Dependence of pulsar death line on the equation of state

Abstract

1 INTRODUCTION

Radio pulsars are unique laboratories for a wide range of physics and astrophysics, which were compellingly identified as ‘classical’ neutron star (NS) not long after their discovery (Glendenning 2000). Its spin period P, and the period derivative |$\dot{P}$|⁠, can be obtained at very high levels of precision through timing measurements (Manchester et al. 2005). Then the pulsars could be placed on the |$P\text{--}\dot{P}$| plane, which is a useful starting point for understanding the diverse flavours of pulsars. Fig. 1 shows the |$P\text{--}\dot{P}$| diagram for the known radio pulsars with black dots.

Figure 1.

The |$P - \dot{P}$| diagram. Blue squares are magnetars (empty squares are radio-loud magnetars, from McGill magnetar catalog: Olausen & Kaspi 2014, http://www.physics.mcgill.ca/∼pulsar/magnetar/main.html). Green diamonds are X-ray dim isolated NSs (XDINSs, from Halpern & Gotthelf 2010, Kaplan & van Kerkwikj 2011 and references therein), cyan circles are the central compact objects (CCOs, from Halpern & Gotthelf 2010; Gotthelf, Halpern & Alford 2013), red stars are rotating radio transients (RRATs, from http://astro.phys.wvu.edu/rratalog/), magenta triangles are intermittent pulsars (from Lorimer et al. 2012, and references therein, Surnis et al. 2013), black dots are rotation powered pulsars (including normal pulsars and millisecond pulsars, which dominate the pulsar population, from ATNF: http://www.atnf.csiro.au/research/pulsar/psrcat/ (version 1.54, Manchester et al. 2005)) and the black open circles are PSR J2144−3933. Lines of characteristic magnetic field (dotted line), characteristic age (dashed line) and typical pulsar death line (dash-dot line) are also indicated. This figure is adapted from the work Tong & Wang (2014).

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With the development of the observational equipment, the number of new radio pulsars with a wide range of parameters dramatically increazed. From optical to γ-ray, there exist plenty of observational phenomena leading to identifications of other classes of NSs have been identified, such as magnetars, X-ray dim isolated neutron stars (XDINS), central compact objects (CCOs) and rotating radio transients (RRATs) (Kaspi 2010; Kaspi & Kramer 2016). Some of them have precise measurements of P and |$\dot{P}$|⁠, and we plot them in Fig. 1 with different symbols and colours.

Magnetars are assumed to be NSs that are powered by their strong magnetic field (Duncan & Thompson 1992). Observationally, they have a rotational period in the range of 2–12 s and their period derivative span a broader range from 10⁻¹⁴ to 10⁻¹⁰. This scatter of period derivative results in a scatter of the characteristic magnetic field of magnetars from less than 10¹³G to higher than 10¹⁵ G. They are young objects with characteristic ages from ∼ 10³ yr to more than 10⁷ yr (Tong 2016; Kaspi & Beloborodov 2017). There are only four magnetars observed in radio band (Camilo et al. 2007; Levin et al. 2012).

XDINSs are sub-classes of NSs because they are isolated NSs and with X-ray emission. The quasi-thermal X-ray emission of XDINSs is relatively low, with great proximity, a lack of radio counterparts and relatively long periodicities (⁠|$P = 3\text{--}11 \,\rm {s}$|⁠). Timing observations of several objects have revealed that they are spinning down regularly, with typical inferred dipolar surface magnetic fields ∼|$(1\text{--}3) \times 10^{13} \,\rm {G}$| and characteristic ages of |${\sim } 1\text{--}4 \,\rm {Myr}$| (Haberl 2007; Kaplan & van Kerkwikj 2009). Up to now, no radio emission of XDINSs is detected. They are located in the upper right-hand side of the pulsar |$P\text{--}\dot{P}$| diagram, between standard radio pulsars and magnetars.

