a new look at anomalous x-ray pulsars

ISSN 1063-7729, Astronomy Reports, 2014, Vol. 58, No. 4, pp. 217–227. c© Pleiades Publishing, Ltd., 2014.Original Russian Text c© G.S Bisnovatyi-Kogan, N.R. Ikhsanov, 2014, published in Astronomicheskii Zhurnal, 2014, Vol. 91, No. 4, pp. 275–286.

A New Look at Anomalous X-Ray Pulsars

G. S Bisnovatyi-Kogan1, 2* and N. R. Ikhsanov3, 4**

1Space Research Institute, Russian Academy of Sciences,ul. Profsoyuznaya 84/32, Moscow, 117997 Russia

2MIFI National Nuclear Research University,Kashirskoye shosse 31, Moscow, 115409 Russia

3Central Astronomical Observatory, Russian Academy of Sciences,Pulkovskoe Shosse 65-1, St. Petersburg, 196140 Russia

4Saint Petersburg State University,Universitetskii pr. 28, Starii Peterhof, Saint Petersburg, 198504 Russia

Received September 25, 2013; in final form, October 10, 2013

Abstract—We explore the possibility of explaining Anomalous X-ray Pulsars (AXPs) and Soft Gamma-ray Repeaters (SGRs) in a scenario with fall-back magnetic accretion onto a young isolated neutron star.The X-ray emission of the pulsar in this case originates due to the accretion of matter onto the surface of theneutron star from a magnetic slab surrounding its magnetosphere. The spin-down rate of the neutron starexpected in this picture is close to the observed value. We show that such neutron stars are relatively youngand are going through the transition from the propeller state to the accretor state. The pulsar’s activity ingamma-rays is connected with its relative youth, and is enabled by energy stored in a non-equilibrium layerlocated in the crust of the low-mass neutron star. This energy can be released due to the mixing of matter inthe neutron star crust with super heavy nuclei approaching its surface and becoming unstable. The fissionof nuclei in the low-density region initiates chain reactions leading to a nuclear explosion. Outbursts areprobably triggered by instability developing in the region where the matter accreted by the neutron staraccumulates in the magnetic polar regions.

DOI: 10.1134/S1063772914040039

1. INTRODUCTION

Anomalous X-ray Pulsars (AXPs) and SoftGamma-ray Repeaters (SGRs) constitute a subclassof X-ray sources with regular pulsations in theirX-ray emission. One of the main features distin-guishing them from other X-ray pulsars is that allof them are isolated objects. None of the presentlyknown 12 AXPs and 11 SGRs is a member of a closebinary system [1]. Persistent pulsations with periodsPs ranging from 2 to 12 seconds and increasing ataverage rates of ν ∼ 10−11−10−14 Hz/s, can easilybe explained in models with a compact star with anon-uniform temperature distribution over its surfaceand a spin frequency ν = 1/Ps. The relatively smallspin period, as well as the soft X-ray spectrum andthe optical identification of some of these pulsars withsupernova remnants, leave little doubt that AXPs andSGRs are isolated neutron stars [2–5].

Most currently known isolated neutron stars be-long to a subclass of radio pulsars. Similar to AXPs

*E-mail: [email protected]**E-mail: [email protected]

and SGRs, these stars are in a spin-down state, andsome are sources of pulsating X-ray emission. How-ever, the similarity between these radio pulsars andAXPs and SGRs ends here. The X-ray luminosityof AXPs and SGRs, LX ∼ 1033−1035 erg/s, signif-icantly exceeds the spin-down rates of these neutronstars. Moreover, the unique signature of this subclassof X-ray pulsars is recurrent gamma-ray outbursts,during which energies of ∼1040−1042 erg are releasedon timescales of one to several seconds [6, 7], aswell as the giant gamma-ray outbursts observed fromSGRs, which are characterized by energy releases of1043−1046 erg during fractions of a second [8]. Dur-ing these events, the pulsar luminosity significantlyexceeds the Eddington limit, which eliminates thepossibility of explaining the gamma-ray outbursts interms of any accretion model. Finally, the spin periodsof AXPs and SGRs range in the narrow interval from2 to 12 s, which fundamentally distinguishes themfrom both ejecting and accreting pulsars, whose peri-ods lie in a huge interval covering six orders of mag-nitude (from 1.5 ms to tens of thousands of seconds).

Two most popular approaches can be distin-

217

218 BISNOVATYI-KOGAN, IKHSANOV

guished among numerous attempts to model AXPsand SGRs. The first associates the uniquenessof these objects exclusively with peculiar qualitiesof the neutron star itself, while the influence ofthe environment on the object’s evolution and thegeneration of high-energy radiation is assumed tobe insignificant. The most popular hypothesis ofthis type is the magnetar picture, which considersAXPs and SGRs to be a subclass of radio pulsarendowed with surface dipole magnetic fields as highas 1014−1015 G. In this case, the spin-down rateof a neutron star due to magneto-dipole emissionis in good agreement with observed values, and thegeneration of high-energy emission from the pulsar isdescribed in terms of the decay and flaring dissipationof its strong magnetic field [9, 10].

