Review Article Physical Environment of Accreting Neutron …downloads.hindawi.com/journals/aa/2016/3424565.pdf · Review Article Physical Environment of Accreting Neutron Stars

Review ArticlePhysical Environment of Accreting Neutron Stars

J Wang

Institute of Astronomy and Space Science Sun Yat-Sen University Guangzhou 510275 China

Correspondence should be addressed to J Wang wangjing6mailsysueducn

Received 8 November 2015 Revised 20 February 2016 Accepted 3 March 2016

Academic Editor Elmetwally Elabbasy

Copyright copy 2016 J Wang This is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Neutron stars (NSs) powered by accretion which are known as accretion-powered NSs always are located in binary systems andmanifest themselves as X-ray sources Physical processes taking place during the accretion of material from their companions forma challenging and appealing topic because of the strong magnetic field of NSs In this paper we review the physical process ofaccretion onto magnetized NS in X-ray binary systems We firstly give an introduction to accretion-powered NSs and reviewthe accretion mechanism in X-ray binaries This review is mostly focused on accretion-induced evolution of NSs which includesscenario of NSs both in high-mass binaries and in low-mass systems

1 Introduction to Accretion-PoweredNeutron Star

Since the discovery for the first extrasolar X-ray sourceScorpius X-1 in 1962 [1] a new field of astronomymdashaccretingcompact objects in the galaxymdashhas arisen which offersunique insight into the physics at extreme conditions Sofar more and more bright galactic X-ray sources have beendiscovered which show a clear concentration towards bothgalactic center and galactic plane They are highly variableon timescales of seconds or less and display typical sourceluminosity of 1034ndash1038 ergs sminus1 in the energy band of 1ndash10 keV It was suggested that the strong galactic X-ray sourcesare NSs accreting material from their companions in closebinary systems [2] The detection of coherent and regularpulsations from the accreting X-ray sources Centaurus X-3[3 4] andHercules X-1 [5] by Uhuru in 1971 provided the firstevidence that the compact objects in many of these sourcesare indeed accreting neutron stars Since then accretion ontorotating NSs have been nicely confirmed as the standardpicture of strongest galactic X-ray sources and the X-raypulsation periods are the spin periods of X-ray pulsars

Accretion remains the only viable source of power to thebinary X-ray pulsars as a whole [6ndash8] For a NS of mass119872 sim 119872

⊙ and of radius 119877 = 10

6 cm the quantity of energy

released bymatter ofmass119898 falling into its deep gravitationalpotential well amounts to

Δ119864acc =119866119872119898

119877 (1)

where 119866 is the gravitational constant This is up tosim1020 ergs gminus1 of accreted matter that is about a tenth of itsrest-mass energy (sim011198882 with the speed of light 119888) whichmakes accretion as an ideal source of power Since each unitof accretedmass releases an amount of gravitational potentialenergy119866119872119877 when it reaches the NS surface the luminositygenerated by the accretion process is given by

119871acc =119866119872

119877 (2)

where is the accretion rate Accordingly to generatea typical luminosity of about 1037ergssminus1 it requires anaccretion rate of sim1017 g sminus1 sim 10minus9119872

⊙yrminus1

During the accretion process radiation passes throughaccretion flow and influences its dynamics in the case ofa sufficiently large luminosity When the outward pressureof radiation exceeds the inward gravitational attractionthe infalling flow will be halted which implies a criticalluminosity Eddington luminosity [9]

119871Edd asymp 13 times 1038 119872

119872⊙

ergs sminus1 (3)

Hindawi Publishing CorporationAdvances in AstronomyVolume 2016 Article ID 3424565 15 pageshttpdxdoiorg10115520163424565

2 Advances in Astronomy

which for spherically symmetric accretion corresponds to alimit on a steady accretion rate the Eddington accretion rate

Edd asymp 10181198776g sminus1 asymp 15 times 10minus8119872

⊙1198776yrminus1 (4)

Lower and higher accretion rates than this critical value arecalled the ldquosub-rdquo and ldquosuper-rdquoEddington accretion respec-tively

Phenomenologically over 95 of the NS X-ray binariesfall into two distinct categories that is low-mass X-ray binary(LMXB) and high-mass X-ray binary (HMXB) [10 11]Wind-fed system in which the companion of NS loses mass inthe form of a stellar wind requires a massive companionwith mass ge10119872

⊙to drive the intense stellar wind and thus

appears to be high-mass system such as Centaurus X-3 NSsin HMXBs display relatively hard X-ray spectra with a peakenergy larger than 15 keV [12] and tend to manifest as regularX-ray pulsars with spin periods of 1ndash103s andmagnetic fieldsof 1011ndash1013 G [13] The companions in HMXBs are brightand luminous early-type Be or OB supergiant stars whichhave masses larger than 10 solar masses and are short livedand belong to the youngest stellar population in the galaxywith ages of sim105ndash107yrs [14ndash16] They are distributed closeto the galactic plane On the other hand X-ray binarieswith companion mass le 119872

⊙ are categorized as low-mass

X-ray binaries (LMXBs) in which mass transfer takes placeby Roche-lone overflow This transfer is driven either bylosing angular momentum due to gravitational radiation (forsystems of very small masses and orbital separations) andmagnetic braking (for systems of orbital periods 119875orb le

2 days) or by the evolution of the companion star (forsystems of 119875orb ge 2 days) Most NSs in LMXBs are X-ray busters [17] with relatively soft X-ray spectra of lt10 keVin exponential fittings [11] In LMXBs the NSs possessrelatively weak surface magnetic fields of 108-9G and shortspin periods of a fewmilliseconds [18] while the companionsof NSs are late-type main-sequence stars white dwarfs orsubgiant stars with F-G type spectra [19 20] In space theLMXBs are concentrated towards the galactic center andhave a fairly wide distribution around the galactic planewhich characterize a relatively old population with ages of(05ndash15) times 1010 yrs A few of strong galactic X-ray sourcesfor example Hercules X-1 are assigned to be intermediate-mass X-ray binaries [21 22] in which the companions of NSshave masses of 1-2119872

⊙ Intermediate-mass systems are rare

since mass transfer via Roche lobe is unstable and would leadto a very quick (sim103ndash105 yrs) evolution of the system whilein the case of stellar wind accretion the mass accretion rateis very low and the system would be very dim and hardlydetectable [23]

2 Physical Processes in AccretingNeutron Stars

21 Mass Transfer in Binary Systems Themass transfer fromthe companion toNS is either due to a stellarwind or toRochelobe overflow (RLOF) as described above [7 24]

