On the X-ray efficiency of the white dwarf pulsar candidate ZTF J190132.9+145808.7 (2024)

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Abstract

1 Introduction

Low-mass stars like our Sun evolve into white dwarfs (WDs) at the end of their lifetimes. One-third of stellar objects are believed to be white dwarfs. These objects are significant not only as major constituents of the Galaxy but also as key entities for understanding the physics under high-density environments, the mechanism of SN Ia explosions, and more. Additionally, WDs close to the Chandrasekhar limit are pivotal for the substantial production of neutronized species like ⁵⁸Ni and ⁵⁵Mn through efficient electron capture processes (Iwamoto etal. 1999; Seitenzahl etal. 2013).

Massive and rapidly rotating WDs are of particular interest because they can form through WD–WD mergers (Dan etal. 2014). As a result of such mergers, these massive WDs possess a smaller radius and higher density, consistent with their equationsof state (MR^1/3 ≈ const., where M and R denote mass and radius respectively; Schwab 2021). A more rapid rotation period is anticipated due to the conservation of angular momentum, although the extent of this conservation remains a topic of study (Schwab 2021). Dipole magnetic fields of such WDs are believed to be intensified by potent dynamo mechanisms during mergers (Tout etal. 2008; García-Berro etal. 2012; Das & Mukhopadhyay 2012).

Identifying and quantifying such massive and rapidly rotating WDs are crucial steps in understanding WD–WD merger rates and their associated nucleosynthesis processes. While massive WDs typically exhibit high-temperature colors in the optical band, most have spin periods around 10⁴ s or longer. Recent deep optical surveys have identified several potential remnants of WD–WD mergers with rapid spin periods. One notable candidate identified by the Zwicky Transient Facility is ZTFJ190132.9+145808.7 (hereafter ZTFJ1901+14). This WD is near the Chandrasekhar limit with a mass between 1.327–|$1.365\, M_{\odot }$|⁠, a measured radius comparable to the Moon, and a notably short spin period of 416 s (Caiazzo etal. 2021).

We propose an innovative approach to investigate these magnetic and rapidly rotating WDs. They are expected to emit nonthermal X-rays as a result of their spin-down, akin to isolated neutron stars, leading to their nickname “white dwarf pulsars.” Such WDs are anticipated to release hard pulsating X-rays, presenting a novel method for their identification. Ostriker etal. (1970) originally proposed this idea, and Sousa et al. (2022) also mentioned this possibility. Observationally, the first such X-ray emission was reported from an accreting magnetized WD, AE Aqr (Terada etal. 2008), which has an incredibly fast rotation period of 33 s. This was followed by discoveries in AR Sco (Buckley etal. 2017; Takata etal. 2018) with a period of 118 s and J191213.72−441045.1 (Pelisoli etal. 2023) with a period of 5.30 min. Their rapid spin, significant magnetic field, and larger radius compared to neutron stars enable them to achieve electrostatic potentials, V∝P⁻²BR^3/2, comparable to those of neutron stars. Consequently, WD pulsars can accelerate particles to energies as high as neutron stars. The X-ray luminosity from these WDs is around |$0.1\%$| of their spin-down energy (Terada etal. 2008), similar in efficiency to neutron stars (Kargaltsev & Pavlov 2008). Given the abundance of WDs compared to neutron stars, they might significantly contribute to the Galactic cosmic ray electron–positron components (Kashiyama etal. 2011; Kamae etal. 2018). Notably, some radio sources labeled “ultra-long period pulsars” could be white dwarfs, providing further evidence of white dwarf pulsars (Hurley-Walker etal. 2022, 2023).