However, the magnetic field is not the only pos-sible energy source for the flaring activity of neutronstars. As was shown in [11–13], a non-equilibriumlayer consisting of super-heavy nuclei is formed inthe interior of the neutron star during the stage ofits rapid cooling. This layer, which is located at thebase of the upper crust of the star, is in static equi-librium at densities of 1010−1012 g/cm3. The nuclearenergy stored in a layer of mass ∼10−4 M� amountsto 1048−1049 erg, and can be released in nuclearexplosions initiated by the mixing of matter of theupper crust that leads to the transport of super-heavynuclei to regions of lower density, where they becomeunstable. During the lifetime of the layer (∼104 yr),this amount of energy is sufficient to provide both theX-ray luminosity of an AXP and more than 1000 re-curring gamma-ray outbursts from a SGR. The spin-down power of the neutron star in the initial versionof this scenario (see [13]) was associated with theoutflow of a relativistic wind generated due to nuclearexplosions on its surface.

The second approach to modeling AXPs andSGRs is based on scenarios with fall-back accretiononto a young neutron star, which can occur undercertain conditions soon after the supernova explosion[14–18]. In this case, the star turns out to besurrounded by a dense, gaseous remnant disk, fromwhich the star accretes material onto its surface [2].The numerical simulations of [19, 20] show thatthe rate of mass accretion from the remnant diskonto the stellar surface remains at levels of M ∼1013−1015 g/s over time spans of ∼104−105 years,sufficient to provide the X-ray luminosities of AXPsand SGRs. Analyzing the X-ray spectra of theseobjects, Trumper et al. [21, 22] showed that they canbe explained well in a model with disk accretion ontoa neutron star with a dipole magnetic moment μ =(1/2)B∗R3

ns in the range 1030−1031 G cm3, where B∗is the surface magnetic field of the neutron star and

Rns is its radius. This interprets AXPs and SGRs as asubclass of accreting pulsars in which the parametersof the isolated neutron star are close to their canonicalvalues.

Scenarios constructed using the second approachprovide the simplest and most consistent descriptionof the basic characteristics of the X-ray emissionfrom AXPs and SGRs. At the same time, theyleave unanswered the question of the nature of theflaring activity of these objects. It is customary toavoid this problem by making additional assumptionsabout very strong small-scale magnetic fields on theaccreting star surface, which could undergo flaringdissipation, causing flaring activity of the pulsar ingamma-rays [21, 22].

In this paper, we show that the age of such an ac-creting “quasi-magnetar” should significantly exceedthe ages of anomalous X-ray pulsars estimated fromtheir spin-down rates and the ages of the associatedsupernova remnants. As an alternative, we suggesta scenario in which the flaring activity of AXPs andSGRs is due to nuclear explosions resulting from therelease of energy stored in the non-equilibrium layer.We begin by analyzing the structure of the remnantdisk that forms in the process of fall-back accretion(Section 2). We conclude that the neutron star is sur-rounded by a remnant magnetic slab with low angularmomentum after the supernova explosion, for a widerange of basic parameters. In this case, the X-rayluminosities and spin-down rates of AXPs and SGRscan be described in a model with accretion from theslab, provided the dipole magnetic moment of the staris ∼1029−1031 G cm3. A star with an initial (at thetime of its birth) spin period of P0 ≥ 65 ms reachesthis stage at a relatively young age (103−104 years),beginning its evolution in the propeller state. It makesa transition to the accretor state as soon as its rotationdecelerates to a period of 2–12 s. Mass accretion ontothe surface of the star may be one of the main factorscontributing to the mixing of matter in its outer crust,leading to the release of nuclear energy stored in thenon-equilibrium layer (Section 3). The amount of thisenergy is sufficient to generate ∼200 giant gamma-ray outbursts at an average recurrence interval ofabout 40 years.

Until recently, long spin periods and strong mag-netic fields have been believed to be exclusive signs ofmagnetars, and have been considered the main prop-erties differentiating these objects from ordinary radiopulsars. However, observational data cast seriousdoubt on the validity of this view. Young et al. [23]reported the discovery of a long-period radio pulsarwith a period of 8.5 s. In addition, the magneticfields of a number of neutron stars which manifestthemselves as ordinary radio pulsars reach valuestypical for hypothetical magnetars. In particular, the

ASTRONOMY REPORTS Vol. 58 No. 4 2014

A NEW LOOK AT ANOMALOUS X-RAY PULSARS 219

magnetic field of the neutron star PSR J1119-6127,rotating with a period of Ps = 0.407 s, is B = 4.1 ×1013 G. The magnetic field B = 5.5× 1013 G was ob-tained for the radio pulsar PSR J1814-1744, spinningwith the period Ps = 3.975 s [24]. Strong magneticfields have also been detected in the radio pulsarsPSR J1847-013 (Ps = 6.7 s, B = 9.4 × 1013 G) andPSR J1718-37 (Ps = 3.4 s, B = 7.4 × 1013 G) [25].On the other hand, the magnetic field of the neutronstar SGR 0418+5729, which exhibits flaring activitythat unambiguously leads to its classification as anAXP/SGR, is only B = 7.5 × 1012 G [26], whichis quite typical for radio pulsars. This means thatstrong magnetic fields do not constitute a necessarycondition for a star’s flaring activity in gamma-rays,and cannot serve as an unambiguous criterion for itsclassification in the subclass of AXPs/SGRs.

