Theoretically Motivated Search and Detection of Non-thermal Pulsations from PSRs J1747-2958, J2021+3651, and J1826-1256

In a recent article we introduced a model for the high-energy emission of pulsars (Torres 2018). According to this model, pulsed spectra detected in gamma-rays and/or X-rays is produced via synchro-curvature radiation, and can be described using just three physical parameters (accelerating electric field, contrast, and magnetic gradient). Interestingly, it was shown that if the model parameters were adjusted to describe the gamma-ray data, the resulting spectral energy distribution (SED) at lower X-ray energies is a relatively good representation of the spectra therein. In particular, in all cases analyzed by Torres (2018) for which both X-ray and gamma-ray data were available, a fit only to the gamma-ray part of the spectra would miss the measured X-ray flux level by less than a factor of ∼2. This was the case even when one of the parameters—on which we comment below—was fixed to an average value (gamma-ray data alone is not constraining enough to fix it), and thus the spectral shape is completely determined by two parameters. Thus, the model was proposed to be a tool to distinguish which of the pulsars detected in gamma-rays could be good candidates to appear in X-rays. A list of pulsars appearing in the Second Fermi-LAT Pulsar Catalog (Abdo et al. 2013, 2PC hereafter) for which the theoretically expected X-ray flux is larger than 10⁻¹³ erg cm⁻² s⁻¹ at 10 keV, was provided in Torres (2018).

Enlarging the non-thermal X-ray pulsar population is clearly an important task: out of more than 200 gamma-ray pulsars known in the Galaxy, only less than 10% have been detected in X-rays (see e.g., Kuiper & Hermsen 2015). Enlarging the sample of X-ray pulsars is crucial for understanding their global properties, and characterizing how these compare with those of other pulsar subsamples.

In this Letter, we take three pulsars out of the list in Torres (2018) and analyze the existing publicly available X-ray data for them. We find that pulsed radiation is indeed found for all of these pulsars, and provide an analysis of their spectral properties, comparing them with model predictions taking into account (or not) the now-determined X-ray data.

We analyze archival XMM-Newton and Chandra data. The Chandra data were reduced using CIAO version 4.7 and CALDB version 4.7.7. We reprocessed the Chandra data to level = 2 and removed periods of high background or flaring appearing in the observations. XMM-Newton data were reduced with the XMM-Newton Science Analysis System (SAS, version 16.1.0). Standard pipeline tasks (epproc for PN and emproc for MOS data) were used to process the raw observation data files (ODFs). XMM-Newton data were also filtered to avoid the periods of hard X-ray background flares. All of the X-ray spectra were rebinned to have at least 25 counts for each channel. Spectral analysis was carried out with XSPEC version 12.8.2.

From the spectral modeling of each pulsar we constructed the corresponding phase-averaged SED. These SEDs are produced by Xspec using the corresponding unabsorbed spectral models (i.e., the corresponding fitted pulsar's power-law models in Table 1) and the real exposure times and background.

Table 1.  Spectral Fits to the X-Ray Pulsars

Source Name	NH	kT	Spectral Index Γ	Unabsorbed Flux in 0.2–10 keV	χ2/D.O.F.
	(1022 cm−2)	(keV)		(10−13 erg cm−2 s−1)
J1747–2958	${2.58}_{-0.24}^{+0.25}$	⋯	1.25 ± 0.12	23.46 ${}_{-1.04}^{+1.24}$	65.08/56
J2021+3651	0.69 (fixed)	0.15 ± 0.01	1.72 ± 0.30	0.78 ${}_{-0.09}^{+0.12}$	38.34/36
J1826–1256	${2.28}_{-0.42}^{+0.49}$	⋯	${1.31}_{-0.20}^{+0.22}$	3.32 ${}_{-0.31}^{+0.47}$	17.29/16
	${0.79}_{-0.26}^{+0.31}$	${1.54}_{-0.12}^{+0.13}$	⋯	2.01 ± 0.12	17.46/16

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For timing analysis, all times of arrivals of X-ray photons were barycenter-corrected using the position of the pulsars and the latest JPL DE405 Earth ephemeris. The pulsations were searched for via an ${Z}_{n}^{2}$-test procedure around the corresponding periods expected from the Fermi-LAT gamma-ray ephemeris or from previous publications, with Fourier resolution (Buccheri et al. 1983), and with the number of harmonics n varied from 1 to 5. We start from ${Z}_{1}^{2}$-test and increase n until a significant signal is detected.

As we shall see below, the face-value periods that we find in X-rays—considering the uncertainty coming only from the Fourier resolution—are close, but not exactly at the gamma-ray pulsation period found earlier with Fermi-LAT. In addition to intrinsic phenomena in the pulsars, and the timing noise being very large in at least two of the sources of our sample, we may consider other possible uncertainties that may lead to such differences. Among them, we consider the particular instrument timing resolution and accuracy, possible fluctuations of observed counts, and also the Fourier resolution.