CCOs are NS-like objects at the centres of supernova remnants. Common properties for CCOs are characterized by steady flux, predominantly surface thermal X-ray emission, lack of a surrounding pulsar wind nebula, and absence of detection at any other wavelength (De Luca 2008; Gotthelf & Halpern 2009; Halpern & Gotthelf 2010). Of the eight most secure CCOs, three are known to be NSs with spin periods of 0.105, 0.424 and 0.112 s (Gotthelf et al. 2013). Spin-down rate has been detected for three CCOs, and they are placed in the middle of the |$P\text{--}\dot{P}$| diagram.

RRATs are a curious class of Galactic radio sources that are characterized by strong radio bursts at repeatable dispersion measures, but not detectable using standard periodicity-search algorithms. Nevertheless, the observed pulses are inferred to occur in multiples with an underlying periodicity that is very radio pulsar like (McLaughlin et al. 2006; Keane & McLaughlin 2011). Their periods distribute over a wide range, from 0.7 to 6.7 s (Keane et al. 2010). RRATs occupy the region partly coinciding with the main radio pulsar cloud in the |$P\text{--}\dot{P}$| diagram (Fig. 1), and partly extending towards longer spin periods and larger magnetic fields, but mainly fall into the end range of the conventional radio pulsars. Intermittent pulsars are having higher slow down rates in the on state (radio-loud) than in the off state (radio-quiet), and suggested to be part-time radio pulsars (Kramer et al. 2006). Their nulling time-scales are very long compared with that of the ordinary pulsars, radiation switching between the on state and the off state. Many models have been proposed for intermittent pulsars (Li 2006; Zhang et al. 2007; Jones 2012; Li et al. 2014), but it is still an open question as to what accounts for the on/off transition and why the rotation slows down faster when the pulsar is on than when it is off.

As discussed before, our knowledge about NSs increases a lot, thanks to rapid development of observational technique during the past two decades. Radio pulsars are just one observational manifestation of NSs. Some other groups of the sources are lying above the death line but they are radio-quiet, which are shown in Fig. 1, such as XDINSs, CCOs and some magnetars. Although none of the XDINSs and CCOs has been detected at radio frequencies, it does not necessarily mean that they have no radio emission. Some authors argued that XDINSs are actually radio pulsars viewed well off from the radio beam (Kaspi & Kramer 2016). CCOs have no optical or radio counterparts or pulsar wind nebulae, and no radio emission is detected; beaming effect cannot explain this lack of radio emission but ultra-low radio luminosities could. Besides the beaming effect and ultra-low radio luminosities, it seems that going back to the basic concept of pulsar death line could help us to understand these phenomena from another perspective.

The standard definition of a death line implicitly assumes that pair formation is a necessary condition for pulsar radio emission and the pulsars become radio-quiet after crossing the death lines during their evolution from the left- to right-hand side in the |$P\text{--}\dot{P}$| diagram. Obviously, this basic condition is not sufficient, and functioning of radio pulsars may imply far more complex physical conditions. However, numerous theoretical attempts to produce satisfactory death lines imply that even the basic necessary condition meets certain challenges (Harding & Muslimov 2001, 2002; Usov 2002). Most of previous works are mainly focused on the configurations of surface magnetic field and the models for pair acceleration. Moreover, the potential drop ΔV, across an accelerator gap, is model dependent. Different particle acceleration model will change ΔV considerably and alter the death lines. The potential drop ΔV is just a few 10¹² V (Ruderman & Sutherland 1975), or has a weak dependence on Ω (Xu & Qiao 2001). Meanwhile, a potential drop around 10¹² V is also a widely accepted result from the pulsar death line criterion (Zhang et al. 2000). In order to minimize the underlying model assumptions, we assume a constant potential drop ΔV = 10¹² V in the polar cap accelerating region. As shown in equations (1) and (2), the pulsar death line is also related to the moment of inertia of NS, which is determined by the EoS. By setting the constant potential drop ΔV, we only need to consider the dependence of pulsar death line on the EoS. Since the deconfinement transition from hadron matter into quark matter is possible at high density, one logically proposes that some pulsars could be strange stars(SSs). But it is still an open question whether these stars are NSs or SSs (Özel & Freire 2016). The most convincing proof for a SS would come from a direct indication of the EoS, and we believe that pulsar death line might provided us the answer.

