A new X-ray census of rotation powered pulsars

Various mechanisms have been proposed to explain the generation of X-ray emission from pulsars. For some young pulsars, surface thermal radiation (e.g. “Magnificent Seven”, “Central Compact Objects”) or magnetic field decay (e.g. “Rotating Radio Transient” and Magnetars) may contribute to the X-ray emission (Torii et al., 1998; Gotthelf et al., 2013; Chang et al., 2023). In certain cases, this results in X-ray luminosity that exceeds the pulsar’s spin-down luminosity. X-ray emission from Be/Gamma-ray Binaries is thought to arise from the shock between the stellar wind and the pulsar wind (Johnston et al., 1992; Miller et al., 2013; Weng et al., 2022). Alternatively, in this study, we focus exclusively on RPPs where rotational energy is the dominant source of X-ray emission. The non-thermal power-law component of their emission can be interpreted as magnetospheric radiation. The origin of the thermal emission, however, remains debated and is likely due to the bombardment of charged particles returning from the magnetosphere onto the polar cap region.

As a leading international facility in radio astronomy, the ATNF catalog offers a comprehensive sample of pulsars, encompassing all published RPPs while excluding accretion-powered pulsars. In this work, we search for X-ray counterparts of 3,630 pulsars listed in the ATNF pulsar catalog v2.3.0³³3https://www.atnf.csiro.au/people/pulsar/psrcat/ (Manchester et al., 2005) by using the XMM-Newton Serendipitous Source Catalog ⁴⁴4https://heasarc.gsfc.nasa.gov/W3Browse/xmm-newton/xmmssc.html (4XMM-DR13 Version, Webb et al., 2020), the Chandra Source Catalog Release 2.0 ⁵⁵5https://cxc.cfa.harvard.edu/csc2.1/index.html (CSC 2.0, Evans et al., 2010, 2024). If the angular separation between a pulsar and an X-ray source, with a detection confidence level greater than 3-sigma, satisfies the condition $\delta=\sqrt{(RA_{\rm X}-RA_{\rm R})^{2}cos^{2}(DEC_{\rm X})+(DEC_{\rm X}-DEC_{\rm R})^{2}}<R_{\rm X}-R_{\rm R}$, we consider the X-ray source to be the counterpart of the pulsar. $RA$ and $DEC$ denote the right ascension and declination of a source, $R_{\rm X}$ and $R_{\rm R}$ represent the positional uncertainties at the 2-sigma confidence level for the X-ray source and the pulsar, respectively. For most radio-loud pulsars, timing procedure leads to achieve positional accuracy down to milliarcseconds, and we use the Third Fermi/LAT Catalog of Gamma-ray Pulsars Catalog (3PC)⁶⁶6https://fermi.gsfc.nasa.gov/ssc/data/access/lat/3rd_PSR_catalog/ (Smith et al., 2023) to obtain more precise positions for radio-quiet and radio-faint pulsars. The typical positional accuracy of XMM-Newton serendipitous point source detections is generally less than 1.57^′′, and the on-axis spatial resolution of Chandra data is sub-arcsecond. However, the X-ray positional accuracy can sometimes be overestimated, leading to some counterparts being missed. We verified the results by cross-referencing with compiled literature tables, identifying 19 counterparts for 8 normal pulsars (NPs) and 11 millisecond pulsars (MSPs) (Li et al., 2008; Lee et al., 2018; Coti Zelati et al., 2020; Chang et al., 2023). In sum, by positional cross-matching, we identify 121 counterparts of pulsars, comprising 65 MSPs and 56 NPs.

In some cases, the positions of X-ray sources in the two X-ray catalogs are imprecise. For sources located in supernova remnants (SNRs) or pulsar-wind nebulae (PWNe), the extended X-ray emission complicates accurate positioning. Because MSPs in globular clusters (GCs) are relatively dense and faint, they are challenging to resolve using the pipelined processing methods applied in the Chandra CSC 2.0 catalog. However, some studies have conducted more detailed analyses of these sources, yielding additional X-ray counterparts (e.g. Hsiang & Chang, 2021; Zhao & Heinke, 2022). We collected data from the literature on 68 MSPs in 29 GCs and 42 NPs associated with SNRs or PWNe, incorporating these into our sample as a significant supplement. However, their luminosities are reported across the entire X-ray band, as detailed spectral analyses are lacking.

