ABSTRACT
Recent research on the classical pulsar B0950+08 demonstrates that the explanation of its high surface temperature by roto-chemical heating encounters some difficulties. We assume that there is a fall-back disc around the newborn neutron star, which originates from the supernova ejecta and influences the spin and magnetic evolution of the star. By taking into account disc accretion and magnetic field evolution simultaneously, the effect of the fall-back disc accretion process on the roto-chemical heating in the neutron star is studied. The results show that there are two roto-chemical deviation phases (spin-up deviation and spin-down deviation), but that only the spin-down deviation leads to heating. The specific cooling curve depends on the accretion disc mass, the initial magnetic field and the magnetic field decay rate. Most importantly, the observations of surface temperature, magnetic field strength and spin period of the classical pulsar B0950+08 are well explained by the accretion roto-chemical heating model. The fall-back accretion process is important in roto-chemical heating for explanations of classical pulsars with high temperature. Given the absence of any evidence of fall-back accretion on to B0950+08, our study is purely hypothetical.
1 INTRODUCTION
Neutron star (NS) thermal evolution is an important tool with which to probe the inner structure of these stars and the nature of dense matter (Yakovlev & Pethick 2004; Page, Geppert & Weber 2006). In nearly all standard cooling models, NSs cool down to a surface temperature of ∼104K over a period of about 106–107yr. However, some ultraviolet thermal emission from millisecond pulsars and from classical pulsars with a higher surface temperature has been observed. Millisecond pulsars J0437 – 4715 (Kargaltsev, Pavlov & Romani 2004) and J2124 – 3385 (Rangelov et al. 2017), whose spin-down age is ∼109 yr, show a surface temperature of about ∼10K, and the classical pulsar 5B0950 + 08 has a surface temperature of ~(1–3) × 105K with a spin-down age of ∼17.5 Myr (Pavlov et al. 2017). These findings naturally raise the idea that heating mechanisms are needed. A number of mechanisms have been studied, such as vortex-creep heating (Alpar et al. 1984; Shibazaki & Lamb 1989; Larson & Link 1999; Gonzalez & Reisenegger 2010), the crust-cracking mechanism (Cheng et al. 1992; Baym & Pines 1971) and the decay of exotic particles (Hannestad, Keranen & Sannino 2002), and in particular the roto-chemical heating mechanism is important for old neutron stars (Reisenegger 1995; Fernandez & Reisenegger 2005).
Gonzalez & Reisenegger (2010) argue that roto-chemical heating and vortex creep can be important both for classical pulsars and for millisecond pulsars. However, recent research into the classical pulsar B0950 + 08 (Pavlov et al. 2017) demonstrates that an explanation of its temperature by roto-chemical heating needs an implausibly short initial rotation period (P0 ≤ 10 ms). Roto-chemical heating encounters difficulties in explaining classical pulsars with high temperature. In this paper, we show that the fall-back accretion process can help to solve the problem in relation to the roto-chemical heating mechanism.
In previous roto-chemical heating studies, the spin-down caused by magnetic dipole radiation, for which the braking index is n = 3, is the main way to trigger the heating process. However, the braking indices have been measured in 13 pulsars (see Espinoza 2013, and references therein), and except for pulsars J0537 – 6910 and B1757 – 24, which have negative n, the pulsars have 0 <n < 3. The possible reasons for the deviation of n from 3 have been studied extensively. In particular, explanations of why most of the measured values are less than 3 include models invoking fall-back disc-assisted spin-down (e.g. Marsden, Lingenfelter & Rothschild 2001; Menou, Perna & Hernquist 2001; Alpar, Ankay & Yazgan 2001; Chen & Li 2006) and magnetic field increase (e.g. Blandford & Romani 1988; Pons, Viganò & Geppert 2012) in pulsars.
The spin evolution of a NS with a surrounding fall-back disc originating from the supernova (SN) ejecta has been investigated by many authors (e.g. Chatterjee, Hernquist & Narayan 2000; Alpar 2001; Menou et al. 2001; Ertan et al. 2009; Yan, Perna & Soria 2012). If the NS experiences accretion from a SN fall-back disc, the newborn pulsar could be accelerated to a period of several milliseconds, and the magnetic field would decay to ∼109G from an initial value of ∼1012 G (Liu & Li 2009). When the disc becomes neutral after ∼103–104 yr (Menou et al. 2001) and accretion ceases, the NS enters the radio phase, and the magnetic field re-diffuses. The fall-back accretion process and magnetic field evolution will affect the thermal evolution of the NS in the non-equilibrium state through their influence on the spin of the star.
