Period clustering of the anomalous X-ray pulsars

# Period clustering of the anomalous X-ray pulsars

G.S. Bisnovatyi-Kogan    N.R. Ikhsanov
###### Abstract

In this paper we address the question of why the observed periods of the Anomalous X-ray Pulsars (AXPs) and Soft Gamma-ray Repeaters (SGRs) are clustered in the range 2–12 s. We explore a possibility to answer this question assuming that AXPs and SGRs are the descendants of High Mass X-ray Binaries (HMXBs) which have been disintegrated in the core-collapse supernova explosion. The spin period of neutron stars in HMXBs evolves towards the equilibrium period, averaging around a few seconds. After the explosion of its massive companion, the neutron star turns out to be embedded into a dense gaseous envelope, the accretion from which leads to the formation of a residual magnetically levitating (ML) disk. We show that the expected mass of a disk in this case is which is sufficient to maintain the process of accretion at the rate  g/s over a time span of a few thousand years. During this period the star manifests itself as an isolated X-ray pulsar with a number of parameters resembling those of AXPs and SGRs. Period clustering of such pulsars can be provided if the lifetime of the residual disk does not exceed the spin-down timescale of the neutron star.

Accepted for publication in Astronomy Reports, vol. 59, No.5 (2015)

address=Space Research Institute of RAS, 84/32 Profsoyuznaya Str, Moscow 117997, Russia, and
National Research Nuclear University “MEPhI”, Kashirskoye shosse 31, Moscow 115409, Russia

address=Pulkovo Observatory, Pulkovskoe Shosse 65, Saint-Petersburg 196140, Russia, and
Saint Petersburg State University, Universitetsky pr., 28, Saint Petersburg 198504, Russia

Period clustering of the anomalous X-ray pulsars

,

missing

Keywords:  Accretion and accretion disks, X-ray binaries, neutron star, pulsars, magnetic field, anomalous X-ray pulsars, Soft gamma-ray repeaters

PACS:  97.10.Gz, 97.80.Jp, 95.30.Qd

## Introduction

In the previous paper McGill (2012) we have explored the possibility of modelling the spin evolution and X-ray emission of Anomalous X-ray Pulsars (AXPs) and Soft Gamma-ray Repeaters (SGRs) within a scenario of magnetic-levitation accretion MCGILL (2012). We have considered a magnetized isolated neutron star accreting material from a residual, slowly rotating magnetically-levitating disk (ML-disk). Using the parameters of this process estimated in MCGILL (2012, 2012, 2012, 2012, 2012, 2012, 2012), we came to a conclusion that the basic features of the X-ray emission from AXPs and SGRs can be explained by the magnetic-levitation accretion model without an assumption about super-strong dipole magnetic field of these neutron stars to be invoked. In particular, the magnetic field strength on the surface of AXPs and SGRs, evaluated within this approach to fit their observed spin-down rates, is in the range  G, and the expected black-body temperature of the pulsar X-ray radiation  keV, which is close to the observed value McGill (2012).

One of the possible mechanisms of flaring activity of these objects in gamma-rays mentioned in our previous paper is a spontaneous release of energy accumulated in a non-equilibrium layer of super-heavy nuclei located at the lower boundary of the upper crust of a low- or moderate-mass neutron star, suggested in MCGILL (2012, 2012, 2012). In the frame of this approach, the AXPs and SGRs can be distinguished by a low mass of the neutron stars that provides the maximum amount of nuclear power in their crust for the minimum energy necessary to trigger its release. The formation of such objects is likely to be a rather rare event, which can account for their relatively low fraction (about 1%) in the family of neutron stars MCGILL (2012, 2012).

At the same time, observational data on the radio-pulsar J 1518-4904, composing a close binary system with another neutron star, favor a possibility for such objects to exist. The pulsar mass measured at the confidence level of 95.4% is , while its companion has a mass of MCGILL (2012). In the previous paper McGill (2012) we have already mentioned that a light neutron star can be formed due to off-center nuclear outburst induced by the core-collapse of the star in the process of supernova explosion. This scenario was studied in MCGILL (2012), presenting the modeling of the SN1b explosion in terms of collapse of the stellar remnant consisting of the iron core of the mass of , surrounded by the helium and carbon-oxygen shells. It has been noted that a rapid temperature rise during the collapse of such a remnant leads to the thermonuclear explosion and, finally, to the ejection of both shells. However, up to now no detailed simulations of this scenario have been performed.