In this work, we are going to discuss the properties of death lines affect by EoSs without considering the magnetosphere and particle acceleration model. Observational constraints on EoSs are also considered. Furthermore, the possible connections between different kinds of NSs will also be discussed. In Section 2, we will introduce the significance of EoSs in astrophysics and the different pulsar death lines with the impacts of EoSs. Conclusion and discussion will be provided in Section 3.

2 EOS AND PULSAR DEATH LINE

The nature of matter at the inside of NS is one of the major unsolved problems in modern science, and this makes NS unparalleled laboratories for nuclear physics and quantum chromodynamics under extreme conditions (Glendenning 2000). Consequently, measurements of the properties of NSs provide constraints on the physics of cold dense matter that complements the constraints obtained from nuclear physics experiments (Özel & Freire 2016; Watts et al. 2016; Lattimer 2017).

The most fundamental property of dense matter is the EoS, the pressure–density–temperature relation of bulk matter. Together with the Tolman–Oppenheimer–Volkoff (TOV) equations, the EoS of NS matter determines the macroscopic properties of the stars and their masses and radii. In fact, solutions provide a unique map between the microscopic pressure–density relation (p–ε or p–ρ) and the macroscopic mass–radius (M–R) relation of the stars (Lindblom 1992). This unique mapping can be used to infer the EoS from astrophysical measurements of their masses and radii (Watts et al. 2016). The cleanest constraints on the EoS to date are derived from radio pulsar timing, where the mass of NSs in compact binaries can be measured very precisely using relativistic effects (Lorimer 2008). Finding the maximum mass of NSs provide additional direct constraints on the EoS. At a minimum, the EoSs that have maximum masses ( ≈ 2 M_⊙, up to now) that fall below the most massive NS can be ruled out (Demorest et al. 2010; Özel et al. 2010; Antoniadis et al. 2013).

Generally, two categories of EoS were widely discussed, which can produce gravity-bound NSs and self-bound SSs (Glendenning 2000; Haensel, Potekhin & Yakovlev 2007). As shown in Fig. 2, the EoSs of self-bound SSs predicted M ∝ R³ for low-mass pulsars, meanwhile the radius is smaller. Moreover, the minimum mass of self-bound SSs can reach as low as planet mass (Xu & Wu 2003; Horvath 2012), while the low-limit mass of gravity-bound NSs is about 0.1 M_⊙ (Akmal & Pandharipande 1997; Glendenning 2000). On the other hand, searching for very low mass pulsars is also an attractive method, because the EoS of self-bound SSs predicted distinct radii at low mass compared with ones by the EoSs of gravity-bound SSs (Li et al. 2015; Wang et al. 2017). Thus, theoretical EoSs could be effectively tested from the accurate measurement of the radii for low-mass pulsars. In Fig. 2, we only showed the typical EoSs that have maximum mass larger than 2.0 M_⊙. The typical nuclear matter EoSs are obtained from Özel & Freire (2016)(http://xtreme.as.arizona.edu/NeutronStars/). The EoSs of SSs are obtained from Li et al. (2016) (for CDDM1, CDDM2) and Bhattacharyya et al. (2016) (for pMIT).

Figure 2.

The mass–radius curves corresponding to different EoSs wherethe maximum mass is larger than 2.0 M_⊙. The corresponding EoSs are get from Özel & Freire (2016) (http://xtreme.as.arizona.edu/NeutronStars/), Li et al. (2016) and Bhattacharyya et al. (2016).

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The pulsar death line is related to the NS moment of inertia I and radius R, which is illustrated in equations (1) and (2). With the equation |$I=\frac{8\pi }{3}\int _{0}^{R}r^4\epsilon \text{d}r$| (Glendenning 2000), we calculated the NS moment of inertia by employing different EoSs, as shown in Fig. 2. The results are presented in Fig. 3. There is no obvious difference between the moment of inertia of NSs and SSs for both low and high mass. The distinction of moment of inertia between the large mass and small mass is very obvious.

Figure 3.

The moment of inertia–mass curves corresponding to Fig. 2.