However, all WD pulsars identified to-date are part of accreting systems. Earlier hard X-ray observations of isolated WDs failed to detect significant nonthermal emissions (Harayama etal. 2013), and the constraints from these observations were somewhat loose. A focused search for isolated WD pulsars is necessary to validate our proposed scenario. In this paper, we present the first nonthermal X-ray search for the massive, high-magnetic field WD, ZTFJ1901+14, using the high-resolution capabilities of the Chandra observatory. In section2, we detail our target selection, observation methods, and data reduction. Section3 outlines our imaging and spectral analysis results, while section4 offers a discussion of our findings.

2 Target selection and observations

Our primary goal is to detect X-rays emitted from isolated, massive, and magnetic WDs that exhibit short spin periods. Recent optical surveys, such as the Zwicky Transient Facility (Bellm etal. 2019) and the Sloan Digital Sky Survey (Eisenstein etal. 2006), have identified several isolated WDs that rotate rapidly. Out of 25 WDs from Kilic etal. (2023), which catalogs ultramassive WDs, we singled out four as potential WD pulsar candidates, based on their strong magnetic fields and rapid rotation. Table1 showcases the physical parameters of these candidates, juxtaposed with EUVEJ0317−855, which has been previously investigated as a WD pulsar candidate (Harayama etal. 2013).

Dipole moments (μ) and spin-down energy (⁠|$\dot{E}$|⁠) were estimated, assuming the relationships μ∝BR³ and |$\dot{E} \propto \mu ^2P^{-4} \propto B^2R^6P^{-4}$|⁠, where R represents their radius and P is their rotation period. We utilized Nauenberg (1972) to determine R, as detailed in table1. It is worth noting that strong magnetic fields might induce minor variations in R, but for our approximate calculations, such effects are negligible. Table1 also presents the derived |$\dot{E}$|⁠, normalized by the value for EUVEJ0317−855.

A crucial parameter for detecting significant X-ray emissions is the spin-down flux, denoted by |$\dot{E}/(4\pi d^2)$|⁠, where d is the distance to the target (refer also to Shibata etal. 2016; Watanabe etal. 2019; Bamba etal. 2020). As observed, ZTFJ1901+14 and SDSS221141.80+113604.5 are poised to exhibit the highest spin-down flux among the five WD pulsar candidates, making them ideal subjects for our investigation. However, only ZTFJ1901+14 has undergone observations with X-ray observatories, leading us to choose it as our primary target.

ZTFJ1901+14 was observed by Chandra ACIS-I (Weisskopf etal. 2002) on 2022/12/09–10 (OBSID: 26496, 27596, and 27597). The data reduction and analysis was done with CIAO 4.15 (Fruscione etal. 2006) and CALDB version 4.10.4. We made the reprocessed cleaned data with the standard method following the CIAO guide, and the resultant exposure time is 39.3 ks.

3 Results

Figure1 displays the 0.5–7 keV image of the ZTFJ1901+14 region. In this figure, the position of our target, as determined by SIMBAD (Wenger etal. 2000), is marked with a white cross. Note that the pixel scale of the CCD onboard ZTF is 1″ and the median delivered image quality is |${2{_{.}^{\prime\prime}}0}$| in full width at half maximum.¹

There is some excess emission around the target position. However, it is not significant, and the wavdetect command in CIAO did not detect any source in this region. In order to estimate the flux of the possible X-ray emission on this position, we cited the coordinate of the pixel which has the maximum count (RA, Dec) |$= ({19^{\rm h}01^{\rm m}32{_{.}^{\rm s}}91},\ +14^{\circ }{58^{\prime }09{_{.}^{\prime\prime}}1})$|⁠. Note that the position is close to the SIMBAD position as shown in figure1. The significance level of this position is estimated with the srcflux command in CIAO to be 0.998 or the 3.5σ level. To convert the count rate to flux, a spectral model is required. We opted for the absorbed power-law model, which is commonly used for both WD pulsars and neutron star pulsars. The absorption column was fixed at 1×10²⁰ cm⁻², based on the presumption that the interstellar medium density is 1 cm⁻³ and considering a distance of 41 pc (Gaia Collaboration 2020). The photon index of WD pulsars ranges from 1–2.5 (Terada etal. 2008; Takata etal. 2018; Schwope etal. 2023). Based on this, we adopted a value of 2. This value is also typical for neutron star pulsars (Kargaltsev & Pavlov 2008). The resulting unabsorbed flux is 2.4(0.9–4.8)×10⁻¹⁵ erg s cm⁻² in the 0.5–7 keV band, where the error range is in |$90\%$| confidence level. Adjusting the photon index to 1.5 did not lead to significant changes (only less than a few |$10\%$|⁠) in our findings.