In this paper we discuss a period clustering of AXPs and SGRs in a relatively narrow interval from 2 to 12 seconds. Analysis of this phenomenon is traditionally performed under assumption about a relatively young age of these objects ( years), which is based on the estimates of their spin-down timescale, , and/or of the age of supernova remnants associated with them. Here is the spin period of a pulsar and is its current spin-down rate. In this case, the model of rotational evolution of AXPs and SGRs can be constructed by making additional assumptions that either these neutron stars have initially super-strong and rapidly decaying dipole magnetic fields (see MCGILL (2012, 2012) and references therein), or they prove to be in the state of exceptionally intensive fall-back accretion just after their birth (see MCGILL (2012) and references therein).

In both scenarios, there is only remote possibility to find an AXP with the period significantly smaller than 1 s, since the characteristic time over which the spin period of a neutron star rises from an initially very short value up to a few seconds is significantly smaller than . A probability to discover a star with a spin period significantly in excess of the observed values is also negligible provided either the decay time of super-strong magnetic field (in the magnetar model) or the life-time of a residual disk in the model of fall-back accretion does not exceed . Thus, in the frame of these scenarios the observed period clustering of AXPs is caused mainly by the selection effect. In this context, the neutron star acquires the capacity to display gamma-ray flares exclusively at the moment of birth and gradually loses this ability on the timescale .

In our previous paper McGill (2012), we have attempted to apply the above approach to study the spin evolution of a neutron star undergoing fall-back accretion from a non-Keplerian magnetic disk. We have pointed out that it is difficult to explain the observed period clustering of AXPs in the frame of this approach without additional assumptions which essentially narrow the range of possible scenarios. This has stimulated us to search for alternative solutions, one of which is discussed in this paper.

A suggested scenario is based on the hypothesis that AXPs and SGRs are descendants of the neutron stars which had been X-ray pulsars in High Mass X-ray Binaries (HMXBs)during a previous epoch. The reason for this is provided by a simple fact that information on the characteristic spin-down timescale of the pulsar, , or on the age of a supernova remnant in which it is located, is not sufficient for correct evaluation of its genuine age. This information can only allow to assert that the time interval from the star transit into the state of regular spin-down (in which it is observed in the present epoch) does not exceed . The true age of the star can, however, substantially exceed .

As an example of such situation one can consider the disintegration of a HMXB due to the core-collapse of its massive component in the process of supernova explosion. The old neutron star which had evolved as a member of a HMXB during the main sequence evolution of its massive companion ( years), after disruption of the binary becomes isolated and embedded into a supernova remnant of significantly younger age. Under the conditions of interest its spin period at the moment of system disintegration is about a few seconds (see Section Pulsar period in a High-Mass X-ray Binary). Capturing matter from the envelope ejected by its exploded companion, this star can find itself in the state of an isolated X-ray pulsar accreting material onto its surface from a residual disk (see Section  Formation of a residual disk). Main features of such sources are briefly discussed and compared with observed parameters of AXPs and SGRs in Section Conclusions.

## Pulsar period in a High-Mass X-ray Binary

The majority of presently known X-ray pulsars are the members of HMXBs, which are close pairs consisting of a massive O/B-star and a neutron star with strong magnetic field. Their X-ray emission is generated due to accretion of matter onto the neutron star surface in the magnetic polar regions. The period of a pulsar corresponds to the spin period of a neutron star, , and its luminosity, , is determined by the rate of mass accretion onto the stellar surface, , where is the mass and is the radius of a neutron star. The period of such a pulsar evolves according to the equation

 2πI˙ν=Ksu−Ksd (1)

towards the equilibrium period, , whose value is defined by the balance of spin-up, , and spin-down, , torques, exerted on the star by the accretion flow. Here is the moment of inertia of a neutron star, is the frequency of its axial rotation and .

According to modern views, HMXBs are the descendants of high-mass binaries, which survived a supernova explosion caused by a core-collapse of their more massive component. A neutron star born in the course of this event (the first supernova explosion) possesses strong magnetic field and rotates with a period of a fraction of a second. Then it is spinning down because of magneto-dipole losses (the ejector state) and later on due to interaction between its magnetic field and the surrounding plasma (the propeller state). When its spin period reaches the critical value determined by the equality of its corotation radius, , and the magnetosphere radius, , the star passes to the accretor state and manifests itself as an X-ray pulsar. Here is the angular velocity of its axial rotation.