In total, we identify 231 X-ray counterparts of pulsars, including 98 NPs and 133 MSPs. Their properties, spanning radio, X-ray, and Gamma-ray bands, are listed in Table 2.

In principle, an association can be unambiguously confirmed as a real counterpart only when X-ray pulsations are detected. However, in most cases, the time resolution of imaging X-ray observations is insufficient for pulsation searches. The time resolution of Chandra data is $\sim$ 3.2 s⁷⁷7https://cxc.harvard.edu/cdo/about_chandra/, and the time resolution of XMM-Newton EPIC-pn’s full frame mode observation data is 73.4 ms⁸⁸8https://heasarc.gsfc.nasa.gov/docs/xmm/uhb/epicmode.html. Nevertheless, X-ray pulsations have been reported for 51 NPs and 18 MSPs in the literature (see Table 2).

We also estimate the probability of positional coincidence using the log$N$ - log$S$ distribution at the Galactic center (Muno et al., 2003). According to the discussion in Section 4.1 of Muno et al. (2003) (see also Figure 10 in their paper), the surface density of sources is extremely high, $\sim$15,000 sources deg^-2 above the flux limit of $3\times 10^{-15}$ erg cm^-2 s^-1 in the 2.0–8.0 keV range. This corresponds to $\sim$0.009 contaminating sources within the 1.57^′′ location error circle (the typical XMM-Newton positional resolution). However, we argue that the chance coincidence probability is likely much lower for the following reasons: 1. The fluxes of pulsars analyzed in this work are mostly greater than $3\times 10^{-15}$ erg cm^-2 s^-1, with a median value of $7\times 10^{-14}/9\times 10^{-15}$ erg cm^-2 s^-1 for NPs and MSPs; 2. The surface density at the pulsar position should be much smaller than at the Galactic center; 3. Chandra sources exhibit even smaller positional uncertainties and a lower likelihood of coincidence. Moreover, more refined analyses have been performed on a large proportion of MSPs in GCs (e.g. Hsiang & Chang, 2021; Zhao & Heinke, 2022, Section 2.1), and sources detected within PWNe can also be considered secure associations. Therefore, we conclude that the probability of positional coincidence is very low, even if there is an absence of X-ray pulsations.

According to the magnetic dipole radiation model for pulsars, rotation power is assumed to be transformed into dipole radiation loss energy. The energy loss rate $\dot{E}$ can be expressed as $\dot{E}=\frac{4\pi^{2}I\dot{P}}{P^{3}}$, where a typical moment of $I=10^{45}$ g cm² is assumed. Other parameters can be derived from the observed rotational period $P$ and its derivative $\dot{P}$, as described by the following relations: the characteristic age $\tau=\frac{P}{2\dot{P}}$, the surface magnetic field strength $B_{\rm surf}=3.2\times 10^{19}(P\dot{P})^{0.5}$ G, the magnetic field at light cylinder $B_{\rm lc}=2.9\times 10^{8}P^{-2.5}\dot{P}^{0.5}$ G.

In this work, distances are estimated using the Dispersion measure (DM) based on the YMW16 electron distribution model (Yao et al., 2017). For radio-quiet or radio-faint Gamma-ray pulsars, we adopt pseudo-distances inferred from the spin-down power $\dot{E}$ and the energy flux $G_{\rm 100}$ above 100 MeV. As a result, upper limits on luminosities across all bands are provided for these sources, with relevant references listed in Table 2. The luminosities of radio-loud pulsars in the radio and Gamma-ray bands are taken from the ATNF catalog and Fermi Pulsar catalog. However, it is important to note that the uncertainty in luminosity is typically dominated by the uncertainty in the DM distance, and could deviate significantly from the actual value. For example, the distance of 93 pc for PSR J1057-5226 is derived using the YMW16 model (Yao et al., 2017), while NE2001 model places it at 720 pc (Cordes & Lazio, 2002), showing an order-of-magnitude difference (Kerr et al., 2018). Moreover, it has been cautioned that the DM distances for some pulsars are over-estimated and questionable, especially for the pulsars in the direction of the Local Arm or the tangential direction of spiral arms (Han et al., 2021). Following Chang et al. (2023), we adopt an uncertainty of 40% in distances in this work. The X-ray luminosity is derived from flux and distance, and its error is calculated based on the error transfer formula.