4 Discussion

In the previous section, we presented that the strongly magnetized and rapidly rotating WD ZTFJ1901+14 has been detected, although it is not prominently bright in the X-ray band. Assuming a distance of 41 pc, the |$90\%$| error range of the 0.5–10 keV luminosity is 5.3(2.3–10.9)×10²⁶ erg s⁻¹. This is three orders of magnitude lower than the nonthermal X-rays detected from other accreting white dwarf pulsars, which have X-ray luminosities of approximately 10²⁹ erg s⁻¹ (Terada etal. 2008; Takata etal. 2018; Schwope etal. 2023), as listed in table2.

In the case of neutron stars, the spin-down energy |$\dot{E}$| and X-ray efficiency η, which is defined as (X-ray luminosity)|$/\dot{E}$|⁠, serve as crucial parameters in understanding them (e.g., Kargaltsev & Pavlov 2008). We thus evaluated |$\dot{E}$| and η for our samples and made comparisons. Caiazzo etal. (2021) measured the surface magnetic field B_s and the radius R of ZTFJ1901+14 to be between 600–900 MG and |$2140^{+160}_{-230}\:$|km, respectively. Meanwhile, Suto etal. (2023) proposed that the magnetic field structure of ZTFJ1901+14 is dipolar, with the magnetic axis inclined at χ=60^○ to the rotation axis, based on optical/UV light curve analyses. Assuming the magnetosphere is force-free, |$\dot{E}$| is estimated to be

where μ=B_dR³|$/$|2 with the dipole magnetic field B_d, with the assumption of B_d=B_s, c is the light speed, and the constant C ∼ 1 (Gruzinov 2005; Spitkovsky 2006; Tchekhovskoy etal. 2013). For accreting WD pulsars, the |$\dot{E}$| of AEAqr was taken to be 6×10³³ erg s⁻¹, as estimated by de Jager (1994) and de Jager etal. (1994). For ARSco, we utilized the spin-down frequency of |$\dot{\nu } = 4.47\times 10^{-17}$| Hz s⁻¹ (Pelisoli etal. 2022), resulting in |$\dot{E} = 5\times 10^{33}\:$|erg s⁻¹ and B_d=900 MG, with assumptions regarding that the spin-down is due to magnetic dipole radiation, |$M = 0.8\, M_{\odot }$|⁠, and R=7000 km. The derived parameters are also listed in table2. Regarding X-ray efficiency, η, AEAqr and ARSco are approximately 1×10⁻⁴, whereas ZTFJ1901+14 is 0.012(0.0022–0.074) In contrast, typical neutron stars have η values between 10⁻⁵ and 10⁻¹ (Kargaltsev & Pavlov 2008). We can thus infer that its X-ray efficiency is similar to typical neutron stars, although the uncertainty is rather large. Figure2 plots the relationship between |$\dot{E}$| and X-ray luminosity for our samples compared to neutron stars, further illustrating this point.