The main parameter determining the differencebetween SGRs and radio pulsars in the flaring-activity model based on a non-equilibrium layer isthe mass of the neutron star [27]. The mass of thenon-equilibrium layer and the energy stored thereincreases as the mass of the neutron star decreases,so that, beginning from a certain neutron-star mass(which is unknown so far), the non-equilibrium layerbegins to exhibit flaring activity, up to the appear-ance of giant flares. This suggests that SGR-typeactivity is characteristic of low-mass neutron starswith massive non-equilibrium layers [27]. The mainconclusions of our analysis are briefly summarized inSection 4.

2. ISOLATED ACCRETING PULSARS

Modeling of supernova explosions (see, e.g., [28,29]) has shown that, at the time of its birth, a neu-tron star is embedded in a dense, gaseous medium,which can substantially influence its further evolu-tion. In particular, under these conditions, the starcan switch into the ejector state only if the pressureof the relativistic wind being ejected from its magne-tosphere, prw(r) = Wej/4πr2c, exceeds the ram pres-sure, pram(r) = ρ(r)v2

ff(r), of the gas located at dis-tances r, exceeding the light-cylinder radius, rlc =c/ωs. Here, ρ and vff = (2GMns/r)

1/2 are the densityand free-fall velocity of the gas surrounding a neutronstar of mass Mns. Wej = fmμ2ω4

s /c3 is the spin-downpower of the neutron star expected in a radio pulsarmodel. μ and ωs = 2π/Ps are the dipole magneticmoment and the angular velocity of a star rotatingwith the period Ps, and fm is a dimensionless pa-rameter, which, in general, is restricted to 1 ≤ fm ≤4 (see [30, 31] and references therein). Combiningthese parameters, we find that prw(rlc) > pram(rlc) if

the initial spin period of the star, P0, satisfies thecondition P0 < Pcr, where

Pcr � 0.24f2/7m μ

4/730 M

−2/715 m−1/7 s. (1)

Here, μ30 and m are the initial dipole magnetic mo-ment and the mass of the neutron star in units iof1030 G cm3 and 1.4 M�, and M15 = M/1015 g/s,where M = 4πr2ρ(r)vff(r) is the mass-transfer ratetowards the neutron star. In this case, the surround-ing gas is blown away by the relativistic wind of thestar beyond its Bondi radius, rG = 2GMns/v

2rel, and,

at early stages of its evolution, the star manifests itselfas a radio pulsar. Here, vrel is the velocity of the starin the reference frame of the surrounding gas.

Rapid rotation is one of the necessary conditionsof the formation of a magnetar, which is expected tohave an initial spin period not exceeding a few mil-liseconds [9]. Under these conditions, the star beginsits evolution in the ejector state for all acceptablevalues of the parameter M given in [18]. The spinperiod in this case gradually increases in accordancewith the canonical radio-pulsar model, and reachesthe critical value at which the star switches to thepropeller state on the timescale [32]

τej � 106f−1/2m I45μ

−130 M

−1/215 v

−1/28 yr, (2)

where I45 = I/1045 g cm2 is the moment of inertiaof the neutron star and v8 = vrel/108 cm/s. Thus, astar with an initial spin period P0 < Pcr can switchto the accretor state (which succeeds the propellerstate) on a timescale comparable to the ages of AXPsand SGRs (e.g., ≤104 yr) only if μ ≥ 1032 G cm3.In this light, the possibility of forming an accret-ing “quasi-magnetar” (i.e., a neutron star whosestrong small-scale surface magnetic field does notmake a significant contribution to its dipole mag-netic moment, which is close to its canonical valueμ ∼ 1030 G cm3, [22]) appears doubtful, since theevolution timescale for these stars in the ejectorstate exceeds the characteristic decay time for asupercritical magnetic field estimated in [33].

A statistical analysis of the observed character-istics of radio pulsars [34–37] shows evidence that,as a rule, the periods of these objects when theyare born are greater than a few tens of milliseconds.This conclusion can be directly obtained theoreticallyin the magnetorotational explosion model for core-collapse supernovae [28, 38]. The condition for astar to switch to the ejector state in this case can bewritten as M < Mcr, where

Mcr � 2 × 1016f7/2m μ2

30m−1/2 (3)

×(

P0

0.1 s

)−7/2

g/s



is a solution of the equation P0 = Pcr(M). Otherwise,a fall-back accretion scenario is realized, in which thegas captured by the neutron star after its birth formsan accretion flow approaching the star to distancessmaller than its light-cylinder radius, and directlyinteracting with the stellar magnetic field.