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As showed in Table 1, we calculate different radii and moment of inertias of the given mass of NSs. We specified the mass of NS, as 0.2, 0.5, 1.0, 1.4 and 2.0 M_⊙. The corresponding radii and moment of inertias are obtained by employing EoSs. The smallest mass of NSs with typical nuclear matter of EoSs named bsk21, wwf1 and h4 (Özel & Freire 2016) sustained is 0.2 M_⊙.

For the SSs, the radius increases with increasing mass. The smallest mass can be determined by the radius measurements. While precise results can be obtained by measuring masses using radio pulsars, measuring the radius is much more challenging and, with the methods currently in use, more model-dependent. Current efforts to constrain the full M–R relation focus primarily on spectroscopic measurements of the surface emission from accreting NSs in quiescence, and when they exhibit thermonuclear (type I) X-ray bursts due to unstable nuclear burning in accreted surface layers (Özel & Freire 2016). Most CCOs have thermal-like X-ray emission, with blackbody temperatures in the range |$0.2\text{--}0.5 \,\rm {keV}$|⁠, with no evidence for additional non-thermal components, and many of them have relatively good distance estimates, which is important for measuring the radius (Gotthelf et al. 2013). Better fit of the observation of CCOs in the supernova remnant Cassiopeia A show the effective temperature |$kT ^{\infty }_{\text{eff}} \approx 0.2 \,\rm {keV}$|⁠, which require a small radius, |$R = 4{-}5.5 \,\rm {km}$| (Pavlov & Luna 2009). XDINSs are characterized by thermal spectra with a broad spectral feature in most of them, which could also be used to measure the radius. The fitted radiation radius of RX J0806.4-4123 is about 3.5 km, and it is suggested to be self-bound SS (Wang et al. 2017). Assuming that the thermal emission comes from the whole surface, we could set the smallest radius of the SSs at 4 km in this work. The corresponding masses and moment of inertias are obtained by employing EoS named pMIT (Bhattacharyya et al. 2016), CDDM1 and CDDM2 (Li et al. 2016), which are shown in Table 2. For comparison, the mass of SSs are set as |$0.03164, 0.02234\, \text{and}\, 0.01829 \,\rm {M_{{\odot }}}$|⁠, respectively (which is calculated from EoSs with R = 4 km), and 0.2, 0.5, 1.0, 1.4 and 2.0 M_⊙.

In Figs 4 and 5, we plot the pulsar death lines for different masses of NSs and SSs with a constant potential drop (ΔV = 10¹² V) and the maximum one defined in equation (1). From the top to bottom panel, the mass of the corresponding pulsar death line increases. For the same mass of NSs or SSs, death lines with different EoSs show no obvious diversity with each other. The region of death lines of SS spreads wider than NS. The death lines of NSs and SSs do not have obvious difference, while the mass is larger than 1.0 M_⊙. Meanwhile, most of the sources are above the death lines. In Fig. 4, the long period (8.5 s) pulsar PSR J2144−3933 is located beyond the death line, but closest to the death lines for 2.0 M_⊙. PSR J2144−3933 just crosses the death lines with 2.0 M_⊙ in Fig. 5. The obvious distinctions of the pulsar death line are for the smallest mass of NS and SS. In the remaining part of this work, we focus on the pulsar death line with the smallest mass of NSs and SSs.

Figure 4.

Pulsar death lines in the |$P - \dot{P}$| diagram for different NS masses and EoSs are shown in dash-dot red lines(bsk21), dash-dot green lines(wwf1) and dash-dot blue lines(h4). From top to bottom groups are for |$0.2\,\rm {M_{{\odot }}}, 0.5\,\rm {M_{{\odot }}}, 1.0\,\rm {M_{{\odot }}}, 1.4\,\rm {M_{{\odot }}}$| and 2.0 M_⊙. Also shown are sources which are the same as the caption of Fig. 1 mentioned. The black open circle is PSR J2144-3933.

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Figure 5.