It has been reported that, the X-ray luminosity of RPPs is strongly correlated with $\dot{E}$, but is orders of magnitude lower than $\dot{E}$ ($L_{\rm X}/\dot{E}\sim 10^{-6}-10^{-1}$, Arzoumanian et al., 2011; Rea et al., 2012; Vahdat et al., 2022; Chang et al., 2023). The correlation was first explored in the soft X-ray band (0.1–2.4 keV) by Becker & Truemper (1997) using ROSAT data, leading to the relation $L_{\rm X}\propto 10^{-3}\dot{E}$. Possenti et al. (2002) later extended this analysis to the 2-10 keV band, obtaining $L_{\rm X}\propto\dot{E}^{1.34}$. Li et al. (2008) made a significant advancement by separating the X-ray luminosity contributions from pulsars and their associated PWNe, reporting $L_{\rm X,psr}\propto\dot{E}^{0.92\pm 0.04}$ and $L_{\rm X,pwn}\propto\dot{E}^{1.45\pm 0.08}$ in 2-10 keV. Recently, Chang et al. (2023) used the non-thermal X-ray luminosity in 0.5–8 keV band for 68 RPPs, finding $L_{\rm X}\propto\dot{E}^{0.88\pm 0.06}$.

In this work, we use X-ray luminosities in 0.3–10.0 keV range for XMM-Newton and in 0.5–7.0 keV range for Chandra, further categorizing the data into Soft X-ray (SX) band ($<$ 2 keV), Hard X-ray (HX) band ($>$ 2 keV). To avoid unscientific results, we exclude pulsars with unavailable or negative values for $\dot{P}$ from the analysis. The upper limits of luminosities for radio-quite or radio faint Gamma-ray pulsars are shown in Figures 1-3, but are not included in the correlation analysis. We plot $\dot{E}$ versus luminosities in the X-ray, Gamma-ray and radio bands in Figure 1, employing Pearson and Spearman correlation tests to assess the strength of linear and monotonic correlations, respectively. The results, summarized in Table 1, show that the X-ray and gamma-ray bands exhibit a strong correlation between $\dot{E}$ and luminosity, approximating $L_{\rm X}\propto 2\times 10^{-4}~{}\dot{E}$. The explicit linear fit across the entire band yields $L_{\rm X}\propto\dot{E}^{0.85\pm 0.05}$, consistent with the findings of Chang et al. (2023). The data align with the broken power-law fitting results in their study.

In general, the thermal component primarily contributes to the soft X-rays, while the non-thermal component dominates hard X-rays and could also contribute significantly to the soft X-rays. Consequently, the correlation in the soft X-ray band becomes noticeably weaker (the Pearson correlation coefficient $r=0.51$) compared to that of Becker & Truemper (1997) due to the mixture of thermal and non-thermal emissions (Figure 1 and Table 1). The correlation in the hard X-ray band remains strong ($r=0.73$), suggesting that harder X-ray emission may provide valuable insights into the process of rotational energy loss being converted into X-rays (Possenti et al., 2002).

Because additional unpulsed X-ray emission may originate from young PWNe or from intrabinary shocks in MSPs (Becker & Truemper, 1997; Becker & Trümper, 1999; Zhang & Cheng, 2003; Chang et al., 2023), the correlation coefficient for MSPs ($r=0.62$) is generally smaller than that of NPs ($r=0.83$), as shown in Table 1. In our sample, 110 sources (MSPs in GCs or NPs associated with PWNe) are adopted from the literature, which only provided the luminosities in the whole X-ray band (Hsiang & Chang, 2021; Zhao & Heinke, 2022). Fitting across the entire X-ray band yields a best-fit slope ($\alpha=0.85\pm 0.05$) higher than those in the soft ($\alpha=0.39\pm 0.08$) and hard ($\alpha=0.79\pm 0.08$) X-ray bands (Figure 1 and Table 1), indicating a discrepancy between the sources from literature and those derived from spatial matching. However, we cannot determine whether this small discrepancy is intrinsic or due to systematic differences in flux calculations. The number of pulsars involved in each fitting is listed in Table 1 as well.