2.1. Accretion Flow Structure

Modeling of fall-back accretion [18] indicates thatthe matter captured by the neutron star after thesupernova explosion possesses relatively low angu-lar momentum (its angular velocity does not exceedthat of the progenitor star and is significantly smallerthan the Keplerian velocity) and its accretion onto thestar takes place in a quasi-spherical regime. How-ever, the duration of this phase is limited by thefree-fall time of matter at the Bondi radius, tff(rG) ∼(r3

G/2GMns)1/2, which, for the parameters of interest

(rG ≤ 1011 cm), is negligibly small comparing to theages of AXPs. That is why it is not possible to con-struct an accretion scenario for these objects with-out the additional assumption that the initial quasi-spherical flow is transformed into a remnant disk overa time span t < tff(rG), with the characteristic timefor radial motion being at least 10 000 years. Indirectevidence for the feasibility of this picture is providedby observations of the planetary system surroundingthe pulsar PSR 1257+12 [39–41].

Until recently, such remnant disks have beenmodeled as Keplerian accretion disks with initialmasses ∼10−5 M� [16]. One possible reason for theirformation is the efficient turbularization of the mattercaptured by the neutron star due to the propagationof a reverse shock [17, 18]. Chatterjee et al. [19]have shown that such a disk can donate matter to theneutron star at the rate 1014−1016 g/s on a timescaleof 104−105 years, which is sufficient to explain theX-ray luminosities of AXPs and SGRs [20]. Atthe same time, a scenario with accretion through aKeplerian remnant disk encounters some difficultiesin the interpretation of the regular spin-down of thesepulsars, which occurs at a fairly high rate. Thisbehavior means that the spin period of the neutronstar is appreciably shorter than its equilibrium pe-riod [42]. However, the estimates presented in [20,43] suggest otherwise. The equilibrium period ofa neutron star with a dipole magnetic moment of1030 G cm3 accreting material from a Keplerian diskat a rate of ∼1015 g/s is close to the range of observedperiods of these pulsars. Moreover, under theseconditions, the Alfven radius of the neutron star,

rA =(μ2/M

√2GMns

)2/7exceeds the corotation

radii of AXPs and SGRs by a factor of two, and turns

out to be an order of magnitude greater than the mag-netosphere radius estimated through temperaturemeasurements of the black-body components in theX-ray spectra of these sources. These inconsistenciesindicate that either the X-ray emission of AXPs andSGRs is not accretion-driven or the accretion-flowstructure deviates from a Keplerian disk.

As first shown in [44, 45], the structure of theaccretion flow onto a gravitating compact object candiffer substantially from a Keplerian disk if the in-falling matter possesses a sufficiently strong mag-netic field Bf. The rapid amplification of the magneticfield in the quasi-spherical flow leads to its decel-eration and further transformation into a magneticdisk-like envelope (magnetic slab) with low angularmomentum. This change in the flow structure occursat the Shvartsman radius [46]

Rsh = β−2/30

(cs(rG)

vrel

)4/3

rG, (4)

where the magnetic pressure in the accretion flow,Em = B2

f (r)/8π ∝ r−4, reaches its ram pressure,Eram = ρ(r)v2

ff(r) ∝ r−5/2. Here, β0 =Eth(rG)/Em(rG), Eth(r) = ρ(r)c2

s (r) is the thermalpressure and cs is the sound speed in the accretionflow.

The material in the magnetic slab is maintained inan equilibrium state by the intrinsic magnetic field ofthe accretion flow, and moves towards the compactstar with a velocity vr ∼ r/τm, as a consequence ofthe dissipation of this field on the timescale τm, whichsignificantly exceeds the free-fall time [44, 45]. Themain conclusions of this scenario were confirmed bynumerical simulations of spherical accretion of mag-netized plasma onto a black hole [47].

Magnetic accretion onto a neutron star was re-cently discussed in [32, 48, 49]. These studies showedthat the transformation of the quasi-spherical flowinto a magnetic slab takes place only if the Shvarts-man radius exceeds the canonical Alfven radius. Thiscondition is valid if vrel < vma, where

vma � 1155β−1/50 μ

−6/3530 M

3/3515 m12/35 (5)

×(

cs(rG)100 km/s

)2/5

km/s.