Numerical simulations of this scenario MCGILL (2012) indicate that a neutron star with a sufficiently strong initial magnetic field at the final stage of its evolution in a HMXB undergoes accretion from a Keplerian disk and its spin period evolves towards the equilibrium period whose value can be estimated as

 P(Kd)eq≃3s × k1/2t μ6/730 m5/7 ˙M−3/717. (2)

Here is a dimensionless parameter of the order of unity, is the dipole magnetic moment of the neutron star in units of , is the neutron star mass in units of and is the mass accretion rate onto the stellar surface in units of  g/s. The age of the neutron star by this moment is

 tms≃6×106 (Mopt20M⊙)−5/2 yr, (3)

that corresponds to the average time of its massive component (of the mass ) evolution on the main sequence MCGILL (2012).

As the massive star begins to evolve off of the main sequence, its radius increases and it fills its Roche lobe. In this case the mass exchange between the system components proceeds on the thermal timescale,  yr, in the form of a stream through the Lagrangian point L1 at the rate  g/s. Here and are the radius and the luminosity of the massive component determined in general case by its mass (see MCGILL (2012) and references therein). For the stars with the mass , the rate of mass exchange between the system components falls in a relatively narrow interval  g/s. Thus, the main parameter determining the spin period of the neutron star at the final stage of the HMXB evolution, , is the strength of its dipole magnetic field.

Dependence for a light () neutron star accreting material from a Keplerian disk at the average rate is presented in the Figure. Dashed region shows the observed period range of AXPs and SGRs. As can be seen in the figure, the spin period of a neutron star at the final stage of a HMXB evolution falls into the dashed region provided its dipole magnetic moment is in the range . For the average stellar radius of 12 km MCGILL (2012) this corresponds to the dipole field strength of  G on the stellar surface in the region of its magnetic poles.

The absolute majority of presently known neutron stars possess the magnetic field within the above mentioned limits. This is valid for both the isolated radio-pulsars and X-ray pulsars in the HMXBs, in which the surface magnetic field was estimated through observations of the cyclotron line in their X-ray spectra (see MCGILL (2012, 2012) and references therein). The periods of these pulsars at the final stage of a HMXB evolution may tend to clustering around an average value, , which is of the order of a few seconds for the parameters of interest.

However, it should be mentioned that the magnetic field strength of only 7 out of 15 pulsars considered in the previous paper McGill (2012), fall within above mentioned interval. According to our estimates, the lower boundary of this interval exceeds the lower limit to the magnetic field strength of the remaining eight X-ray sources (McGill (2012), Table 3). This can be connected with oversimplification of our model in which the rotation velocity of matter in the residual magnetically-levitating disk was adopted to be zero. One cannot totally exclude a possibility of enhanced dissipation of the neutron star magnetic field and/or change of its configuration during the period of its flaring activity. This can be, in particular, stimulated by the anomalous heating of the neutron star crust, if the source of its magnetic field is located relatively close to its surface. Finally, it is worthwhile to mention a large diversity of possible scenarios of the final stage of the HMXBs evolution, and to notice that realization of our hypothesis is expected predominantly within a compact model of pre-supernova, which was used for interpretation of the SN 1987A MCGILL (2012).

## Formation of a residual disk

Evolution of the binary system at the phase of a HMXB ends up with a core-collapse of its massive component accompanied with a supernova explosion and ejection of a massive shell (). As a result, the second neutron star or a black hole is born. Depending on the energetics and geometry of the explosion, a HMXB can either become a system of two degenerate objects or disintegrate into two isolated compact stars with the latter case being more probable MCGILL (2012). The characteristic time of system disintegration, , is determined by the initial orbital separation, , and the value of their spatial velocity, , achieved in the process of supernova explosion.