To consider an uncertainty according to the corresponding timing resolution and accuracy, the arrival time of each event was uniformly sampled within its corresponding uncertainty to produce a simulated event list, which was later analyzed via the ${Z}_{n}^{2}$-test. This procedure was repeated 10,000 times for each source, leading to the estimation of a related period uncertainty. Similarly, to estimate the period uncertainty resulting from fluctuations of observed counts, we sampled within the phase bins of the corresponding folded lightcurves, assuming a Poisson distribution. Then the phases of sampled events are converted into arrival times in the observation to produce a simulated event list, which was again later analyzed via the ${Z}_{n}^{2}$-test. This process was also repeated 10,000 times for each source, leading to the estimation of a related period uncertainty. The errors in the periods were estimated as the sum in quadrature of the simulation obtained uncertainties and the Fourier resolution, with the former dominating the period error. In the folded pulse profile, an arbitrary reference epoch T₀ was set to the start of respective observations. The pulse profile was modeled by a sinusoidal function with n harmonics, while n was decided by the ${Z}_{n}^{2}$-test. The pulse fraction is defined as the ratio between the semi-amplitude of fundamental sinusoid and the average count rate. All of the uncertainties quoted in the spectral analysis are at 1σ confidence level. Considering these uncertainties we find that the Fermi-LAT periods found earlier and the X-ray periods that we detect are compatible in all three cases studied.

2.1. PSR J1747–2958: Detection of the Pulsar in the "Mouse" Nebula

J1747–2958 is a 98.8 ms young pulsar, detected in radio and gamma-rays (see, e.g., Camilo et al. 2002; 2PC) associated with the axisymmetric nebula G359.23-0.82 (referred to as the Mouse). The Mouse nebula was discovered and studied in detail with Chandra (Gaensler et al. 2004; Klingler et al. 2018). In Gaensler et al. (2004), the head region of the Mouse nebula was decomposed into two Gaussian components. The first Gaussian was identified as a point-like source possibly associated with the pulsar itself, albeit no X-ray pulsations were earlier reported.

Chandra/High Resolution Camera (HRC) observed PSR J1747–2958 with 58 ks exposure on 2008 February 7 (obsID 9106), providing sufficient timing accuracy (∼11.1 ms, which is the median of time differences between observed events).⁷ We considered this data set and searched for pulsations via the ${Z}_{n}^{2}$-test procedure. Photons were extracted with a radius of 1 arcsec using the position of the first Gaussian reported in Gaensler et al. (2004), which was proposed to be the pulsar magnetospheric emission (Gaensler et al. 2004). We found a peak at P = 0.09878(10) s (90% uncertainty) with a ${Z}_{2}^{2}$ statistic of 39.33 (Figure 1), which corresponds to a significance ∼5.3σ after trials correction. The trials are the different values of n (1 and 2) considered in ${Z}_{n}^{2}$-test. The latest Fermi-LAT gamma-ray ephemeris⁸ covers a range from 2008 August 13 to 2013 October 7. We extrapolated the gamma-ray ephemeris to the epoch of the X-ray observation. The expected pulse period is P = 0.098823858(4) s, which is compatible within the uncertainty of the period that we detected in X-rays. Additionally, we note that the timing noise is not considered in the pulse period prediction from gamma-ray ephemeris. Based on current LAT Gamma-ray Pulsar Timing Models, the timing noise of PSR J1747–2958 may reach values as large as ∼2 × 10⁻⁵ s during the X-ray observation, leading to additional uncertainties in the predicted period. The HRC data were folded at the detected period and the pulse profile is also shown in Figure 1, yielding a pulse fraction of 20.9% ± 4.7%. Klingler et al. (2018) carried out a search of X-ray pulsations in the same Chandra/HRC data, but reported no detection. This is possibly due to different extraction regions, and/or a different number of events considered in the timing analysis. However, we note that the pulse fraction derived in our analysis is consistent with the upper limit calculated by Klingler et al. (2018; 34%).

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We carried out the spectral analysis of PSR J1747–2958 using archival Chandra observations (obsIDs 14519, 14520, 14521, 14522). The pulsar spectrum was extracted from an elliptical region centered on the first Gaussian component reported in Gaensler et al. (2004), having a major axis of 1.1 arcsec, an ellipticity of 0.25, and a position angle of 118° (see Figure 2, top-left panel). To exclude the contamination of the nebula, the background spectrum was extracted from an elliptical region centered on the second Gaussian component with a major axis of 2.4 arcsec, an ellipticity of 0.12, and a position angle of 91°, excluding the region used for the pulsar spectrum extraction. To increase the statistics, spectra from different observations are combined using the task combine_spectra. We have tested that a simultaneous fitting to the spectra from different observations leads to consistent results. The combined spectra of PSR J1747–2958 could be well fitted with an absorbed power law (see Figure 3, reduced χ² = 1.16, D.O.F = 56). The fitted N_H and flux level (Table 1) are consistent with the value reported in Gaensler et al. (2004). An absorbed blackbody could not lead to an acceptable fit (reduced χ² = 1.7, D.O.F = 56). We also tested an absorbed power law plus blackbody model. However, the blackbody component is not significantly required according to the F-test (significance below 3σ). Thus, we conclude that the spectrum that we observe from PSR J1747–2958 is non-thermal.