A massive companion star at some evolutionary phaseejects much of its mass in the form of intense stellar wind

driven by radiation [25 26]The wind material leaves surfaceuniformly in all directions Some of material will be capturedgravitationally by the NS This mechanism is called stellarwind accretion and the corresponding systems are wind-fedsystems [7]

In the course of its evolution the optical companion starmay increase in radius to a point where the gravitational pullof the companion can remove the outer layers of its envelopewhich is the RLOF phase [9 27] In addition to the swelling ofthe companion this situation can also come about due to forexample the decrease of binary separation as a consequenceof magnetically coupled wind mass loss or gravitational radi-ation Some of material begins to flow over to the Roche lobeof NS through the inner Lagrangian point without losingenergy Because of the large angular momentum the matterenters an orbit at some distance from the NS Subsequentbatches of outflowingmatter from companion have nearly thesame initial conditions and hence fall on the same orbit As aresult a ring of increasing density is formed by the gas nearthe NS Because of the mutual collisions of individual densityor turbulence cells the angular momentum in the ring isredistributed The ring spreads out into a disk in which thematter rotates differentially Gradually the motion turns outinto steady-state disk accretion on a time scale determined bythe viscosity Some optical observations suggested that masstransfer by ROLF is taking place or perhaps mixed with thatby stellar wind accretion in some systems

22 Accretion Regimes Because of the different angularmomentum carried by the accreted material and the motionof the NS relative to the sound speed 119888

119904in the medium for

accretion onto a NS several modes are possible [28]

221 Spherically Symmetric Accretion [29] The NS hardlymoves relative to the medium in its vicinity that is V

infin≪ 119888119904

The gas of uniform density and pressure is at rest and doesnot possess significant angular momentum at infinity andfalls freely with spherically symmetrical and steady motiontowards the NS

222 Cylindrical Accretion [30] Although the angularmomentum is small the NS velocity is comparable with orlarger than the sound speed in the accreted matter that isVinfin≳ 119888119904 In this case a conical shock form will present as the

result of pressure effects

223 DiskAccretion [6 31] In a binary system if thematerialsupplied by the companion of the NS has a large angularmomentum due to the orbital motion we must take both thegravitational force and centrifugal forces into considerationAs a result an accretion disk ultimately forms around a NS inthe binary plane

In a number of astrophysical cases [32] the motion ofaccreted material in the vicinity of a compact object presentstwo-stream accretion that is a quasi-spherically symmetricflux of matter in addition to the accretion disk [33]

23 Accretion from a Stellar Wind Mass loss in the case ofOB supergiant stars is driven by radiation pressure [34] The

Advances in Astronomy 3

windmaterial is accelerated outwards from stellar surface to afinal velocity V

infin which is related to the escape velocity Vesc =

radic2119866119872OB119877OB for an OB companion with mass 119872OB andradius119877OB Inmost cases V

infinasymp 3Vesc In these circumstances

the stellar wind is intense characterized by a mass loss rate119908sim 10minus6ndash10minus5119872

⊙yrminus1 and highly supersonic with a wind

velocity of V119908sim 1000ndash2000 km sminus1 which is much larger than

the orbital velocity Vorb sim 200 km sminus1 at 2119877OB

231 Characteristic Radii In thewind accretion theory threecharacteristic radii [7 8 35] can be defined as follows

Accretion Radius Only part of stellar wind within a certainradius will be captured by the gravitational field of the NSwhereas the wind material outside that radius will escapeThis radius called the accretion radius 119903acc [28] can bedefined by noting that material will only be accreted if ithas a kinetic energy lower than the potential energy in thegravitational potential well of NS that is

119903acc =2119866119872

V2119908

sim 1010 119872

119872⊙

Vminus21199088

cm (5)

where V1199088

is a wind velocity in units of 1000 km sminus1 (V1199088

=

V119908(108 cm sminus1)) We consider now two cases 119903mag smaller

than 119903acc and 119903mag larger than 119903acc

Magnetospheric Radius (i) 119903mag lt 119903acc when accreting theelectromagnetic field of NS will hinder the plasma fromfalling all the way to its surface The inflow will come tostop at certain distance at which the magnetic pressure ofNS can balance the ram pressure of accreting flow Herethe magnetosphere forms at a magnetospheric radius 119903mag[36 37] which is defined by

119861 (119903mag)2

8120587= 120588 (119903mag) V (119903mag)

2

(6)

where 119861(119903mag) sim 1205831199033

mag is the magnetic field strength at 119903magand 120588(119903mag) and V(119903mag) are wind density and wind velocityat 119903mag respectively 120583 is the magnetic moment of the NS (ii)119903mag gt 119903acc here the inflows cannot experience a significantgravitational field Assuming a nonmagnetized sphericallysymmetric wind [36] the wind density near the NS 120588(119903mag)

given by 120588(119903mag) asymp 11990841205871198862V119908[37 38] where 119886 is the

orbital separation between two components and 119886 ≫ 119903magis assumedTherefore the magnetospheric radius in this caseis given by setting 120588(119903mag)V

2

119908= 119861(119903mag)

28120587 which yields

119903mag sim 1010minus16

119908minus6Vminus161199088

11988613

1012058313

33cm (7)

where 119908minus6

= 119908(10minus6119872⊙yr) 120583

33= 120583(10

33 Gcm3)and 11988610is related to the orbital separation as 119886 sim 10

1211988610cm

sim 101211987523

1198871011987213

30cm 119875

11988710is the orbital period 119875

119887in units

of 10 days and 11987230

is the companion mass in units of 30solar masses When 119903mag lt 119903acc the gravitational field of NSdominates the falling of wind matter The wind density andwind velocity at 119903mag can therefore be taken as 120588(119903mag) =

V(119903mag)41205871199032

mag and V(119903mag) = radic2119866119872119903mag respectivelyThe accretion rate depends on 119903acc and 119886 according to [9]

119908

sim1199032

acc(41198862)

sim 10minus5(119872

119872⊙

)

2

Vminus41199088119886minus2

10 (8)

Then the magnetospheric radius is given by using (6) whichyields

119903mag sim 1010(119872

119872⊙

)

minus57

minus27

119908minus6V87119908811988647

1012058347

33cm (9)

Corotation Radius A corotation radius 119903cor can be definedwhen the spin angular velocity (Ω

119904= 2120587119875

119904) of NS is equal

to the Keplerian angular velocity (Ω119896= radic1198661198721199033) of matter

being accreted

119903cor sim 1010(119872

119872⊙

)

13

11987523

1199043cm (10)

where 1198751199043

is the spin period of the neutron star 119875119904in units of

103 s

232 Accretion onto a Magnetized NS The stellar wind flowsapproximately spherically symmetric outside the accretionradius When approaching the NS according to the relativedimensions of three characteristic radii different accretionregimes can be identified