In this paper, we assume that there is a fall-back disc around a newborn NS. Depending on the accreted mass, the NS magnetic fields decays to some extent, and increases after accretion stops. By taking into account the disc accretion and field evolution simultaneously, we model the evolution of the surface temperature, chemical deviation, magnetic field and spin period of the NS, and compare the calculated distributions with observations of the classical pulsar B0950 + 08. The structure of this paper is as follows. The thermal evolution and chemical evolution theory of the NS are described in Section 2. The fall-back disc accretion model including consideration of the magnetic field evolution is introduced in Section 3. We present the calculated results for various parameters and compare them with the observations in Section 4. The discussion and conclusion are given in Section 5.
2 THERMAL EVOLUTION OF THE NS IN THE NON-EQUILIBRIUM STATE
In previous studies, pure magnetic dipole radiation was assumed to be the braking mechanism for the NS’s spin-down and the magnetic field was constant. However, consideration of the fall-back disc accretion changes the spin evolution and magnetic field evolution, which affects the cooling process of the star definitively.
3 FALLBACK DISC ACCRETION WITH MAGNETIC FIELD EVOLUTION
where B0 is the initial magnetic field strength, ΔM is the mass already accreted on the NS, and M0 is a parameter that determines the rate of decay, with a typical value of ∼10−5–10−4 M⊙ (Taam & van den Heuvel 1986; Shibazaki et al. 1989; Romani 1990).
The propeller phase begins when |$\dot{M}$| decreases to Rc < Rm. The infalling matter is assumed to be accelerated outwards owing to the centrifugal barrier, taking away the angular momentum of the NS (Illarinov & Sunyaev 1975). The same formula as in the accretor phase was used to estimate the propeller spin-down torque.
The magnetic field re-diffusion equation (11) couples to the thermal evolution equation (5) through the spin evolution equation (10) and chemical evolution equation (4) in our model.
4 RESULTS
We consider a uniform star model for the simulation and take a baryon number density of n = 1.0 fm−1. The spin angular velocity and surface magnetic field of the NS change with the accretion evolution, which affects the cooling process of the star. We integrate the chemical, thermal, magnetic field and spin evolution equations, and the accretion model parameters are listed in Table1. Figs 1–4 show the magnetic field evolution, spin angular velocity evolution, chemical evolution and thermal evolution for different parameters. In order to analyse the effect of the accretion clearly, we choose a set of parameters for comparison in eachfigure.
Surface magnetic field, spin frequency, chemical deviation and surface temperature as a function of time, with models with differents values of the parameters. Model 1, model 2 and model 3 are represented by the solid line, dashed line and dotted line, respectively. The units of the temperature, chemical deviation, magnetic field, spin angular velocity and time are K, K, G, s−1 and s respectively.
Surface magnetic field, spin frequency, chemical deviation and surface temperature as a function of time, with different models of parameters. Model 1 and model 4 are represented by the solid line and dashed line, respectively. The units of the temperature, chemical deviation, magnetic field, spin angular velocity and time are K, K, G, s−1 and s respectively.
Surface magnetic field, spin frequency, chemical deviation and surface temperature as a function of time, with different model of parameters. Model 1 and model 5 are represented by the solid line and dashed line, respectively. The units of the temperature, chemical deviation, magnetic field, spin angular velocity and time are K, K, G, s−1 and s respectively.
Surface magnetic field, spin frequency, chemical deviation and surface temperature as a function of time, with different model of parameters. Model 1 and model 6 are represented by the solid line and dashed line, respectively. The units of the temperature, chemical deviation, magnetic field, spin angular velocity and time are K, K, G, s−1 and s respectively.