The linear correlation between X-ray luminosity and spin-down power strongly supports the magnetic dipole model, in which magnetic dipole radiation extracts rotational kinetic energy and causes the spin-down. We further investigate the correlation between X-ray luminosity and five timing parameters, as shown in Figures 2, 3, and Table 1. For correlations where the Pearson correlation coefficients $r>0.6$, we use curve_fit module in Scipy package to plot fitting lines showing the trend, and provide fitting errors at 1-sigma confidence level. In Figure 2 panels a and b, the best fits for $P$ and $\dot{P}$ of NPs are $L_{\rm X}$ $\propto{P}^{-2.95\pm 0.27}$ and $L_{\rm X}$ $\propto\dot{P}^{1.08\pm 0.12}$, respectively. Since pulsar timing parameters are functions of $P$ and $\dot{P}$, the best fit in two-dimension is found to be $L_{\rm X}$ $\propto P^{-2.55\pm 0.16}\dot{P}^{0.85\pm 0.04}$. In panel c, the surface magnetic strength ($B_{\rm surf}$) shows no significant correlation with $L_{\rm X}$, consistent with the result from Chang et al. (2023). In panel d, the grey regions represent theoretical lines for $logB_{\rm surf}=6,7,8,...,14$ G under the assumption that $L_{\rm X}\propto\dot{E}^{0.85}$. The distribution ranges for MSPs and NPs fall between 6-10 and 10-14, consistent with the values calculated from detected $P$ and $\dot{P}$. It is worth noting that MSPs are old neutron stars “recycled” by the accretion of mass and angular momentum from a companion star in a mass-transfer binary (Bhattacharya & van den Heuvel, 1991), so the characteristic age $\tau$ might deviate from the real age for MSPs.

In Figure 3 panel a, we report for the first time a strong correlation between $B_{\rm lc}$ and $L_{\rm X}$ for MSPs. The fitting line reveals a consistent relationship for both NPs and MSPs, approximately $L_{\rm X}\propto B_{\rm lc}^{1.14}$. This strong correlation is also found in the Gamma-ray band, with Pearson correlation coefficients $r=0.71$ for MSPs and $r=0.79$ for NPs (Table 1.c). This result provides valuable constraints for high-energy emission models of pulsars. The outer-gap model offers an explanation for these emissions, suggesting that Gamma-rays are produced in the outer-gap by electron-positron pairs ($e^{\pm}$) through inverse Compton scattering or curvature radiation processes (Cheng et al., 1986). As these gamma-rays travel back toward the neutron star surface, they convert into secondary electron-positron pairs via photon-photon pair creation. These secondary pairs then emit non-thermal X-rays through synchrotron radiation near the light cylinder (Cheng et al., 1998; Takata et al., 2012).

We also note that with the same $B_{\rm lc}$, X-ray luminosity of NPs tends to exceed that of MSPs. Therefore, two-dimensional fitting is also performed, as shown in panel b of Figure 3. The best fitting between $L_{\rm X}$ and $B_{\rm lc}$, $\tau$ is $L_{\rm X}$ $\propto B_{\rm lc}^{0.86\pm 0.06}\tau^{-0.42\pm 0.03}$. This result is consistent with the $L_{\rm X}-\dot{E}$ correalation, as $L_{\rm X}\propto B_{\rm lc}~{}\tau^{-0.5}\propto P^{-3}\dot{P}\propto\dot{E}$.