(I) Magnetic Inhibition Propeller In a system in which themagnetospheric radius is larger than the accretion radius(119903mag gt 119903acc) the stellar wind flowing around the NS maynot experience a significant gravitational field and interactsdirectly with the magnetosphere In this case very littlematerial penetrates the magnetospheric radius and will beaccreted by NS [24] Most of wind material is ejected bythe rotation energy of the NS This is called the propellermechanism The NS in this situation behaves like a propellerand spins down due to dissipation of rotational energyresulting from the interaction between the magnetosphereand stellar wind The spin-down torque is expressed as [3940]

119879119898pro = minus

1205811199051205832

1199033mag (11)

where 120581119905le 1 is a dimensionless parameter of order unity [37]

The equation governing the spin evolution can be written as[7 41]

2120587119868119889

119889119905

1

119875119904

= 119879 (12)

where 119868 is the moment of inertia of NS and 119879 is the totaltorque imposed on NS Therefore the spin-down rate will be

119904119898

sim 10minus612058111990512058333119868minus1

451198752

1199043119886minus1

1012

119908minus6V121199088

s sminus1 (13)


where 11986845

is the moment of inertia in units of 1045 g cm2Accordingly if the NS can spin down to a period of 1000 sunder this torque the spin-down timescale is given by

120591119898pro = 119875119904119904pur

sim 103120581minus1

119905120583minus1

3311986845119875minus1

119904311988610minus12


yr(14)

In this regime the spin period 119875119904must satisfy

119875119904gt 112

119872

119872⊙

Vminus21199088

s (15)

In addition the mechanism of magnetic inhibition of accre-tion only occurs before the centrifugal barrier acts that is thecondition 119903cor gt 119903mag must be satisfied which gives

119875119904gt 64 times 10

3 119872

119872⊙

Vminus31199088

s (16)

By imposing 119903mag = 119903acc we get the minimum X-ray lumi-nosity

119871119909119898 (min) sim 1033 ( 119872

119872⊙

)

minus3

119877minus1

61205832

33V71199088

ergs sminus1 (17)

and the corresponding maximum orbital period

119875orb119898 (max)

sim 103(119872

119872⊙

)

92

119872minus12

OB30120583minus32

3334

119908minus6Vminus3341199088

days(18)

For very fast rotating X-ray pulsars (119875119904≪ 1 s) the magnetic

pressure exerted by pulsar onwindmaterial provides an addi-tional mechanism for inhibiting accretion through 119903acc [24]The condition that magnetic pressure at 119903acc counterbalancesthe wind ram pressure gives the minimum X-ray luminosity119871119909119903(min) possible for accretion through 119903acc to take place

119871119909119903(min) sim 10431205832

33(119875119904

01 s)

minus4119872

119872⊙

119877minus1

6Vminus11199088

ergs sminus1 (19)

Once the mass capture rate at 119903acc is higher than that of(19) the flow will continue to be accreted by the NS untilreaching the centrifugal barrierThis scenario is themagneticinhibition of accretion like that of a radio pulsar [35]

(II) Centrifugal Inhibition Quasi-Spherical Capture If themagnetospheric radius is smaller than the accretion radiusthe wind material penetrates through the accretion radiusand halts at 119903mag The captured material cannot penetrate anyfurther because of a super-Keplerian drag exerted by themag-netic field of NS The inflow in the region between 119903acc and119903mag falls approximately in a spherical configuration The NSbehaves like a supersonic rotator that is the linear velocity ofmagnetosphere is much higher than the sound speed in windclumps which strongly shocks the flow at magnetosphericboundary and ejects some of material beyond the accretionradius via propeller mechanism This scenario corresponds

to the propeller mechanism and imposes a spin-down torqueon the NS which is expressed by [32]

119879119888pro = minus

V2119908

Ω119904

(20)

In addition due to the high speed of stellar wind theaccumulation rate of accreted matter is higher than theejection rate Consequently more and more wind clumpsare deposited outside the magnetosphere [42] This regimeis in connection with the spin-down evolution of young X-ray pulsars [24] The minimum X-ray luminosity at whichaccretion possibly occurs is given by

119871119909119888 (min)

sim 1043119877minus1

6(119872

119872⊙

)

minus23

1205832

33(119875119904

1 s)

minus73

ergs sminus1(21)

An X-ray pulsar with luminosity below this value turns offas a result of the centrifugal mechanism Correspondingly athreshold in wind parameters is given by a relation betweenspin period and orbital period [35]

119875119904119888(min)

sim (119872

119872⊙

)

minus117

11987227

3012058367

33(119875orb1 day

)

47

minus37

119908minus6V1271199088

s(22)

which is obtained by using the condition 119903mag = 119903cor Keplerrsquosthird law and a free-fall approximation for the wind materialwithin 119903acc

In the vicinity of the NS the deposited wind matter isdisordered because of the supersonic rotation which causesturbulent motion around the magnetosphere The velocityof the turbulent wind at the magnetospheric boundary isclose to the sound speed [32] Consequently the accumulatedmatter can rotate either prograde or retrograde [43] Whenflowing prograde an accretion torque will transfer someangular momentum onto the NS and spin it up The spin-uptorque is

119879119888acc119901 = Ω

1199041199032

mag (23)

Therefore the total torque is written as

119879119888119901= 119879119888pro + 119879119888acc119901 = Ω

1199041199032

mag minusV2119908

Ω119904

(24)

The NS then behaves as a prograde propellerIf the wind flows retrogradely it imposes an inverted

torque and spins down the NS which is

119879119888acc119903 = minus119879119888acc119901 = minusΩ

1199041199032

mag (25)

Accordingly the total torque imposed on the NS in thisscenario is

119879119888119903= 119879119888pro + 119879119888acc119903 = minus

V2119908

Ω119904

minus Ω1199041199032

mag (26)


The NS is then a retrograde propeller and can spin down to avery long spin period [43] Substituting (26) into (12) we canobtain the spin evolutionary law in the regime of retrogradepropeller

119904119888119903

sim 10minus7119868minus1

45[119908minus6


101198753

1199043+ (

119872

119872⊙

)

97

sdot 57

119908minus6119886minus107

10Vminus2071199088

12058347

331198751199043] s sminus1

(27)

The NS can spin down to 1000 s during a timescale of

120591119888119903sim 10411986845[119908minus6


101198752

1199043

+ (119872

119872⊙

)

97

57

119908minus6119886minus107

10Vminus2071199088

12058347

33]

minus1

yr

(28)

Therefore the retrograde propeller can be an alternativemechanism for NSs with very long spin periods such assupergiant fast X-ray transients [43]