Initial model parameters for the fall-back disc.
| Model No. | Md(M⊙) | Rd, 10 | P0(ms) | B0(1012G) | M0(10−5 M⊙) |
|---|---|---|---|---|---|
| 1 | 0.28 | 0.1 | 14 | 3 | 1 |
| 2 | 0.1 | 0.1 | 14 | 3 | 1 |
| 3 | 0.02 | 0.1 | 14 | 3 | 1 |
| 4 | 0.28 | 0.1 | 14 | 3 | 10 |
| 5 | 0.28 | 0.1 | 14 | 10 | 1 |
| 6 | 0.28 | 0.01 | 14 | 3 | 1 |
| Model No. | Md(M⊙) | Rd, 10 | P0(ms) | B0(1012G) | M0(10−5 M⊙) |
|---|---|---|---|---|---|
| 1 | 0.28 | 0.1 | 14 | 3 | 1 |
| 2 | 0.1 | 0.1 | 14 | 3 | 1 |
| 3 | 0.02 | 0.1 | 14 | 3 | 1 |
| 4 | 0.28 | 0.1 | 14 | 3 | 10 |
| 5 | 0.28 | 0.1 | 14 | 10 | 1 |
| 6 | 0.28 | 0.01 | 14 | 3 | 1 |
Initial model parameters for the fall-back disc.
| Model No. | Md(M⊙) | Rd, 10 | P0(ms) | B0(1012G) | M0(10−5 M⊙) |
|---|---|---|---|---|---|
| 1 | 0.28 | 0.1 | 14 | 3 | 1 |
| 2 | 0.1 | 0.1 | 14 | 3 | 1 |
| 3 | 0.02 | 0.1 | 14 | 3 | 1 |
| 4 | 0.28 | 0.1 | 14 | 3 | 10 |
| 5 | 0.28 | 0.1 | 14 | 10 | 1 |
| 6 | 0.28 | 0.01 | 14 | 3 | 1 |
| Model No. | Md(M⊙) | Rd, 10 | P0(ms) | B0(1012G) | M0(10−5 M⊙) |
|---|---|---|---|---|---|
| 1 | 0.28 | 0.1 | 14 | 3 | 1 |
| 2 | 0.1 | 0.1 | 14 | 3 | 1 |
| 3 | 0.02 | 0.1 | 14 | 3 | 1 |
| 4 | 0.28 | 0.1 | 14 | 3 | 10 |
| 5 | 0.28 | 0.1 | 14 | 10 | 1 |
| 6 | 0.28 | 0.01 | 14 | 3 | 1 |
Fig. 1 shows the evolution of the star with different accretion disc masses Md. According to the assumption of the accretion process that |$\dot{M}_{\rm d}(t_{0})=M_{\rm d}/(5t_{0})$|, a larger total accretion disc mass means a higher accretion rate, which leads to a faster spin-up and higher angular velocity at the end of the accretor phase. Meanwhile, more matter accretion on to the NS surface results in a smaller surface magnetic field. Although there is an obvious spin-up process and a large chemical deviation (spin-up deviation), the roto-chemical heating effect does not appear at this time, because the temperature of the star is too high. When the star goes into the radio pulsar phase and spins down, another chemical deviation (spin-down deviation) occurs. A faster spin angular velocity at the beginning of the radio phase leads to a subsequently larger chemical deviation. The magnetic field re-diffusion affects the chemical deviation process simultaneously. Owing to the re-diffusion, the surface magnetic field increases in the radio phase, which leads to a faster spin-down process and strengthens the chemical deviation. The start time of the magnetic re-diffusion influences the chemical deviation curve. If the re-diffusion begins before the chemical deviation reaches the quasi-equilibrium state (model 3), the chemical deviation will have a larger peak value but shorter duration. In contrast, the chemical deviation will have a plateau (model 1 and 2). Furthermore, the heating effect appears in the spin-down stage because the temperature of the star is now low enough, which has the same effect as the chemical deviation. Although the heating effect is not observable in the accretor phase, the accretion process influences the chemical deviation and cooling through the effect on the spin and magnetic field evolution, and the cooling curve is sensitive to the value of the accretion rate.
Figs 2 and 3 display the evolution of the star with different initial magnetic field strengths B0 and values of the magnetic decay parameter M0. Both B0 and M0 affect the evolution of the magnetic field in the accretor phase, but they have no influence on the spin angular velocity at this phase, because the spin evolution in the accretor phase depends on the accretion rate. Only when the star is in the radio phase does the evolution of the magnetic field affect the spin evolution. As discussed above, only the spin-down process has a heating effect. A larger magnetic field (model 4 in Fig.2 and model 5 in Fig. 3) at the end of the accretor phase causes an earlier occurrence of the radio phase, which leads to an earlier appearance of spin-down heating.