The interaction between the slab and the dipole mag-netic field of the neutron star results in the formationof a magnetosphere, whose minimum radius is esti-mated as (see Eq. (12) in [50])

rma =

(cm2

p

16√

2ekB

)2/13

(6)



Table 1. Anomalous X-ray Pulsars∗

Name∗∗ Ps, s P , 10−11 s/s τ , 103 yr LX, 1034 erg/s TBB, keV d, kpc

1E 1547.0-5408 2.07 4.7 0.7 0.08 0.43 4.5

CXOU J174505.7-381031 3.83 6.4 0.95 6.0 0.38 13.2

PSR J1622-4950 4.33 1.7 4.0 0.063 0.4 9

XTE J1810-197 5.54 0.8 11 3.9 0.2 3.5

1E 1048.1-5937 6.45 2.3 4.5 0.6 0.51 2.7

1E 2259+586 6.98 0.048 230 2.2 0.4 3.2

CXOU J010043.1-721134 8.02 1.88 6.8 6.1 0.38 60

4U 0142+61 8.69 0.2 70 11 0.4 3.6

CXO J164710.2-455216 10.61 0.07 230 0.3 0.5 4

1RXS J170849.0-400910 11.0 1.9 9.1 5.9 0.45 3.8

1E J1841-045 11.8 3.93 4.8 19 0.45 8.5∗ Data from the McGill SGR/AXP Online Catalog http://www.physics.mcgill.ca/∼pulsar/magnetar/main.html; ∗∗ Ps is the spinperiod of the pulsar and P = dPs/dt derivative. τ = Ps/2P , LX is the X-ray luminosity, TBB the temperature of the blackbodycomponent, and d the distance to the object.

× α2/13μ6/13(GMns)1/13

T2/130 M4/13

,

where mp and kB are the proton mass and Boltz-mann’s constant, and T0 is the gas temperature atthe inner radius of the slab in the region of its in-teraction with the magnetic field of the neutron star.The dimensionless parameter α = Deff/DB in thisexpression is the ratio of the effective coefficient forthe diffusion of the accretion flow into the stellarmagnetic field at its magnetosphere boundary, Deff,and the Bohm diffusion coefficient DB = ckT/16 eV[51]; according to Gosling et al. [52], its value is0.1−0.25. The plasma penetrating into the stellarmagnetic field reaches the stellar surface in the regionof the magnetic poles, flowing along the field lines.The radius of the base of the resulting accretion col-umn in a dipol field approximation can be estimated

as ap ∼ Rns (Rns/rma)1/2 [42].

The spin evolution of the neutron star in the mag-netic accretion scenario depends strongly on the an-gular velocity of the matter at the inner radius of theslab, which, in general, is limited to 0 ≤ Ωsl < ωk,

where ωk(r) =(GMns/r

3)1/2 is the Keplerian angu-

lar velocity. Spin-down can be observed if Ωsl(rm) <ωs [53]. Absolute value of the spin-down torqueapplied to the neutron star from the magnetic slab inthis case can be estimated using the expression (see

Eq. (8) from [42])

|K(sl)sd | =

kmμ2

(rmarcor)3/2

(1 − Ωsl(rm)

ωs

), (7)

where rcor =(GMns/ω

2s

)1/3 is the corotation radiusof the neutron star and km is a dimensionless param-eter of the order of unity.

2.2. Parameters of the Sources

The parameters of the AXPs and the best studiedSGRs are collected in Tables 1 and 2. We considerthe hypothesis that these are isolated neutron starsundergoing mass accretion onto their surfaces frommagnetic slabs. The magnetosphere radius of a neu-tron star is given by (6) in this case. Solving theinequality rma ≤ rcor for μ, we find that the centrifugalbarrier at the magnetosphere boundary should notprevent plasma from reaching the stellar surface ifμ ≤ μmax, where

μmax � 1030 G cm3 (8)

× α−10.1T

1/36 m−1/9L

2/334 R

2/36 P

13/9s ,

R6 = Rns/106 cm, α0.1 = α/0.1 and Ps is the pulsarspin period in seconds. This condition determines themaximum possible dipole magnetic moment of theneutron star in this scenario.

A lower limit to the dipole magnetic moment ofa neutron star spinning down at the rate νsd can be



Table 2. Soft gamma-ray repeaters∗

Name∗∗ Ps, s P , 10−11 s/s τ , 103 yr LX, 1034 erg/s TBB, keV d, kpc

SGR 1627–41 2.6 1.9 2.2 1.0 0.5 11

SGR 1900+14 5.2 9.2 0.9 9.0 0.43 12–15

SGR 1806–20 7.6 75 0.16 16 0.6 8.7

SGR 0526–66 8.05 3.8 2.2 14 0.53 50∗ Data from the McGill SGR/AXP Online Catalog http://www.physics.mcgill.ca/∼pulsar/magnetar/main.html; ∗∗ Parameters arethe same as in Table 1.

obtained by solving the inequality |Ksd| ≥ 2πI|νsd|.Taking |Ksd| = |Ksl

sd| [see (7)] and solving this in-equality for μ, we find μ ≥ μmin, where

μmin � 7 × 1028 G cm3 × P13/17s L

6/1734 ν

13/17−12 (9)

× k−13/17m α

9/170.1 I

13/1745 m14/17T

3/176 R

2/36 .