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2.2. J2021+3651: Confirmed Detection of the Pulsar in the "Dragonfly" Nebula

2.3. Detection of PSR J1826–1256

Figure 4 shows the derived non-thermal SEDs for all three pulsars in comparison with models. The first two rows show the initial prediction. The model is described in detail in Torres (2018) and references therein. It encompasses two essential ingredients. On the one hand, it contains a dynamical, time/position-dependent description of particle trajectories in an accelerating region in the outer part of the magnetosphere, where particles are subject to radiative losses. On the other hand, it features a computation of the spectrum emitted at each position, while particles traverse the accelerating region. For both losses and radiation, the full synchro-curvature process is considered (see Viganò et al. 2015b for a description). Just three physical parameters (which we mention below), together with the timing properties of the pulsar (i.e., the measured period and period derivative), define the shape of spectrum.

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The red line in each panel of Figure 4 is the fitted model in all cases. The panels in the second row show a zoom of the region in the X-ray band for each of the corresponding first-row panels. In all three panels of the first row the model is obtained as a fit to the gamma-ray data only. Such data are lying at energies that are six orders of magnitude larger than the spectrum that we have now determined. These red lines represent the theoretical prediction that we used in order to consider that these three pulsars were actually detectable in X-rays. In deriving such predictions, the model used only two physical parameters: the accelerating electric field, ${E}_{| | }$, and the contrasting ${({x}_{0}/{R}_{l})}^{-1}$. The latter is a description of the uniformity of the particle distribution along the accelerating region. The third physical parameter of the model, the magnetic gradient, b, representing a measure of how fast the magnetic field declines along the particle trajectory, was kept fixed. The value of b in these initial fits was assumed to be 2.85, and not fitted against. This value is the average found for the pulsars detected in non-thermal X-rays (above 20 keV) and gamma-rays studied by Torres (2018). As it was discussed earlier (Viganò et al. 2015a; Torres 2018), having only gamma-ray above 100 MeV makes for a difficult distinction among different values of b. The values of these parameters are shown in Figure 4. The agreement between the X-ray predictions of these models and the determined spectral data is impressive, which confirms that the model works well to select which pulsars among those detected in gamma-rays are detectable in X-rays. This fact can then be used to further enlarge, as we do here, the sample of non-thermal pulsars detected in the X-rays, which is still small (see e.g., Kuiper & Hermsen 2015).

The third row of Figure 4 shows the model fits obtained considering also the X-ray data, whereas the fourth row zooms into the X-ray region. These fits have a free value of magnetic gradient, and were obtained spanning uniformly on ${E}_{| | }$, b, and x₀. These three physical parameters provide a correct description of the whole multiwavelength data set. All three fits can cope well with both sets of data in such different energy regimes, confirming that the model is generally applicable. In all three cases, whereas the values of the accelerating electric field ${E}_{| | }$ and the contrast ${({x}_{0}/{R}_{l})}^{-1}$ are roughly unchanged from the gamma-ray-only fits, the chosen magnetic gradients are larger. The value for J2021+3651 (b = 3.30) is in fact the largest of the magnetic gradients found till now for all X-ray and gamma-ray detected pulsars (see Torres 2018), with all three being comparable to Vela (b = 3.25) or PSR J2022+3842 (b = 3.10). Larger values of b make the spectrum softer at X-ray energies (predicting larger fluxes at soft X-ray energies; see Supplementary Figure 2 of Torres 2018). If more normal pulsars would be better described by values of magnetic gradients larger than 2.85, the current average value, the number of possible X-ray detectable pulsars will increase. This is something that future studies using new samples of gamma-ray pulsars (e.g., the forthcoming Third Fermi Pulsar Catalog) should take into account.

We acknowledge the support from The National Key Research and Development Program of China (2016YFA0400800), the Spanish grants AYA2015-71042-P, SGR2017-1383, AYA2017-92402-EXP, iLink 2017-1238, and the National Natural Science Foundation of China via NSFC-11473027, NSFC-11503078, NSFC-11673013, NSFC-11733009, NSFC-U1838201, XTP project XDA 04060604 and the Strategic Priority Research Program "The Emergence of Cosmological Structures" of the Chinese Academy of Sciences, grant No. XDB09000000. Jian Li acknowledges the support from the Alexander von Humboldt Foundation. Alessandro Papitto acknowledges the agreements ASI-INAF I/037/12/0 and ASI-INAF 2017-14-H.O, and the Marie Skłodowska-Curie grant agreement 660657-TMSP-H2020-MSCA-IF-2014. We acknowledge the support of the PHAROS COST Action (CA16214). We acknowledge discussions with Dr. Long Ji, and help from the XMM-Newton & Chandra Helpdesk staff. Work at NRL is supported by NASA.