Pulsar death lines in the |$P\text{--}\dot{P}$| diagram for different EoSs and mass of SSs with the dash-dot red lines(pMIT), dash-dot green lines(CDDM1) and dash-dot blue lines(CDDM2). From the top to bottom groups are for M(R = 4 km), 0.2, 0.5, 1.0 , 1.4 and 2.0 M_⊙. Also shown are sources that are the same as mentioned in the caption of Fig. 1. The black open circle is PSR J2144−3933.

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In Fig. 4, most of the magnetars and XDINSs are above the death lines, and some of them are close to the death lines. In Fig. 5, all of the XDINSs are below or across the death lines, which agrees with the observational fact that XDINSs do not have detected radio emission until now. All of the radio-loud magnetars are above the death lines. And some of the radio-quiet magnetars are below the death lines. For the CCOs, in the NS case(Fig. 4), all of them are above the death lines. And in the case of SS (Fig. 5), they are near or below the death lines. RRATs are located around the normal pulsars, dispersed on the both side of the death lines. All of the intermittent pulsars are above the death lines for both NSs and SSs.

3 DISCUSSION AND CONCLUSION

Pulsars could radiate radio and high-energy photons. There must be certain kinds of particle acceleration and radiation mechanisms in the pulsar magnetosphere. Previous studies showed that different polar gap models and geometry of the magnetic field will result in different pulsar death lines (Harding & Muslimov 2002; Zhang et al. 2000; Wu et al. 2003). Moreover, the potential drop across the accelerator and maximum potential drop are relying on the moment of inertia and radius of the stars (Xu & Qiao 2001), to some degree. The M–R relations of the NS and SS are different, especially at low mass. EoSs affect the location of the pulsar death line in the |$P\text{--}\dot{P}$| diagram. The results show that different EoSs results in different pulsar death lines.

In Figs 2 and 3, we plotted the mass-radius and moment of inertia-mass curves with different EoSs. NSs and SSs are differentiated from each other at the low-mass region, because the radius of self-bound SSs is increasing with mass but gravity-bound NSs do not. Moreover, Figs 4 and 5 show that the death lines of SSs spread across larger areas than that of NSs. Particularly, the location of pulsar death lines with the smallest mass of SS is much higher than others. Comparing with Fig. 4, the distribution of sources, such as magnetars, XDINSs, CCOs and RRATs, is much more different for the death line with the smallest mass of SS.

Magnetars have an X-ray luminosity as high as 10³⁵ erg s^{− 1}, and their strong surface dipole field may lie in the range of |$10^{13}\text{--}10^{15} \,\rm {G}$|⁠. And four of the magnetars are detected in the radio band. XDINSs are nearby sources with a surface dipole field of around 10¹³ G, have very faint X-ray emission and without radio activities. In Viganò et al. (2013), the authors studied magneto-thermal evolution of isolated NSs. In this model, magnetars evolve and their magnetic fields dissipate, their observational properties (both timing and luminosities) appear compatible with those of the XDINSs. They suggested that XDINSs are supposed to be dead magnetars. In addition to these, two magnetars may have passive fallback discs (Wang, Chakrabarty & Kaplan 2006; Kaplan et al. 2009), and Ertan et al (2014) interpret that XDINSs have gone through a past accretion phase with spin-down and emerged in their current state. In Fig. 5, we could find that the radio-loud magnetars are above the death line. Meanwhile, all of the XDINSs and some radio-quiet magnetars are below the death line. There might be some links between XDINSs and magnetars. We could suggest that XDINSs might be the result of magnetar evolution, even could be dead magnetars. From the study of pulsar death lines, the results here are in line with these previous speculations. Furthermore, well-fitted spectra would provide the radii of the sources. The fitted radiation radius of RX J0806.4-4123 is about 3.5 km, and it could be a low-mass SS candidate (Wang et al. 2017).