We also drew comparisons with another isolated WD pulsar candidate, EUVEJ0317−855. Harayama etal. (2013) estimated its |$\log \dot{E} 29.0$|–30.8 with the assumption of the radius of 5000–10000 km. On the other hand, the mass of EUVEJ0317−855 is 1.31–|$1.37\, M_{\odot }$| (ka*wka etal. 2007), which is quite similar to that of ZTFJ1901+14, and its radius should be as small as that of ZTFJ1901+14 according to the mass–radius relation of WDs. We thus re-estimated its |$\log \dot{E}$| to be 27.1 (or |$\dot{E} = 1.3\times 10^{27}\:$|erg s⁻¹) with the assumed radius of 2417 km (see table1), following that the magnetic dipole moment μ∝BR³ and |$\dot{E} \propto \mu ^2P^{-4} \propto B^2R^6P^{-4}$|⁠. The re-estimated parameters for EUVEJ0317−855 are also listed in table2. With this |$\dot{E}$|⁠, the 3σ upper-limit of the X-ray efficiency for EUVEJ0317−855 is 33 in the 0.5–8 keV band, which is much larger the value with larger radius assumptions. Figure2 also shows EUVEJ0317−855 with the assumptions of our new radius estimation. The figureshows clearly that the X-ray efficiency of ZTFJ1901+14 is well constrained compared with that for EUVEJ0317−855, although the significance is not so high.

Here, we note the uncertainty of our results. One of the concerns is the uncertainty of the derived radius with the model by Nauenberg (1972), which ignored the effect of magnetic field and rotation: the strong magnetic field makes degenerate pressure smaller and the stellar radius becomes larger, and as a result, the derived spin-down energy becomes larger and the X-ray efficiency smaller. This effect clearly appear only with the magnetic field of much larger than ∼10⁹ g, thus in this sense, our conclusion (not very bright in X-rays) does not change. The error range of the radius for ZTFJ1901+14 is just |$10\%$| (Caiazzo etal. 2021), which is already included in our discussion. The distance uncertainty is negligible, less than 0.1 pc (Gaia Collaboration 2020).

Assuming that the X-ray detection from ZTFJ1901+14 is the case, let us explore the potential emission mechanism. Given the measured radius, magnetic field strength, and rotation period, ZTFJ1901+14 is likely below the death line for WD pulsars (Kashiyama etal. 2011), i.e., electron–positron pair multiplication process in the magnetosphere will not be operational. Nevertheless, a charged particle could still undergo acceleration through the electric field induced by the unipolar induction, especially along the open field lines. The maximum voltage can be estimated as

Hereafter, we adopt the maximum values of the radius and magnetic field strength within uncertainties (Caiazzo etal. 2021) as our fiducial values. Accordingly, electrons could be accelerated to sub-TeV energies with Lorentz factors of γ_max ≈ eV_max|$/$|m_ec² ∼ 8.1×10⁵. Assuming that electrons with a Goldreich–Julian density n_GJ ≈ B_d|$/$|ceP are supplied quasi-steadily into the polar cap region r_cap ≈ R×(2πR|$/$|cP)^1/2, the kinetic luminosity of the electrons becomes comparable to the spindown luminosity, i.e., |$L_\mathrm{e} \approx 2\pi r_\mathrm{cap}^2 c m_\mathrm{e} n_\mathrm{GJ} \gamma _\mathrm{max} c^2 = 2 \mu ^2 (2\pi /P)^4c^{-3} \approx \dot{E}$|⁠. These electrons partially lose their energies through the curvature radiation. The emission frequencies can be estimated as

where h is the Planck constant, and R_c is the curvature radius of the open field lines. In the case of ZTF1901|$+$|14, the emission is predominantly in the X-ray band for R ≲ R_c ≲ 100R, i.e., in the near surface region. The X-ray luminosity can be estimated as |$L_X \approx (t_\mathrm{c}/t_\mathrm{dyn}) L_\mathrm{e} \approx (t_\mathrm{c}/t_\mathrm{dyn}) \dot{E}$|⁠, where t_dyn ≈ R|$/$|c is the dynamical timescale of the electrons and |$t_\mathrm{c} \approx 3m_\mathrm{e}cR_\mathrm{c}^2/2e^2\gamma _\mathrm{max}^3$| is the energy-loss timescale through the curvature radiation. Then, the X-ray radiation efficiency is given as

Hence, within the uncertainties, the observed X-ray luminosity of ZTFJ1901+14 may be attributed to the curvature radiation from sub-TeV electrons accelerated along the open field lines.