If the stellar wind is not strong enough the accretingmatter is approximately in radial free fall as it approachesthe magnetospheric boundary The captured material cannotaccumulate near the magnetosphere due to the supersonicrotation of NS Consequently the accretion torque containsonly the spin-down torque imposed by the propeller mecha-nism and the details are discussed by Urpin et al [44 45]

In the retrograde propeller phase both interaction withmagnetic fields by the propeller mechanism and invertedaccretion contribute to the luminosity which results in

119871119888119903= V2

119908+119866119872

119903magsim 1032[10(

119872

119872⊙

)

2

sdot 119908minus6


10+ (

119872

119872⊙

)

267

sdot 97

119908minus6Vminus3671199088

119886minus187

10120583minus47

33] ergs sminus1

(29)

In addition energy is also released through the shock formednear 119903mag and the corresponding luminosity is discussed byBozzo et al [38]

(III) Direct Wind Accretion Accretor When 119903acc gt 119903mag and119903cor gt 119903mag the captured material flows from the accretionradius and falls down directly toward the magnetospherewhere it is stopped by a collisionless shock Because of asub-Keplerian rotation at the magnetospheric boundary therotating NS cannot eject the wind material in this case andthe inflows experience a significant gravitational field andaccumulate near themagnetosphere Some of them penetratethe NS magnetosphere presumably in a sporadic fashion bymeans of Rayleigh-Taylor and Kelvin-Helmholtz instabilities[46] Along the open field lines above the magnetic polesmatter can also flow in relatively freely Most accumulatedmatter forms an extended adiabatic tenuous and rotating

atmosphere around the magnetosphere The time-averagedX-ray luminosity is governed by the wind parameters at 119903acc

119871119909sim 1033119877minus1

6(119872

119872⊙

)

3

119872minus23

30(119875orb1 day

)

minus43

sdot 119908minus6

Vminus41199088

ergs sminus1

(30)

assuming all gravitational potential energy is converted intoX-ray luminosity

When a gradient of density or velocity is present itbecomes possible to accrete angular momentum and theflows become unstable instead of a steady state [47] Theshock cone oscillates from one side of the accretor to theother side allowing the appearance of transient accretiondisks This instability generally known as flip-flop oscillation[48] produces fluctuations in the accretion rate and givesrise to stochastic accretion of positive and negative angularmomentum leading to the suggestion that it is the source ofthe variations seen in the pulse evolution of some supergiantX-ray binaries [49 50] If the accretion flow is stable theamount of angularmomentum transferred toNS is negligibleThe more unstable the accretion flow the higher the transferrate of angular momentum [48] A secular evolution canbe described by sudden jumps between states with counter-rotating quasi-Keplerian atmosphere around the magneto-sphere which imposes an inverted accretion torque on NSexpressed as (25) In addition the reversal rotation betweenthe adiabatic atmosphere and the magnetosphere may alsocause a flip-flop behavior of some clumps in the atmosphereejecting a part of material and leading to the loss of thermalenergy which imposes a torque with the same form as(11) [32 37] Accordingly the total spin-down torque in theregime of 119903acc gt 119903mag and 119903cor gt 119903mag is

119879119886= minus120581119905

1205832

1199033magminus Ω

1199041199032

mag (31)

Substituting it into (12) we obtain the spin evolution [43]

119904acc sim 10

minus6119868minus1

45[12058111990512058327

331198752

1199043(119872

119872⊙

)

157

sdot 67

119908minus6Vminus2471199088

119886minus127

10+ 12058387

331198751199043(119872

119872⊙

)

minus147

sdot 37

119908minus6V1671199088

11988687

10] s sminus1

(32)

The spin-down timescale reads

120591119904acc

sim 10311986845[12058111990512058327

331198751199043(119872

119872⊙

)

157

67

119908minus6Vminus2471199088

119886minus127

10

+ 12058387

33(119872

119872⊙

)

minus147

37

119908minus6V1671199088

11988687

10]

minus1

yr

(33)


The corresponding X-ray luminosity in this scenario is

119871119909acc

sim 1031(119872

119872⊙

)

267

97

119908minus6Vminus3671199088

119886minus187

10120583minus47

33ergs sminus1

(34)

24 Accretion from an Accretion Disk

241 Disk Formation In some systems the companionevolves and fills its Roche lobe and a RLOF occurs Aconsequence of RLOF is that the transferred material has arather high specific angular momentum so that it cannot beaccreted directly onto the mass-capturing NS Note that thematter must pass from the Roche lobe of the companion tothat of the NS through the inner Lagrangian point Withinthe Roche lobe of the NS the dynamics of high angularmomentum material will be controlled by the gravitationalfield of the NS alone which would give an elliptical orbitlying in the binary plane A continuous stream trying tofollow this orbit will therefore collide with itself resultingin dissipation of energy via shocks and finally settles downthrough postshock dissipation within a few orbital periodsinto a ring of lowest energy for a given angular momentumthat is a circular ring Since the gas has little opportunity torid itself of the angular momentum it carried we thus expectthe gas to orbit the NS in the binary plane Such a ring willspread both inward and outward as effectively dissipativeprocesses for example collisions of gas elements shocksand viscous dissipation and convert some of the energy ofthe bulk orbital motion into internal energy which is partlyradiated and therefore lost to the gas The only way in whichthe gas can meet this drain of energy is by sinking deeperinto the gravitational potential of the NS which in turnrequires it to lose angular momentum The orbiting gas thenredistributes its angular momentum on a timescale muchlonger than both the timescale over which it loses energy byradiative cooling and the dynamical timescale As a result thegas will lose as much energy as it can and spiral in towardsthe NS through a series of approximately Keplerian circularorbits in the orbital plane of binary forming an accretion diskaround the NS with the gas in the disk orbiting at Keplerianangular velocityΩ

119870(119903) = (119866119872119903

3)12

To be accreted onto the NS the material must somehowget rid of almost all its original angular momentum Theprocesses that cause the energy conversion exert torques onthe inspiralingmaterial which transport angularmomentumoutward through the disk Near the outer edge of the disksome other process finally removes this angular momentumIt is likely that angular momentum is fed back into theorbitalmotion of a binary system by tidal interaction betweenthe outer edge of disk and companion star An importantconsequence of this angular momentum transport is that theouter edge of disk will in general be at some radius exceedingthe circularization radius given by Frank et al [9] and it isobvious that the outer disk radius cannot exceed the Rochelobe of the NS Typically the maximum and minimum diskradii differ by a factor of 2-3 However the disk cannot extendall the way toNS surface because of themagnetic field which

stops the accreting plasma at a position where the pressure offield and the plasma become of the same order and leads tothe truncation and an inner boundary of accretion disk