The surface magnetic-field strengths of AXPs andSGRs, B∗ = 2μmin/R

3ns, magnetosphere radius,

rma(B∗), radius of the accretion column base, ap �[R3

ns/rma(B∗)]1/2, and the blackbody temperature,

Tbb, calculated in the magnetic accretion model arepresented in Table 3. The parameters of the objectslisted in this table support the idea that their X-rayradiation has an accretion origin.

The presented surface magnetic fields are in goodagreement with the observational data on their X-rayspectra. The magnetosphere radii of these stars inour scenario do not exceed their corotation radii, andthe expected spin-down rates are consistent with theobserved values. Finally, the expected temperaturesfor blackbody radiation emitted at the base of theaccretion column are also close to observational esti-mates. The somewhat higher temperatures estimatedfor some of the objects could be connected with ad-ditional heating of the neutron-star surface by hardradiation generated due to the Comptonization of softphotons on electrons in the accretion flow.

2.3. Age and Period Clustering

As was shown above, under the conditions of in-terest, the lifetime of a neutron star in the ejector statesignificantly exceeds the ages of AXPs and SGRsestimated from their spin-down rates and the ages ofthe associated supernova remnants. However, muchhigher spin-down rates are expected if the neutronstar starts its evolution in the propeller state. This sit-uation can be realized if the initial mass accretion ratetowards the neutron star, M0, satisfies the inequalityM0 ≥ Mcr [see (3)]. In this case, the pulsar age, τ ,

is limited solely by the spin-down time of the neu-tron star in the propeller state, τ ≥ τpr = πI/P0K

prsd,

which can be estimated for the case of maximumspin-down torque, K

prsd = Mωsr

2m, as

τpr =I

2Mr2m

. (10)

Since the magnetosphere radius of a neutron star inthe propeller state is rm ≥ rcor, we find

τpr � 1870I45

(M0

1017 g/s

)−1

(11)

×(

rcor

3 × 108 cm

)−2

yr.

Thus, in the scenario considered, AXPs and SGRsare relatively young neutron stars, whose periods ap-parently correspond to the transition of the neutronstar from the propeller to the accretor state [20].In this case, the observed period clustering indi-cates that the magnetosphere radii of these neutronstars in the propeller state, r

(pr)m , were in the range

(2.8−8.9)× 108 cm, and hence significantly exceededtheir values at the present epoch (see Table 3). Thisconclusion does not contradict our results presentedin the previous sections. Instead, it suggests that thediffusion coefficient at the magnetosphere boundaryof a star in the propeller state significantly exceedsthe Bohm diffusion coefficient. This could be con-nected with plasma turbularization [45] by a shockgenerated at the magnetosphere boundary of the starin a supersonic propeller state. As the diffusion co-efficient grows, the radius of stellar magnetosphereincreases to the Alfven radius, rA, whose value forthe parameters of interest (M ∼ 1016−1017 g/s andB ∼ 1012−1013 G) corresponds to the above intervalfor r

(pr)m .

3. FLARING ACTIVITY OF AXPs AND SGRsThe results of the previous sections make it possi-

ble to explain the high X-ray luminosities and rapid



Table 3. Parameters of AXPs and SGRs in the magnetic-accretion model∗

Name Ps, s|νsd|,

10−12 Hz/sLX,

1034 erg/sB∗,

1012 Grcor,

108 cmrma,

108 cmap,

105 cmTbb,keV

SGR 1627–41 2.6 2.8 0.25 3.9 3.2 1.2 0.9 0.5

SGR 1900+14 5.2 3.4 9.0 2.7 5.2 1.0 1.0 1.1

SGR 1806–20 7.6 13 16 12 6.6 1.7 0.8 1.4

SGR 0526–66 8.05 0.59 14 1.2 6.9 0.6 1.3 1.1

1E 1547.0–5408 2.07 11 0.08 0.63 2.8 2.1 0.7 0.4

CXOU J174505.7–381031 3.83 4.4 6.0 2.3 4.2 1.0 1.0 0.98

PSR J1622–4950 4.33 0.9 0.063 1.5 4.6 1.2 0.9 0.33

XTE J1810–197 5.54 0.26 3.9 0.3 5.4 0.46 1.5 0.73

1E 1048.1–5937 6.45 0.55 0.6 0.31 5.95 0.83 1.1 0.53

1E 2259+586 6.98 0.01 2.2 0.024 6.3 0.17 2.4 0.49

CXOU J010043.1–721134 8.02 0.29 6.1 0.51 6.9 0.51 1.4 0.83

4U 0142+61 8.69 0.03 11 0.11 7.26 0.21 2.2 0.77

CXO J164710.2–455216 10.61 0.006 0.3 0.011 8.3 0.22 2.1 0.32

1RXS J170849.0–400910 11.0 0.16 5.9 0.4 8.5 0.46 1.5 0.81

1E J1841–045 11.8 0.28 19 0.99 8.9 0.49 1.4 1.1∗|νsd| is the observed spin-down rate of the pulsar, B∗ the magnetic field at the neutron-star surface, estimated as B∗ = (1/2)μminR