Fig. 4 shows the influence of the initial disc radius Rd on the evolution of the star. The initial disc radius has a slight effect on the accretion rate of the star, which leads to a small difference in the angular velocity in the early stages. There is no obvious influence on the thermal evolution of the star.
Surface temperature as a function of spin period, with an initial magnetic field strength B0 = 3 × 1012G, initial spin period 14 ms, accretion disc mass 0.01M⊙, magnetic decay parameter M0 = 10−5M⊙, and initial disc radius 109cm. The error bar shows the observations for PSR B0950 +08 (Pavlov et al. 2017). The units of the temperature and spin period are K and ms, respectively.
Surface magnetic field as a function of spin period, with the initial magnetic field strength B0 = 3 × 1012G, initial spin period 14 ms, accretion disc mass 0.01 M⊙, magnetic decay parameter M0 = 10−5 M⊙, and initial disc radius 109cm. The error bar shows the observations for PSR B0950+08 (ATNF pulsar data base http://www.atnf.csiro.au/research/pulsar/psrcat/ and Manchester et al. 2005) The units of the magnetic field and spin period are G and ms respectively.
We compare the theoretical accretion roto-chemical heating model with the observations of PSR B0950+08 in Figs 5 and 6, and the initial parameters are shown in the figures. The theoretical results of the model for the spin period and surface temperature accord well with the observations of the classical PSR B0950 + 08, and the magnetic field strength is 1.41 per cent higher than the observed value. The accretion roto-chemical heating model can explain the observations of this classical pulsar well with reasonable initial parameters.
5 CONCLUSION AND DISCUSSION
In the present work, we have demonstrated the effect of the fall-back disc accretion process on roto-chemical heating in a NS. The accretion leads to spin-up in the early stage, together with spin-down in the radio pulsar phase, so there are two roto-chemical deviation phases (spin-up deviation and spin-down deviation), but only the spin-down deviation leads to a heating effect in the star. The specific cooling curve is strongly related to the accretion disc mass, initial magnetic field and the magnetic decay rate. Furthermore, we tested the model with the observations of the classical PSR B0950+08. The theoretical surface temperature, surface magnetic field strength and spin period explain the observations well, which means that roto-chemical heating is still a reasonable heating mechanism for classical pulsars and the fall-back accretion process is important. Because most recent observations do not find an IR counterpart of a debris disc around B0950+08 (Wang et al. 2014), our study is a purely hypothetical.
In order to explain the behaviour of an old pulsar (∼109 yr) with higher surface temperature (∼105K), the roto-chemical heating mechanism has been investigated extensively. Nucleon superfluidity (Petrovich & Reisenegger 2010) and the effect of a deconfinement phase transition (Wei & Zheng 2011) have been suggested to promote the heating efficiency and increase the theoretical surface temperature of the star, which improves the agreement between theory and observations. However, roto-chemical heating encounters a difficulty in explaining classical pulsars with high surface temperatures, such as pulsar B0950+08. The traditional roto-chemical heating in classical pulsars is driven by the chemical imbalance built up during their early spin-down, and is therefore strongly sensitive to the initial rotation period. A short initial period (P0 ≤ 10 ms) is needed to explain the temperature of B0950, in order to allow the pulsar to accumulate a large enough chemical imbalance by its current age. The implausibly short initial period in the traditional model makes it difficult to explain the observed period of the star (P= 253 ms). However, the fall-back accretion process affects the thermal, spin and magnetic field evolution of the star simultaneously, with which roto-chemical heating reasonably explains the observations of B0950. In the accretion roto-chemical heating model, the heating is still driven by the chemical imbalance built up during the spin-down process; however, it is sensitive to the spin and spin-down rate when the spin-down begins, which depends on the initial period and the accretion process. Accretion leads the star to spin up and spin fast for a long time (∼104yr), which guarantees that the spin-down process can accumulate enough heating energy. Meanwhile, the magnetic field’s re-diffusion provides stronger spin-down in the later stage, which enhances the heating efficiency and slows down the rotation.
