SGR), having emitted several X-ray bursts in recent years. In one of our previous articles, we investigated the contribution to high-energy and very high-energy gamma-ray emission (VHE, $E > 100$ GeV) due to cosmic-ray acceleration of SNR G57.2+0.8 hosting SGR J1935+2154 using the GALPROP propagation code. However, follow-up observations of SGR 1935+2154 were made for 2 hours on April 28, 2020, using the High Energy Stereoscopic System (H.E.S.S.). The observations coincide with X-ray bursts detected by INTEGRAL and Fermi/Gamma-ray Burst Monitor (GBM). These are the first high-energy gamma-ray observations of an SGR in a flaring state, and upper limits on sustained and transient emission have been derived. Now that new H.E.S.S. observations have been made, it is interesting to update our model with respect to these new upper limits. We extend our previous results to a more general situation using the new version of GALPROP. We obtain a hadronic model that confirms the results discussed by H.E.S.S.. This leads to an optimistic prospect that cosmic ray gamma rays from SGR J1935+2154 can contribute to the overall gamma energy density distribution and in particular to the diffusion gamma rays from the Galactic center.

astro-ph.HE↗

arXiv2021

Mass limits of the extremely fast-spinning white dwarf CTCV J2056-3014

CTCV J2056--3014 is a nearby cataclysmic variable with an orbital period of approximately $1.76$ hours at a distance of about $853$ light-years from the Earth. Its recently reported X-ray properties suggest that J2056-3014 is an unusual accretion-powered intermediate polar that harbors a fast-spinning white dwarf (WD) with a spin period of $29.6$ s. The low X-ray luminosity and the relatively modest accretion rate per unit area suggest that the shock is not occurring near the WD surface. It has been argued that, under these conditions, the maximum temperature of the shock cannot be directly used to determine the mass of the WD (which, under the abovementioned assumptions, would be around $0.46$ $M_\odot$). Here, we explore the stability of this rapidly rotating WD using a modern equation of state (EoS) that accounts for electron--ion, electron--electron, and ion--ion interactions. For this EoS, we determine the mass density thresholds for the onset of pycnonuclear fusion reactions and study the impact of microscopic stability and rapid rotation on the structure and stability of WDs, considering them with helium, carbon, oxygen, and neon. From this analysis, we obtain a minimum mass for CTCV J2056--3014 of $0.56~M_\odot$ and a maximum mass of around $1.38~M_\odot$. If the mass of CTCV J2056--3014 is close to the lower mass limit, its equatorial radius would be on the order of $10^4$~km due to rapid rotation. Such a radius is significantly larger than that of a nonrotating WD of average mass ($0.6\, M_\odot$), which is on the order of $7\times 10^3$~km. The effects on the minimum mass of J2056-3014 due to changes in the temperature and composition of the stellar matter were found to be negligibly small.

astro-ph.SR↗

arXiv2021

High-energy gamma-ray emission from SNR G57.2+0.8 hosting SGR J1935+2154

In recent years some efforts have been made to identify active sources capable of accelerating particles up to $10^{15}$ eV, known as PeVatrons. Measurements of TeV ($10^{12}$ eV) gamma-rays from supernova remnants (SNRs) have shown that efficient particle acceleration can occur in SNR diffusive shocks. In this paper, we obtain the contribution to the high energy and very-high-energy gamma-ray (VHE, $E > 100$ GeV) emission due to cosmic-ray acceleration from SNR G57.2+0.8 hosting the Soft Gamma Repeater (SGR) J1935+2154 with the use of the GALPROP code. To do so, we take into account the SNR + SGR association as a single source close to the Galactic center. We propose that the above setting can provide a more comprehensive scenario for the generation of GeV-TeV gamma-rays. We also discuss the contribution from the SNR G57.2+0.8 and SGR J1935+2154 region to the diffusive TeV energy gamma-ray emission from the Galactic center.

astro-ph.HE↗

arXiv2021

Gravitational Wave Emission by Accretion of Matter and Magnetic Deformation in Fast Rotating White Dwarfs

We discuss some aspects of Sousa et al.(2020a, 2020b) concerning two mechanisms of gravitational wave (GW) emission in fast-spinning white dwarfs (WDs): accretion of matter and magnetic deformation. In both cases, the GW emission is generated by an asymmetry around the rotation axis of the star. However, in the first case, the asymmetry is due to the amount of accreted matter in the magnetic poles, while in the second case it is due to the intense magnetic field. We have estimated the GW amplitude and luminosity for three binary systems that have a fast-spinning magnetized WD, namely, AE Aquarii, AR Scorpii and RX J0648.0-4418. In addition, we applied the magnetic deformation mechanism for SGRs/AXPs described as WD pulsars. We found that, for the first mechanism, the systems AE Aquarii and RX J0648.0-4418 can be observed by the space detectors BBO and DECIGO if they have an amount of accreted mass of $δm \geq 10^{-5}M_{\odot }$. For the second mechanism, the three systems studied require that the WD has a magnetic field above $\sim 10^{9}$ G to emit GWs that can be detected by BBO. Furthermore, we found that some SGRs/AXPs as WD pulsars can be detected by BBO and DECIGO, whereas SGRs/AXPs as highly magnetized neutron stars are far below the sensitivity curves of these detectors.

astro-ph.SR↗

arXiv2021

Observational Constraints on the Pulsar Wind Model: The Cases of Crab and Vela

As is well known, pulsars are extremely stable rotators. However, although slowly, they spin down thanks to brake mechanisms, which are in fact still subject of intense investigation in the literature. Since pulsars are usually modelled as highly magnetized neutron stars that emit beams of electromagnetic radiation out of their magnetic poles, it is reasonable to consider that the spindown has to do with a magnetic brake. Although an interesting and simple idea, a pure magnetic brake is not able to adequately account for the spindown rate. Thus, many alternative spindown mechanisms appear in the literature, among them the pulsar wind model, where a wind of particles coming from the pulsar itself can carry part of its rotational kinetic energy. Such a spindown mechanism depends critically on three parameters, namely, the dipole magnetic field $B$, the angle between the magnetic and rotation axes $(ϕ)$, and the density of primary particles $(ζ)$ of the pulsar's magnetosphere. Differently from a series of articles in this subject, we consider for the first time in the literature a statistical modelling which includes a combination of a dipole magnetic and wind brakes. As a result, we are able to constrain the above referred parameters in particular for Crab and Vela pulsars.

astro-ph.HE↗