3ns

[see (9)], rcor the corotation radius, rma the magnetosphere radius for a neutron star accreting matter from a magnetic slab [see (6)], ap

the radius of the accretion-column base, and Tbb the effective blackbody temperature, estimated as T =

(LX

2πa2p σSB

)1/4

.

spin-downs of AXPs and SGRs in a scenario withaccretion of matter from a remnant magnetic slabonto the surface of a young neutron star whose ba-sic parameters are close to their canonical values.This excludes the possibility of describing the pulsargamma-ray activity in terms of flaring dissipation ofits magnetic field. Models involving non-stationaryaccretion turn out not to be efficient, due to the ex-tremely high (super-Eddington) luminosities of thesources during flares. This circumstance unambigu-ously indicates that the source of the energy releasedduring flares is located in the star itself, and hence isconnected with properties of its internal structure.

3.1. Energy of the Non-Equilibrium Layer

One way to form a considerable reservoir of energyin the crust of a young neutron star was pointed out byBisnovatyi-Kogan and Chechetkin [11], who showedthat a non-equilibrium layer of super-heavy nucleiwith a large energy excess relative to the equilibriumstate can be generated in the envelope of a youngneutron star. This non-equilibrium layer forms at

the interface between the inner and outer crust of theneutron star, and remains in quasi-equilibrium at thedensities ρnel ∼ 1010−1012 g/cm3. The initial massof the layer is Mnel ∼ 10−4 M� [12], and its lifetime inthe absence of instabilities can be infinite. The totalenergy accumulated in the non-equilibrium layer canreach ∼1049 erg (see Eqs. (5.6)–(5.9) in [13]).

The release of free energy stored in the non-equilibrium layer could occur during mixing of thematter in the neutron star crust, when heavy nucleifrom the non-equilibrium layer reach the outer layersof the star where the density of the material is<1010 g/cm3. Under these conditions, super-heavynuclei become unstable to β decay, which initiatenuclei fissions and α decays, leading to chain reac-tions and ultimately to a nuclear explosion (see [54]for a detailed discussion). The duration of the energyrelease in this case is determined by the characteristictime for the β decay (the slowest process in thischain), which is τβ ∼ 10−4−50 s for the parameters ofinterest (see Eq. (7.5) in [13]). This covers the rangeof emission times of gamma-ray outbursts observed



0.0005

0.5

M

ns

/

M

�

0

0.0010

0.0020

0.0015

1.0 1.5 2.0

M

nel

/ M

�

Dependence of the mass of the non-equilibrium layer on the neutron-star mass. The lines show the top and bottom boundariesof the layer mass measured from the stellar surface. The equation of state of the equilibrium matter was used to construct themodel of the neutron star, with the boundaries of the layer specified by the densities. Using a non-equilibrium equation of statewill increase the mass of the layer, but should not fundamentally change the values given in the figure.

from SGRs, and, at the same time, the spectrumof the primary radiation generated in the nuclearexplosion lies in the energy range 0.002−3 MeV.The energy efficiency of neutron transmutation ina nucleus comprises 1% of their rest energy, and∼0.1% of the rest energy of super-heavy nuclei canbe released via fission. Taking the efficiency for amixture of these components to be ζ ∼ 3 × 10−3, wefind that the mass of non-equilibrium matter requiredto explain a gamma-ray outburst with energy εγ inthe nuclear-explosion scenario can be estimated as

Mγ � 4 × 1026

(ζ

0.003

)−1 (εγ

1045 erg

)g. (12)

Thus, the energy accumulated in the non-equilibriumlayer of a young neutron star proves to be sufficientfor ∼250 giant (∼1045 erg) gamma-ray outbursts,which can occur with an average recurrence time of∼40 years over a time span of ∼104 years.

3.2. Trigger of the Flare

According to the calculations presented in [27],the mass of the non-equilibrium layer forming in thecrust of a light neutron star exceeds the mass of thislayer in a more massive star (figure). This suggestsneutron stars of moderate mass as the most probablecandidates for SGRs, and implies the possibility ofdistinguishing supernova remnants associated withthe birth of these objects. This is in agreement withthe results by Marsden et al. [55] indicating that

AXPs and SGRs have relatively compact and higher-density supernova remnants. They noted that theobservational appearance of these remnants suggestsa relatively low energy for the supernova explosions,which is typical for the birth of low-mass neutronstars.

An analysis of the Galactic population of neutronstars leads to the conclusion that youth is not asufficient condition for the realization of an AXP orSGR. Only these 23 objects among a large num-ber of neutron stars with parameters satisfying theabove criteria demonstrate flaring activity in gamma-rays. This implies that energy release from the non-equilibrium layer cannot proceed spontaneously, andis not connected exclusively with the properties of thestructural evolution of the star itself (in particular,starquakes, volcanic activity, and tensions arising inthe stellar crust in the presence of liquid–gas phasetransitions). Flares seem to be triggered by the actionof external factors on the star, one of which may bethe accretion of matter onto its surface.