242 Magnetosphere We consider a steady and radial free-fall accretion flow with constant accretion rate approach-ing the magnetized NS Thus the inward radial velocity andmass density near the boundary are given by

V119891119891equiv (

2119866119872

119903)

12

= 16 times 109119903minus12

8(119872

119872⊙

)

12

cm sminus1

120588119891119891equiv

(V11989111989141205871199032)

= 49 times 10minus10119903minus32

817(119872

119872⊙

)

minus12

g cmminus3

(35)

where 1199038is the distance 119903 in units of 108 cm

The magnetic field of the NS begins to dominate the flowwhen the magnetic pressure 11986128120587 becomes comparable tothe pressure of accretingmatter sim120588

119891119891V2119891119891 which includes both

ram pressure and thermal pressure The Alfven radius 119877119860is

then given by

119861 (119877119860)2

8120587= 120588119891119891(119877119860)

V119891119891(119877119860)2 (36)

for a dipolar field outside the NS 119861(119903) = 1205831199033 So the Alfven

radius reads

119877119860sim 108minus27

1712058347

30(119872

119872⊙

)

minus17

cm


3712058347

30(119872

119872⊙

)

minus17

119877minus27

6cm

(37)

where 12058330

is the magnetic moment of NS in units of1030 Gcm3 and 119871

37is the accretion luminosity in units of

1037 ergs sminus1Initially the radial flow tends to sweep the field lines

inward due to the high conductivity of plasma Since theplasma is fully ionized and therefore highly conducting wewould expect it to move along the field lines close enoughto the NS surface However it is the high conductivity ofaccreting plasma that complicates its dynamics since it deniesthe plasma ready access to the surface [51] The plasma tendsto be frozen to the magnetic flux and be swept up buildingup magnetic pressure which halts the flow and truncatesthe inner disk Finally an empty magnetic cavity is createdand surrounded by the plasma that is piling up drivingthe boundary inward as the plasma pressure there keepson increasing Such transition behavior may possibly occurat the very beginning of the accretion flow but a steadystate is quickly reached When the magnetic pressure of theNS magnetic field can balance the ram pressure of inflowsthe radial flow eventually halts creating a restricted regioninside which the magnetic field dominates the motion of the


plasmaThis is the magnetosphere [36] whose formation is aconsequence of the transition of the plasma flow

The boundary of the static magnetosphere is determinedby the condition of static pressure balance between the insideand outside of the magnetosphere including both plasmapressure 119875 and magnetic pressure 11986128120587 that is

(1198612

119905

8120587)

in+ 119875in = (

1198612

119905

8120587)

out+ 119875out (38)

where 119861119905is the tangential component of magnetic field and

the subscripts ldquoinrdquo and ldquooutrdquo denote ldquoinsiderdquo and ldquooutsiderdquo ofthe magnetospheric boundaryThis pressure balance relationcan be used to estimate the scale size of magnetosphere ina static configuration 1199030mag which depends on the particularphysical conditions near the magnetospheric boundary [36]However although the supersonic flow of plasma towardthe boundary is halted [8] some plasma can enter themagnetosphere via particle entry through the polar cuspsdiffusion of plasma across themagnetospheric boundary andmagnetic flux reconnection [52] as well as the Rayleigh-Taylor instability at the boundary If there is a plasma flowinto the magnetosphere its static structure is broken andthe static pressure balance (38) is no longer valid Insteadthe momentum balance and continuity of magnetic fieldcomponents normal to the magnetospheric boundary imply

(1198612

119905

8120587)

in+ 119875in + (120588V

2

119899)in

= (1198612

119905

8120587)

out+ 119875out + (120588V

2

119899)out

(39)

where 120588V2119899is the dynamic pressure of plasma in terms of its

velocity components V119899normal to the boundary Obviously

the scale of magnetosphere is in the order of the Alfvenradius 119903mag sim 119877119860

243 Transition Zone and Boundary Layer Theregionwherethe field lines thread the disk plasma forms a transitionzone between the unperturbed disk and the magnetospherein which the disk plasma across the field lines generatescurrents The currents confine the magnetic field inside ascreening radius 119903

119904sim (10ndash100)119903cor which gives the extent

of transition zone Within this region the accreted matteris forced to corotate with the NS by means of transportingangular momentum via stresses dominated by the magneticfield According to the angular velocity of accreting materialthe transition zone is divided into two parts (Figure 1) anouter transition zone with Keplerian angular velocity anda boundary layer in which the plasma deviates from theKeplerian value significantly which is separated at the radius1199030The structure of outer transition zone between 119903

0and

119903119904is very similar to that of a standard 120572-disk [25] at the

same radius with threemodifications that is the transport ofangularmomentumbetween disk andNSdue to themagnetic

Boundary

Outertransition

Transition

zonelayer

Magnetosphericflow

regionUnperturbed

disk flow

2h

rA

rsr0

120575

Figure 1 Schematic representation of the transition zone [41]which consists of an outer transition zone and a boundary layer

stress associated with the twisted field lines the radiativetransport of energy associated with the effective viscousdissipation and resistive dissipation of currents generated bythe cross-field motion of the plasma [53] and the dissipativestresses consisting of the usual effective viscous stress and themagnetic stress associated with the residual magnetosphericfield In the outer transition zone the poloidal field isscreened on a length-scale sim119903 by the azimuthal currentsgenerated by the radial cross-field drift of plasma Theazimuthal pitch of the magnetic field increases from 119903

0to a

maximum at the corotation point 119903cor and begins to decreaseThus the field lines between 119903cor and 119903119904 are swept backwardand exert a spin-down torque on the NS

The Keplerian motion ends at radius 1199030 and then the

angular velocity of the plasma is reduced from the Keplerianto corotate with NS which resorts to the transportation ofangular momentum through magnetic stresses [54] Accord-ingly the accreted matter is released from the disk and startsflowing toward the NS along the magnetic field lines Thetransition of the accretion flow requires a region with someextent This is the boundary layer with thickness of 120575 equiv

1199030minus 119877119860≪ 1199030 in which the angular momentum is conserved

and the angular velocity of plasma continuously changes fromΩ119870(1199030) toΩ

119904between 119903

0and 119903cor Because of a sub-Keplerian

angular velocity the close balance between centrifugal forceand gravitational force is broken and the radial flow attains amuch higher velocity increasing inward continuously from1199030where it must equal the slow radial drift characteristic

of the outer transition zone due to continuity Since therising magnetic pressure gradient opposes the centrifugalsupport the radial velocity passes through amaximum and isreduced to zero at the inner edge of the boundary layer Theboundary layer is basically an electromagnetic layer in thesense that the dominant stresses are magnetic stress and thedominant dissipation is through electromagnetic processeswhich obeyMaxwellrsquos equations However themass flow alsoplays an essential role in this layer since the cross-field radialflow generates the toroidal electric currents that screens themagnetic field of the NS