Because of the envelope ejection in the supernova explosion, both stars turn out to be embedded into a dense gaseous medium. The velocity of the main part of the ejecta at the initial phase of its expansion significantly exceeds  km/s and reaches 10 000 km/s in the outermost layers. This powerful plasma flow interacts with the old neutron star and disrupts the accretion structure formed around its magnetosphere in the previous epoch (e.g. a Keplerian accretion disk) before final dispersion into an extended supernova remnant. However, the initial expansion velocity of the inner portion of the ejecta is significantly smaller. Calculations MCGILL (2012, 2012) have shown that the innermost parts of the ejecta with the mass under remain gravitationally bound with the young compact object, and soon after the supernova explosion return to a new-born star in the form of fall-back accretion flow. The mass of this portion of matter which is in the state of free expansion with a relatively slow velocity  km/s does not exceed MCGILL (2012). However, it is this material which can form a massive magnetically-levitating disk surrounding the magnetosphere of the old neutron star. Accreting from this disk, the star shows itself as an isolated X-ray pulsar with the period . In this Section we show that the life-time of such a pulsar can be as long as a few thousand years.

The amount of mass with which an old neutron star interacts in a unit time moving with a relative velocity can be estimated as follows

 ˙Mcap=πr2Gρenvvrel≃1020gs−1×m2v−38a−313(M010−3M⊙). (4)

Here is the Bondi radius of the neutron star, is the mean density of the material in the inner part of the envelope with the mass contained inside radius ,  cm and  cm.

The total amount of matter which can under favorable conditions be captured by the neutron star over the time span from this part of the envelope is

 Mcap≃10−8M⊙ × m2a−213v−38(M010−3M⊙)(vkick300km/s)−1. (5)

The parameter in this expression is normalized to its average value derived in MCGILL (2012, 2012).

The structure of an accretion flow forming by the captured material inside Bondi radius is determined by the relative velocity of the star, , as well as by physical parameters of the gas and magnetic field strength in the surrounding envelope. As has been recently shown by Ikhsanov et al. (see MCGILL (2012) and references therein), a scenario of quasi-spherical accretion can be realized under the condition , where

 vma≃3000kms−1×β−1/50 μ−6/3530 ˙M3/3520 m12/35 c2/57. (6)

Here is the ratio of the thermal, , to magnetic, , energy in the material captured by the star at its Bondi radius. Parameters , and denote the density, the magnetic field strength and the sound speed in the accretion flow, respectively. If , where

 vkd≃30kms−1×ξ3/70.2β1/70m3/7c−2/77(Porb100days)−3/7, (7)

the material captured by the star is accumulated in a Keplerian accretion disk McGill (2012). Here is the orbital period of the HMXB and is a parameter accounting for the angular momentum dissipation in the quasi-spherical non-magnetic flow, normalized on its average value obtained in MCGILL (2012).

Finally, in the intermediate case , we expect realization of magnetic-levitation accretion scenario in which the matter captured by the neutron star is accumulated around its magnetosphere in a form of non-Keplerian magnetically levitating disk and moves towards the star in the diffusion regime. The outer radius of the ML-disk is determined by the Shvartsman radius MCGILL (2012),

 Rsh=β−2/30rG(cs(rG)vrel)4/3, (8)

at which the magnetic pressure in the free-falling gas reaches its ram pressure. In the presence of large-scale magnetic field with the inhomogeneity scale in excess of Bondi radius, the initially quasi-spherical flow rapidly decelerates and transforms its geometry into a ML-disk MCGILL (2012, 2012).

The inner radius of a ML-disk corresponds to the magnetosphere radius of the neutron star and equals MCGILL (2012, 2012)

 rma=(cm2p16√2ekB)2/13α2/13Bμ6/13(GMns)5/13T2/130L4/13XR4/13ns. (9)

Here and are the proton mass and the electron charge, and is the Boltzmann constant. is the gas temperature in the diffusion layer at the magnetosphere boundary (magnetopause) , and is a dimensionless parameter, expressing the ratio of the effective coefficient of the accretion flow diffusion into the magnetic field of the star, , to the Bohm diffusion coefficient.

The mass of a ML-disk forming in this scenario is

 Md=4πRsh∫rmaρ(r)hz(r)rdr, (10)

where is the density of the disk, and is its half-thickness.These parameters can be estimated taking into account that the gaseous (as well as the magnetic) pressure in the ML-disk reaches its maximum,

 ρ(rma)c2s(rma)=μ22πr6ma, (11)

at the inner radius of the disk and decreases with distance from the star as MCGILL (2012, 2012). Taking into account that the gas temperature in the disk is

 T(r)=(˙MGMns4πr3σSB)1/4, (12)

and, correspondingly, the sound speed is , the density distribution in the radial direction is