Since η ≪ 1, sub-TeV electrons escape into the interstellar medium without significant energy loss, thus supporting the idea that strongly magnetized, rapidly rotating massive WDs like ZTFJ1901+14 are efficient factories for electron cosmic rays (Kashiyama etal. 2011). The lower X-ray efficiency compared to X-ray-bright neutron star pulsars can be attributed to its relatively low-density environment. Relativistic winds and cosmic rays launched from the magnetosphere of ZTFJ1901+14 would be directly injected into the interstellar medium, whereas those from a young neutron star pulsar are surrounded by a supernova remnant (e.g., Gelfand etal. 2009; Bamba etal. 2010) and can efficiently dissipate at and beyond the wind termination shock (e.g., Meintjes etal. 2023). Accreting WD pulsars are typically found in environments with a relatively high density of materials from their companion stars, as observed in the case of AEAqr (Terada etal. 2008), or they may encounter stellar winds from their companion stars, as seen in the case of ARSco (Buckley etal. 2017; Takata etal. 2018).

Further investigations, such as deeper X-ray observations of ZTFJ1901+14 and measurements of the magnetic field of J191213.72−441045.1, will provide more insights into clarifying the differences in non-thermal emissivity among these types of compact star systems. We need deeper X-ray observations with large effective area missions such as Athena (Nandra etal. 2013) and Lynx (Gaskin etal. 2018), in order to make quantitative comparison of X-ray efficiencies of other compact star systems. The requirement of point source sensitivity of the Wide Field Imager (WFI) onboard Athena is of the order of 10⁻¹⁷ erg s⁻¹ cm⁻² (Nandra etal. 2013), resulting η ∼ 10⁻⁴ for ZTFJ1901+14, which is enough to judge if isolated massive and rapid-rotating WDs are efficient X-ray emitters or not.

5 Summary

We conducted an X-ray search for ZTFJ1901+14, one of the most massive and rapidly-rotating white dwarfs, using Chandra and found X-ray emission with the significence of 3.5σ level. Assuming a photon index of 2, we derived its intrinsic flux to be 2.3(0.9–4.7)×10⁻¹⁵ erg cm⁻² s⁻¹ and luminosity 4.6(2.0–9.5)×10²⁶ erg s⁻¹ for a 0.5–7 keV band in the |$90\%$| confidence range, given its distance of 41 pc. We derived an X-ray efficiency (η) concerning the spin-down luminosity to be 0.012(0.0022–0.074), which is similar to typical neutron star pulsars. The observed X-ray emission might suggest that such WDs have the capability to accelerate electrons to sub-TeV energies. However, conclusive evidence requires more in-depth observations. For a comprehensive comparison with other WD pulsars that have companion stars, like AEAqr, ARSco, and SDSSJ191213.72−441045.1, further X-ray/optical observations are essential to pinpoint physical parameters including the magnetic field and spin-down luminosity.

Acknowledgements

We thank the anonymous referee for their constructive comments. We thank Shinpei Shibata for the fruitful discussions. This research has made use of the SIMBAD database, operated at CDS, Strasbourg, France. This work was financially supported by Japan Society for the Promotion of Science Grants-in-Aid for Scientific Research (KAKENHI) Grant Numbers, JP19K03908 (AB), JP23H01211 (AB), JP20K04009 (YT), JP20H01904 (KK), JP22H00130(KK), JP23H04899 (KK), JP21H01078 (SK), JP22H01267 (SK), JP22K03681 (SK), and JP20H00153 (TT).

References

© The Author(s) 2024. Published by Oxford University Press on behalf of the Astronomical Society of Japan.

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