CCOs are known as well-fitted surface thermal X-ray emission and without emissions in the radio band. Due to its particular properties of thermal X-ray emission, CCOs are useful targets for measuring the temperatures and radii of the pulsars. The apparent blackbody radii of CCOs are only about a few kilometres, which is significantly smaller than the 10–15 km canonical NS radii. The better fitted radius of CCO we used in this work are R = 4–5.5 km. Actually, some of CCOs with well-fitted blackbody radii are much smaller than we used here (data can be found from Viganò et al. 2013 and the table in Potekhin et al. 2015, as well as from www.neutronstarcooling.info). In Fig. 5, the CCOs are near to or below the death line for smaller mass of self-bound SS. The CCOs supposed to be low-mass SSs. Moreover, it is suggested that the radio-quiet CCO, 1E 1207.4−5209, could be a low-mass bare SS with a radius of a few kilometres according to its peculiar timing behaviour (Xu 2005). If the smallest mass or radius of pulsars are detected in future, we could also suggest that they should be SSs. Well-determined fitting model and sufficient observational data of thermal X-ray emissions would allow us to learn more about these thermal emitting sources.

RRATs generally have longer periods than those of the normal pulsar population, and several have high magnetic fields, similar to those other NS populations like the X-ray bright magnetars. However, some of the RRATs have spin-down properties very similar to those of normal pulsars, making it difficult to determine the cause of their unusual emission and possible evolutionary relationships between them and other classes of NSs. In the wind braking model (Kou & Tong 2015; Tong & Kou 2017), as the pulsar tends to death, the radio emission will not stop immediately, the pulsar will be observed if the condition of electron pair production is met. Moreover, Zhang et al. (2007) suggested that the RRATs as well as the nulling pulsar and intermittent pulsars are located slightly below the death line and become occasionally active only when the conditions for pair production and coherent emission are satisfied. In Fig. 5, RRATs are dispersed on the both side of the death line for small mass SS. We might take RRATs as NSs on the verge of death.

PSR J2144−3933 is a radio-loud pulsar with long period of 8.5 s (Young, Manchester & Johnston 1999), and located far beyond all conventional death lines. And it seems to challenge the emission models. Many works discussed this problem by considering different types of the acceleration models (Zhang et al. 2000; Gil & Mitra 2001). In Figs 4 and 5, the location of this pulsar in |$P\text{--}\dot{P}$| diagram is close to and just cross to the pulsar death lines for 2.0 M_⊙. In the traditional radio emission model, we could suggest that PSR J2144−3933 may have a high mass, which would be larger than 2.0 M_⊙.

As discussed in Section 2, constraints on EoSs will give us more information about the pulsar death line that would help us to understand the observational properties of various classes of NSs. The maximum and minimum mass of the NSs measured from observational data provides additional direct constraints on the EoS, and the moment of inertia is a powerful probe of its internal structure and its EoS (Lattimer & Schutz 2005; Raithel, Özel & Psaltis 2016). The dense matter EoS is the main scientific objective for a number of different telescopes over the next decade, operating in very different wavebands, such as the Square Kilometre Array (SKA, Bourke et al. 2015), the Five hundred metre Aperture Spherical radio Telescope (FAST, Nan et al. 2011), Neutron Star Interior Composition Explorer Mission (NICER, Gendreau, Arzoumanian & Okajima 2012), Enhanced X-ray Timing and Polarization mission (eXTP, Zhang et al. 2016), etc. For pulsar death lines, if the moment of inertia of the pulsar is well measured, they will provide a new avenue into our understandings of the pulsar death line. Till now, the most precise measurement of the mass of compact objects is in the binary systems, and isolated objects are very difficult to determine their masses. In this work, we used the X-ray luminosity to estimate the radii of the compact objects. The emission area that could give us only a lower limit of the surface area of the star. Comparing current radius measurements with those obtained from other techniques will further reduce the uncertainties in the NS radii. In this work, the pulsar death line is discussed with multiwavelength observational properties, such as radio, X-ray. We expect that multiple observational facts would combine with each other to help us know more about pulsars.

Acknowledgements

We gratefully thank Dr. Li Ang for useful discussions and comments. We are also grateful to Dr. Rai Yuen for a careful reading of the manuscript, who made a number of valuable suggestions, and to an anonymous reviewer for the thoughtful comments that helped significantly to improve the clarity of this work. This work is supported by the National Natural Science Foundation of China (No. 11373006) and the West Light Foundation of Chinese Academy of Sciences (No. ZD201302).

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