8

9

10

11

12

13

14

15

16

log[B

(G)]

minus25 minus20 minus15 minus10 minus05 00 05 10 15minus30

log[P(s)]

Spin-up line

Death lin

e

Normal pulsarsMSPsBinaries

Figure 2Magnetic fields and spin periods of observed pulsars (datataken from the ATNF pulsar catalogue) Black dots denote normalpulsars Red dots represent MSPs Green triangles are pulsars inbinaries The ldquospin-up linerdquo represents the minimum spin periodto which a spin-up process may proceed in an Eddington-limitedaccretion while the ldquodeath linerdquo corresponds to a polar cap voltagebelow which the pulsar activity is likely to switch off [17]

244 Accretion-Induced Field Decay and Spin-Up

Formation of the Question Observationally pulsars are usu-ally assigned to be of two distinct kinds whose spin periodsand surface magnetic fields are distributed almost bimodallywith a dichotomy formagnetic fields and spin periods whichare between 119861 sim 1011ndash13G and 119861 sim 108-9 G respectively andbetween 119875

119904sim 01ndash10 s and 119875

119904sim 1ndash20ms respectively [55]

The large population of pulsars with high surface magneticfields of 119861 sim 10

11ndash13 G and long spin periods of 119875119904sim a

few seconds represents the overwhelmingmajority of normalpulsars Most of them are isolated pulsars with only veryfew sources in binary systems On the other hand there isa smaller population of sources which display much weakermagnetic fields 119861 sim 10

8-9 G and short spin periods 119875119904le

20ms andmost of this population is associated with binaries(Figure 2) These two typical populations are connected witha thin bridge of pulsars in binaries The abnormal behaviorof the second group particularly the fact that these aremostly found in binaries had led to extensive investigationsconcerning the formation of millisecond pulsars (MSPs)The currently widely accepted idea is the accretion-inducedmagnetic field decay and spin-up that is to say the normalpulsars in binary systems have been ldquorecycledrdquo [56ndash58] Theobjects formed by such a recycling process are called recycledpulsars [17 59]

Accretion-Induced Field Decay and Spin-Up The accretedmatter in the transition zone begins to be channeled ontothe polar patches of the NS by the field lines where the

compressed accreted matter causes the expansion of polarzone in two directions downward and equatorward [60]Therefore the magnetic flux in the polar zone is diluted withmore and more plasma piling up onto the polar caps whichsubsequently diffuse outwards over the NS surface Thisprocess expands the area of the polar caps until occupyingthe entire surface and the magnetic flux is buried under theaccreted matter Finally the magnetosphere is compressed tothe NS surface and no field lines drag the plasma remainingan object with weak magnetic field The minimum field isdetermined by the condition that the magnetospheric radiusequals the NS radius which is about sim108 G Meanwhile theangular momentum carried by the accreted matter increasesto the spin angular momentum of the NS and spins itup leaving a MSP with spin period of a few millisecondsThe combined field decay and spin-up process is called therecycling process of the NS

During the recycling process the accretion-inducedmag-netic field evolution can be obtained analytically for an initialfield 119861(119905 = 0) = 119861

0and final magnetic field 119861

119891by [60]

119861 (119905) =

119861119891

(1 minus [119862 exp (minus119910) minus 1]2)74 (40)

Here we define the parameters as follows 119910 = (27)(Δ119872119872cr) the accreted mass Δ119872 = 119905 the crust mass 119872cr sim

02119872⊙ and 119862 = 1 + radic1 minus 119909

2

0sim 2 with 1199092

0= (1198611198911198610)47 The

bottom magnetic field 119861119891is defined by the magnetospheric

radius matching the NS radius that is 119903mag(119861119891) = 119877 Themagnetospheric radius 119903mag is taken as 119903mag = 120601119877

119860with a

model dependent parameter 120601 of about 05 [9 13 61] Usingthe relation 119903mag(119861119891) = 119877 we can obtain the bottommagneticfield

119861119891= 132 times 10

8(

18

)

12

119872

119872⊙

14

119877minus54

6120601minus74 G (41)

where 18= 10

18 g sminus1 For details see [60]In the meantime the NS is spun up by the angular

momentum carried by accreted matter according to [61]

minus119904= 58 times 10

minus5[(

119872

119872⊙

)

minus37

119877127

6119868minus1

45]

times 11986127

12(11987511990411987137

37)2

119899 (120596119904) s yrminus1

(42)

where 11986112

is the magnetic field in units of 1012 G and thedimensionless torque 119899(120596

119904) is a function of the fastness

parameter [36] which is introduced in order to describe therelative impotance of stellar rotation and defined by the ratioparameter of the angular velocities [36 54]

120596119904equiv

Ω119904

Ω119870(119877119860)= 135 (

119872

119872⊙

)

minus27

119877157

611986167

12119875minus1

119904119871minus37

37 (43)

Therefore the dimensionless accretion torque on NS is givenby [61]

119899 (120596119904) = 14 times (

1 minus 120596119904120596119888

1 minus 120596119904

) (44)


Spin-up line

Death lin

e

Normal PSRsMSPsBinaries

log[B0] = 137

log[B0] = 127

log[B0] = 117

8

9

10

11

12

13

14

15

16lo

g[B

(G)]

0 1 2minus2 minus1minus3

log[P(s)]

(a)

Spin-up line

Death lin

e


8

9

10

11

12

13

14

15

16

log[B

(G)]


log[P(s)]

P0 = 100 sP0 = 10 sP0 = 1 s

(b)

Figure 3 Magnetic field and spin period evolutionary tracks In (a) tracks are plotted for different initial magnetic fields 1198610= 5 times 10

13 G(solid line) 5 times 1012 G (dash line) and 5 times 1011 G (dot line) at fixed initial spin period 119875

0= 100 s and accretion rate 1017 g sminus1 In (b) with

different initial periods 1198750= 1 s (dotted line) 10 s (dashed line) and 100 s (solid line) at fixed initial field 119861

0= 5 times 10

12 G and accretion rate1017 g sminus1

where 120596119888sim 02ndash1 [62ndash66] with an original value of 035

[61 67] For a slowly rotating star 120596119904≪ 1 the torque is

119899(120596119904) asymp 14 so that 119879disk asymp 14119879

0 For fast rotators 119899(120596

119904)

decreases with increasing 120596119904and vanishes for the critical

value 120596119888 For 120596

119904gt 120596119888 119899(120596119904) becomes negative and increases

with the increasing 120596119904 until it reaches a maximum value

typically 120596max sim 095The NS cannot be spun up to infinity because of the

critical fastness120596119888 at which this torque vanishes and theNS is

spun up to reach the shortest spin period Then the rotationreaches an equilibrium spin period 119875eq