 ρ(r)=ρ(rma)(rrma)−7/4, (13)

where is the Stefan-Boltzmann constant. Finally, the half-thickness of the disk can be estimated as MCGILL (2012, 2012)

 hz(r)=(kBT(r)r3mpGMns)1/2. (14)

Substituting (9), (8) and (1214) to (10) and taking into account that under the conditions of interest , we find

 Md≃10−7M⊙×α−7/30.1β−11/120μ5/1330 ˙M99/10420 m25/52c11/67v−55/128, (15)

where parameter is normalized following MCGILL (2012). Thus, if an old neutron star captures material from the envelope having been ejected in the collapse of its massive companion, this can result in formation of a ML-disk with the mass sufficient to provide the process of accretion at the rate  g/s over a time span of a few thousand years. Within this scenario, the ML-disk is composed of the matter located in the inner parts of the ejecta where the expansion velocity does not exceed 1000 km/s.

## Conclusions

We show that isolated X-ray pulsars with the periods of a few seconds can be descendants of the HMXBs. Within our scenario, these objects are old ( years) neutron stars with the dipole magnetic field in the range  G, which accrete matter onto their surface from a residual slowly rotating ML-disk. Under these conditions, the spin periods of the pulsars are increasing at a high rate and their life-time is limited by that of a residual disk. If this time does not exceed the spin-down timescale of a neutron star, their periods are expected to cluster around an average value of a few seconds. If the life-time of a residual disk is smaller than the typical dissipation time of a supernova remnant, these neutron stars can be embedded into the nebulosity forming in the explosion of their massive HMXB companion, which resulted in the disintegration of the binary system during the previous epoch.

It is quite noticeable that the objects described above are similar to AXPs and SGRs in some of their manifestations. In particular, the periods of isolated X-ray pulsars which are descendants of HMXBs are tending to clustering around an average value which is of the order of a few seconds for the classical range of the neutron stars magnetic fields ( G). At the same time, these stars are regularly spinning down at a rather high rate due to accretion of matter from the ML-disk onto their surface. The sources arising in this scenario can be situated in the supernova remnants in spite of the fact that their age exceeds by an order of magnitude the average life-time of such nebulosities. On the other hand, the epoch during which a neutron star is a member of a HMXB is long enough for its spin period to increase from an arbitrarily small initial value to a few seconds. Finally, as was first mentioned by Mereghetti & Stella MCGILL (2012), the model of accretion onto a neutron star provides the simplest explanation of the spectral characteristics of the X-ray emission observed from AXPs and SGRs (see, also, MCGILL (2012, 2012, 2012, 2012); McGill (2012)).

An attempt to apply our scenario in order to associate AXPs with the descendants of HMXBs rises, however, a number of questions. First, within this approach one cannot eliminate a possibility that in the vicinity of AXP there exists a young compact supernova remnant formed in the core-collapse of the massive HMXB component. Discovery of such a source or its contribution to the energetics and/or structure of the nebulosity could help to estimate the prospects of a hypothesis that AXPs are descendants of HMXBs. Another question remaining open is why an old neutron star exhibits activity in the gamma-rays only after disintegration of the HMXB and merely during the phase of accretion from the ML-disk.

In the previous paper McGill (2012) we have mentioned that accretion of matter onto the stellar surface could be considered as one of possible triggers of the flares. On the other hand, the star was also in the state of accretion during its evolution as a part of the HMXB. One of the factors discriminating these phases of accretion could be the chemical composition of matter captured by the star from its surroundings. In particular, after the supernova explosion the accretion flow can be expected to be enriched with heavy elements. Accumulation of this material on the stellar surface may cause a thermonuclear flare which exceeds in power the super-flares in X-ray bursters MCGILL (2012, 2012, 2012) and can trigger reactions leading to the release of nuclear energy stored in the non-equilibrium layer of the neutron star crust. The degree of impact which this factor would have on the ability of neutron stars to produce gamma-ray flares will be addressed in one of the upcoming papers.

## Acknowledgments

The authors thank an anonymous referee for constructive criticism and stimulating comments, as well as N.G. Beskrovnaya and V.Yu. Kim for useful discussions and help in preparation of this manuscript. The work was partly supported by RFBR under the grants No. 13-02-00077 and No. 14-02-00728, SPbSU under the grant No. 6.38.18.2014, the President Support Program for Leading Scientific Schools NSH-261.2014.2, and the RAS Presidium Program No. 41.

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