3.3. Possible Model for the Formationof a Low-Mass Neutron Star

The birth of a low-mass neutron star should bea fairly rare event, since the fraction of AXP/SGRsamong currently known neutron stars is only about1% (see, e.g. [1, 56]). The detection of binary pulsarsin systems with neutron stars has enabled accuratedetermination of their masses. For instance, ob-servations of the binary pulsar system J1518-4904



indicated the masses of the components to be mp =0.72+0.51

−0.58 M� and me = 2.00+0.58−0.51 M� [57] with a

95.4% probability.The most accurate measurements of neutron-star

masses have been carried out for two close bina-ries consisting of neutron stars, in which general-relativistic effects are most clearly displayed. In oneof them, containing the first binary pulsar discovered,the mass of the visible, recycled (spun-up) pulsar is1.4424 M�, while the mass of its invisible companionis 1.386 M� [58]. In another double-pulsar system,the mass of the recycled pulsar is 1.3381 M�, and themass of the young “normal” pulsar is 1.2489 M� [59].In both systems, the mass of the recycled pulsarexceeds the mass of its companion, which has notundergone mass accretion. In the process of diskaccretion, the recycled pulsar mass is increased byan amount that does not depend on its initial period(exceeding ∼0.2 s) and is related to the final spinperiod of the pulsar Pms (in milliseconds) as [60]

Δm = 0.22 M�(M/M�)1/3

P4/3ms

. (13)

A pulsar with a final mass of 1.4 M� and a recycledperiod of 2, 5, 10, or 50 ms requires the accretion of0.1, 0.03, 0.01, and 0.001 M�, respectively. Thus,in double neutron star systems with spin periods inexcess of 23 ms (see, e.g., [61]), the mass added to anaccreting pulsar during spin-up is insignificant.

Neutron stars are born in core-collapse supernovaexplosions. The collapse results from the loss ofhydrodynamic stability in the final stage of the evolu-tion of massive stars. The majority of neutron starsobserved as radio pulsars and X-ray sources wereprobably born in this process. We assume that singlelow-mass neutron stars, which are suggested to beSGRs, are formed through another, rarely realizedchannel. As an example, let us consider the evolutionof a single star of moderate mass, which ends withthe loss of thermal stability of the star and an off-center explosion of its degenerate core. Models withoff-center explosions and collapsing cores were dis-cussed in [62] for Type Ib supernovae. The outcome ofthis process should depend on the mass ratio betweenthe iron core and the He and C–O shell layers. Themass of the collapsing remnant should be close to thatof the iron core, which can reach 0.4−0.6 M�. Whenthe whole star (core + shell) looses its hydrodynam-ical stability, the collapse starts, and the temperaturein He and C–O shells exceeds the critical value thatgives rise to thermonuclear explosions, leading to theejection of both shells. As a result, a low-mass neu-tron star can be born. Note, however, that verificationof this scenario requires numerical simulations of thestellar evolution and, in particular, of the explosiondynamics.

4. CONCLUSIONS

The main conclusion of this paper is that it is notnecessary to invoke the magnetar hypothesis in orderto explain the spin evolution of AXPs and SGRs.The expected spin-down rate of a neutron star ac-creting matter from a non-Keplerian magnetic slabcorresponds to the observed values if the magnetic-field strength at its surface is 1010−1013 G. In thisapproach, the blackbody temperature of the pulsarX-ray radiation is also close to observed estimates,supporting the feasibility of the proposed scenario.

Gamma-ray flaring activity of a neutron star inthe absence of a super-strong magnetic field can beexplained if there exists an energy reservoir locatedbeneath its surface. We consider a non-equilibriumlayer of super-heavy nuclei which forms at earlystages of the neutron star evolution as such a reser-voir. In this picture, AXPs and SGRs are relativelylight neutron stars undergoing accretion onto theirsurfaces, leading to the development of instabilities intheir crusts that can trigger flaring activity.

The formation of a low-mass neutron star could beconnected with an off-centre nuclear explosion andthe induced collapse of the central region of the starin the supernova explosion. The discovery of neutronstars with masses close to a solar mass testifies thatthe mass of the remnant formed during the collapseand supernova explosion can be significantly lowerthan the initial mass of the collapsing core. Thisprovides indirect confirmation of the possibility offorming low-mass neutron stars.

ACKNOWLEDGMENTS

The authors thank S.O. Tarasov and N.G. Bes-krovnaya for their help in this work.

This work was supported by the Program No. 21of the Presidium of the Russian Academy of Sci-ences “Non-stationary phenomena in the Universe”and “Formation and evolution of stars and galaxies,”the Russian Foundation for Basic Research (projects11-02-00602, 13-02-00077) and by the PresidentialProgram of Support for Leading Scientific Schoolsof the Russian Federation (grants NSh-1625.2012.2,NSh-5440.2012.2).

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Translated by N. Beskrovnaya


a new look at anomalous x-ray pulsars

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