To discover pulsars within the Galactic latitude of $\pm 10^{\circ}$ from the Galactic plane, the GPPS survey team designed the snapshot observation mode. In this mode, a sky patch of approximately 0.1575 square degrees is surveyed by four pointings using three-beam switching of the 19 beams from the L-band 19-beam receiver on FAST (see Han et al., 2021, for more details). The L-band resolution is about $2.9^{\prime}$ (Jiang et al., 2019), so the initial position of a pulsar detected from one beam has an accuracy of $\leq$ 1.5^′. This accuracy can be significantly improved ($\leq 0.1^{\prime\prime}$) with a long-term timing campaign (Su et al., 2023). We search for candidate X-ray counterparts within a circle centered on the position of GPPS pulsars with a radius of 1.5^′. Only point-like X-ray sources are selected by setting the extent flag $EP\_Extent=0$ for XMM-Newton sources and $extent\_flag=FALSE$ for Chandra sources. The spatial match between the GPPS pulsars and the XMM-Newton and Chandra catalogs yields 48 associations (Table 3).

We convert the fluxes listed in the catalogs to the isotropic X-ray luminosities in the 0.3–10 keV, 0.5–7 keV bands for XMM-Newton and Chandra catalogs, respectively. The X-ray point sources exhibit X-ray luminosities ranging from $3.6\times 10^{29}$ erg s^-1 to $6.8\times 10^{33}$ erg s^-1. Assuming that the GPPS pulars share the same $L_{\rm X}\propto\dot{E}^{0.85}$ (Figure 1) correlation with the ATNF pulsars, the spin-down power can be calculated with the observed X-ray luminosity. The period $P$ of GPPS pulsars is provided in the catalog, and the period derivative $\dot{P}$ can be derived using the formula $\dot{P}=\frac{\dot{E}P^{3}}{4\pi^{2}I}$. For most sources, the $\dot{E}$ and $\dot{P}$ derived from the $L_{\rm X}-\dot{E}$ relation appear excessively high. It is illogical for many sources to have $\tau$ values smaller than 1 kyr or magnetic fields greater than the critical value of $4.4\times 10^{13}$ G (Figure 4). Excluding these suspicious sources, there remain 27 X-ray point sources located within 16 pulsar’s positional error circles. However, it is important to note that a large proportion of these X-ray point sources are not actual X-ray counterparts of the GPPS pulsars. Due to FAST’s angular resolution of 2.9^′, the probability to find an unrelated X-ray source within a positional error circle is $10^{3}-10^{4}$ times higher than that for the ATNF pulsars. Hence, there may be multiple X-ray sources surrounding the GPPS pulsars within 1.5^′ (e.g. J1855+0139g and J2021+4024g).

As discussed above, X-ray luminosity of RPPs is positively correlated with their rotational energy loss (Becker & Truemper, 1997; Li et al., 2008; Arzoumanian et al., 2011; Vink et al., 2011; Rea et al., 2012; Vahdat et al., 2022; Chang et al., 2023). Assuming $L_{\rm X}\sim 2\times 10^{-4}~{}\dot{E}$ (Figure 1), the derived values of $\dot{E}$ are larger than $10^{34}~{}{\rm erg~{}s^{-1}}$ when $L_{\rm X}>2\times 10^{30}~{}{\rm erg~{}s^{-1}}$. In contrast, more than 60 newly discovered pulsars by FAST have timing solutions, and their spin-down powers are mostly less than $10^{34}~{}{\rm erg~{}s^{-1}}$ (Li et al., 2018; Su et al., 2023; Wu et al., 2023). We estimate the fluxes of these pulsars using the detected $P$ and $\dot{P}$, finding that almost 87% are lower than $10^{-15}~{}{\rm erg~{}cm^{-2}~{}s^{-1}}$. However, most sources detected by XMM-Newton and Chandra have fluxes over $10^{-14}~{}{\rm erg~{}cm^{-2}~{}s^{-1}}$ and $10^{-15}~{}{\rm erg~{}cm^{-2}~{}s^{-1}}$, respectively (Webb et al., 2020; Primini et al., 2011). Only the MSP, PSR J1953+1844, is found to be associated with a faint X-ray point source, 2CXO J195337.9+184454 (R.A. = 19:53:38.0, Dec. = +18:44:54.4) with a separation of less than 0.2^′′ (Pan et al., 2023). Thus we conclude that the X-ray counterparts of FAST newly discovered pulsars are less likely to be found in the mentioned XMM-Newton and Chandra catalogs compared to ATNF pulsars, and the point sources with high X-ray luminosity are likely to be coincidences with GPPS pulsars.