119875eq = 2120587120596minus1

119888Ω119870119903mag = 189119861

67

9ms (45)

where 1198619is 119861 in units of 109 G and the second relation is

obtained by taking119872 = 14119872⊙and119877

6= 1 and an Eddington

limit accretion rate [17] This line in field-period diagram isreferred to as the ldquospin-up linerdquo

At the end of the accretion phase (with an accreted mass≳02119872

⊙) the magnetic field and spin period of recycled

pulsars arrive at bottom values which cluster in a range of119861 = 10

8-9G and 119875119904lt 20ms [68 69] and the NSs remain to

beMSPs [70 71]Theminimum spin period and bottom fieldare independent of the initial values ofmagnetic field and spinperiod see Figure 3 [72] The bottom fields mildly vary withthe accretion rates and the accretedmass (Figure 4) while theminimum period is insensitive to them (Figure 5) [72 73]

The direct evidence for this recycled idea has been foundin LMXBs with an accreting millisecond X-ray pulsar forexample SAX J 18084-3658 [74] and in observing the linkbetween LMXBs and millisecond radio pulsars in the form

of the transition from an X-ray binary to a radio pulsar PSRJ 1023+0038 [75] The NSs in LMXBs are the evolutionaryprecursors to ldquorecycledrdquo MSPs [57] It is evident that X-raypulsars and recycled pulsars are correlated with both theduration of accretion phase and the total amount of accretedmatter [57 76] When a small amount of mass (≲0001119872

⊙) is

transferred the spin period mildly changes which may yielda HMXBwith a spin period of some seconds such as Her X-1[77] and Vela X-1 [78] If the NS accretes a small quantity ofmass from its companion for example sim0001119872

⊙ndash001119872

⊙

a recycled pulsar with a mildly weak field (119861 sim 1010 G) and

short spin period (119875119904sim 50ms) will be formed [79] like PSR

1913+16 and PSR J 0737-3039 [80] After accreting sufficientmass (Δ119872 ≳ 01119872

⊙) the lower magnetic field and shorter

spin period (119875119904≲ 20ms) of a MSP form for example SAX J

18084-3658 [74] and PSR J 1748-2446 [81]

245 Rotation Effects and Quasi-Quantized Disk The orbitalmotion of an accreting flow controlled by the potential ofa rotating NS is different from that in a flat space-timebecause of the rotation effects and the strong gravity of theNS Firstly stellar oblateness arises from the rotation and thusa quadruple term in the gravitational potential appears [82]Secondly a rotating massive object will impose a rotationalframe-dragging effect on the local inertial frame which isknown as gravitoelectromagnetism [83ndash86]

Gravitoelectromagnetism Gravitoelectromagnetism (GEM)is based on an analogy between Newtonrsquos law of gravitationand Coulombrsquos law of electricity The Newtonian solution ofthe gravitational field can be alternatively interpreted as a


7

8

9

10

11

12

13lo

g[B

(G)]

minus30 minus25 minus20 minus15 minus10 minus05 00 05minus35

log[P(s)]

(a)

7

8

9

10

11

12

13

log[B

(G)]

minus8 minus6 minus4 minus2 0minus10

log[ΔM(M⊙)]

(b)

minus3

minus2

minus1

0

log[P

(s)]


log[ΔM(M⊙)]

(c)

1

2

3

4

5

P (m

s)

04 06 08 1002

ΔMM⊙

(d)

Figure 4 119861 and 119875 evolution in the recycling process (a) shows the joint evolution of 119861 and 119875 (b) and (c) are their evolution as a functionof accreted mass Δ119872 The solid dashed and dotted lines are plotted for an accretion rate of 1018 g sminus1 1017 g sminus1 and 1016 g sminus1 respectivelyInitial values 119861

0= 5 times 10

12 G and 1198750= 1 s were taken (d) is a zoom-in view of (c) in a linear scale for spin periods shorter than 5ms

gravitoelectric field It is well known that a magnetic field isproduced by the motion of electric charges that is electriccurrent Accordingly the rotating mass current would giverise to a gravitomagnetic field In the framework of thegeneral theory of relativity a non-Newtonian massive mass-charge current can produce a gravitoelectromagnetic field[87ndash89]

In the linear approximation of the gravitational field theMinkowski metric 120578

120583] is perturbed due to the presence of agravitating source with a perturbative term ℎ

120583] and thus thebackground reads

119892120583] = 120578120583] + ℎ120583]

10038161003816100381610038161003816ℎ120583]10038161003816100381610038161003816≪ 1 (46)

We define the trace-reversed amplitude ℎ120583] = ℎ120583]minus(12)120578120583]ℎ

where ℎ = 120578120583]ℎ120583] = ℎ

120572

120572is the trace of ℎ

120583] Then expandingthe Einsteinrsquos field equations 119866

120583] = 8120587119866119879120583] in powers of ℎ120583]

and keeping only the linear order terms we obtain the fieldequations (in this part we set the light speed 119888 = 1) [82]

◻ℎ120583] = minus16120587119866119879120583] (47)

where the Lorentz gauge condition ℎ120583]] = 0 is imposed In

analogy with Maxwellrsquos field equations ◻119860]= 4120587119895

] we canfind that the role of the electromagnetic vector potential 119860]

is played by the tensor potential ℎ120583] while the role of the 4-

current 119895] is played by the stress-energy tensor119879120583]Therefore

the solution of (47) in terms of the retarded potential can bewritten as [90 91]

ℎ120583] = 4119866int

119879120583] (1199020 minus

10038161003816100381610038161003816119902119894minus 1199021015840

119894

10038161003816100381610038161003816 1199021015840

119894)

1003816100381610038161003816119902119894 minus 1199021015840

119894

1003816100381610038161003816

11988931199021015840

119894 (48)

with coordinates 119902120583

= (1199020 119902119894) (Note that we choose

Greek subscripts and indices (ie 120583 ] 120572 120573) to describethe 4-dimension space-time components (0 1 2 3) for test


Spin-up line

Death lin

e


2minus2 minus1 0 1minus3minus4

log[P(s)]

7

8

9

10

11

12

13

14

15

16

log[B

(G)]

M16 120596c = 09M17 120596c = 09M18 120596c = 09

Figure 5 Magnetic field and spin period evolutionary tracks fordifferent accretion rates = 10