It has been shown (Fernandez & Reisenegger 2005) that stars with roto-chemical heating will reach a quasi-steady state in which their temperature and chemical imbalance are determined only by their current spin and spin-down rate, almost independently of their temperature evolution history, but still sensitive to their spin evolution history. In particular, the roto-chemical heating in the classical pulsars is strongly sensitive to their initial period (Gonzalez & Reisenegger 2010). The fall-back accretion process influences the spin of the star, which means that it will affect the roto-chemical heating definitively. A millisecond pulsar with high temperature, such as PSR J2124-3358, can also be explained by the accretion roto-chemical heating model. With the same initial spin period (14 ms) and magnetic field (∼3 × 1012G) as the classical pulsar, but a much larger accretion disc mass (∼0.42 M⊙), accretion roto-chemical heating provides a high temperature of about 105 K at ∼109yr. The higher accretion rate makes the star spin rapidly (∼5 ms) and leads to a lower surface magnetic field (∼10G) in old age. 8Accretion roto-chemical heating is similar to the traditional model for the explanation of millisecond pulsars. But fall-back accretion is important for classical pulsars.
Joule heating, caused by the Ohmic decay of the magnetic field with Hall drift in the neutron star crust, may hold a neutron star as hot as 0.5 MK after 2 Myr if the crustal field is strong enough (Miralles, Urpin & Konenkov 1998; Pons, Miralles & Geppert 2009; Geppert & Viganò 2014), and can maintain the observed surface temperature of PSR B0950+08 too. Certainly, Joule heating depends on the NS model and initial field configurations. For a surface temperature of 0.1–0.3 MK as in B0950 with a spin-down age of 17.5 Myr, an initial dipole magnetic field of 1013 G is needed for the Joule heating model (Miralles et al. 1998; Geppert & Viganò 2014). In our accretion roto-chemical heating model, the initial dipole magnetic field is about 1012G and the magnetic field of B0950 is about 2.4 × 1011G, which probably means that the Joule heating effect is not substantial in the accretion roto-chemical heating model. In addition, compared with roto-chemical heating with an isotropic heating effect, a particularly impressive example for the Joule heating is the creation of small hot spots at the surface of NSs as a consequence of the Hall drift appearing in the crust of highly magnetized neutron stars (Viganò et al. 2013; Geppert & Viganò 2014). Until now only one thermal spectrum emitted from the bulk of B0950 surface has been observed (Pavlov et al. 2017), which probably means that the heating coming from the core is dominant.
We used a very simplified model to investigate the effect of the fall-back accretion on the roto-chemical heating mechanism, and some details are ignored, such as accretion-driven crustal heating by pycno-nuclear processes (the deep-crustal heating), magneto-thermal coupled effects and so on. The deep-crustal heating provides an efficient heating process in the neutron stars crust with thermal luminosity Lq ≈ 1031–1033 erg s−1 (Wijnands, Degenaar & Page 2013). However, in our accretion roto-chemical heating model, the accretion process ceases 105–106 s after birth. Thus the deep-crustal heating does not have an obvious heating effect at the time the roto-chemical heating appears, which is at about 0.1–1 Myr. For explanations of old classical pulsars (t ∼ 107 yr), such as pulsar B0950, the deep-crustal heating can be ignored in the accretion roto-chemical heating model.
2D magneto-thermal modelling will provide more details for our accretion roto-chemical heating, such as the temperature dependence of the electrical conductivity, which, in turn, controls the magnetic field evolution in the crust (see Geppert 2017 and references therein). Therefore a more detailed model with a consistent magneto-thermal simulation modelling is worth pursuing. Finally, our calculation is based on a uniform-density model without general relativity, which provides a rough understanding of the effect of the accretion process on roto-chemical heating. Taking the structure of the star into account in the frame of general relativity, using realistic equations of state of dense matter, and a more realistic cooling model could provide more accurate constraints, but this is left to our future work.
In conclusion, the fall-back disc accretion process is important for understanding high-temperature classical pulsars. However, the formation scenario of fall-back disc accretion and the magnetic field decay induced by it are subject to many uncertainties and further study is needed.
ACKNOWLEDGEMENTS
This work is supported by the Natural Science Foundation of China under grant numbers 11547021,11347108 and 11773011. WW thanks the Department of Physics and Astronomy at California State University Long Beach for a visiting scholar position and is grateful for financial support from the China Scholarship Council.