18 g sminus1 (solid line) = 1017 g sminus1

(dashed line) and = 1016 g sminus1 (dotted line) at fixed initial field

1198610= 5 times 10

12 G and period 1198750= 100 s

particles while the subscripts and indices 119894 119895 119896 denote the3-dimension space coordinates 119909 119910 119911 (or 119903 120579 120601))

Neglecting all terms of high order and smaller terms thatinclude the tensor potential ℎ

119894119895(1199020 119902119894) [82] we canwrite down

the tensor potential ℎ120583] [91]

ℎ00= 4Φ

ℎ0119894= minus2119860

119894

(49)

HereΦ and119860119894are theNewtonian or gravitoelectric potential

and the gravitomagnetic vector potential respectively Theycan be expressed as

Φ(1199020 119902119894) = minus

119866119872

119903

119860119894(1199020 119902119894) = 119866

119869119895119902119896

1199033120576119895119896

119894

(50)

where 119860119894is in terms of the angular momentum 119869

119894of the NS

Accordingly the Lorentz gauge condition reduces to

120597Φ

1205971199020

+1

2nabla119894119860119895120575119894119895= 0 (51)

Then we can define the gravitoelectromagnetic field thatis the gravitoelectric field119892

119894and the gravitomagnetic fieldR

119894

as [90]

119892119894= minusnabla119894Φ minus

120597

1205971199020

(1

2119860119895) 120575119894119895

R119894= nabla119895119860119896120576119895119896

119894

(52)

In the approximation of weak field [82 92] and slow rotationof the NS 119892

119894contains mostly first-order corrections to

flat space-time and denotes the Newtonian gravitationalacceleration whereas R

119894contains second-order corrections

and is related to the rotation of the NSUsing (47) (51) and (52) and analogy with Maxwellrsquos

equations we get the gravitoelectromagnetic field equations[90]

nabla119894119892119895120575119894119895= minus4120587119866120588

119892

nabla119895119892119896= minus

120597

1205971199020

(1

2R119894)

nabla119894R119895120575119894119895= 0

nabla119895R119896=120597119892119894

1205971199020

minus 4120587119866119895119892119894

(53)

Here 120588119892is the mass density and 119895

119892= 120588119892times (velocity of the

mass flow generating the gravitomagnetic field) denotes themass current density or mass flux These equations includethe conservation law for mass current 120597120588120597119902

0+ nabla119894119895119892119895120575119894119895= 0

as they should beIn analogy with the Lorentz force in an electromagnetic

field the motion of a test particle 119898 in a gravitoelectromag-netic field is subject to a gravitoelectromagnetic force

119865GEM119894 = 119876119892119892119894 + 119876R119895R119896120576119895119896

119894 (54)

where 119876119892= minus119898 and 119876R = minus2119898 are the gravitoelectric

charge and gravitomagnetic charge of the test particle [90]respectively Accordingly the Larmor quantities in the gravi-toelectromagnetic field would be

119871= minus

119876119892

119898

119871=119876RR

2119898

(55)

Therefore the test particle has the translational acceleration119871= and the rotational frequency 120596

119871= |R|

For a rotating NS with angular velocity Ω119904 the gravita-

tional Larmor frequency of test particles can bewritten as [93]

120596119871=10038161003816100381610038161003816R10038161003816100381610038161003816=4

5

1198661198721198772

1199033Ω119904 (56)

which causes a split of the orbital motion of accreted particlesand changes the circular orbital motion on the binary planein the vertical direction of disk

Axisymmetrically Rotating Stars The stationary and axisym-metric space-timemetric arising from a rotating object in thepolar coordinates has a form of [94]

1198891199042= 1198902](119903120579)

1198891199052minus 1198902120582(119903120579)

1198891199032

minus 1198902120583(119903120579)

[11990321198891205792+ 1199032sin2120579 (119889120601 minus 120596 (119903 120579) 119889119905)2]

(57)


As a new feature of an axisymmetric structure the nondiag-onal elements appear

119892119905120601= 119892120601119905= 1199032sin21205791198902120583(119903120579)120596 (119903 120579) (58)

For a test particle at a great distance fromNS in its equatorialplane (120579 = 1205872) falling inwards from rest the 120601 coordinate at120579 = 1205872 obeys the equation [95]

119889120601

119889119905=119892120601119905

119892119905119905=

minus119892120601119905

119892120601120601

=minus11990321198902120583(119903120579)

120596 (119903)

minus11990321198902120583(119903120579)= 120596 (119903) (59)

Therefore 120596(119903) is referred to as the angular velocity of thelocal inertial frame which can be expressed in terms of theKeplerian velocity at radius 119903 [96]

120596 (119903) =2

5

radic119866119872

11990331198772Ω2

119904 (60)

Consequently the accreted matter experiences an increasingdrag in the direction of the rotation of NS and possesses acorresponding angular velocity 120596(119903)

Closure Orbits and the Quasi-Quantized Disk The accretionflow in a turbulentKeplerian diskwill flowhelical trajectoriesthat is open circular orbits at each radius and is endowedwith two frequencies that is angular frequency in thedirection of rotation and gravitational Larmor frequency invertical direction of the disk The former deviates the orbitalmotion from a circular orbit and the latter leads to somevertical oscillation If and only if the gravitational Larmorfrequency and angular velocity of accreted plasma satisfy [97]

119897120596 (119903) = 119899120596119871 (61)

where 119899 and 119897 are integers with 119899 119897 ge 1 the vertical splitalong disk and deformation in the direction of rotation of NScan be harmonious [98] that is the vertical oscillation anddeformed orbital circularmotion are syntonic As a result theorbital motion of accreted matter on separated and deformedopen circular orbits returns to a closed circular motion Theradii of closure circular orbits are give by

1199033=41198992

1198972

119866119872

Ω2119904

(62)

We call such a disk structure that is tenuous spiraling-ingas with the appearance of a band of closure circular orbitsas a quasi-quantized disk Moving on the closure circularorbits the accreted material is in a stable state with the firstderivative of angular momentum being larger than or equalto zero which correspond to a minimum of the effectivepotential With a slight perturbation the test particle willoscillate around the minimum manifesting as drift of theorbital frequency If the perturbation is strong enough totransfer sufficient angular momentum outwardly and drivethe particles to leave this state the material will continue tofollow the original helical track and spirals in In a turbulentand viscous accretion disk dissipative processes for exampleviscosity collisions of elements and shocks are responsiblefor the perturbation

3 Final Remarks

In NS X-ray binary systems accretion is the only viableenergy source that powers theX-ray emission Accretion ontoa NS is fed by mass transfer from the optical companionstar to the NS via either a stellar wind or RLOF Stellarwind accretion always occurs in NSHMXBs with high-mass OB supergiant companions or Be stars with radiation-driven stellar winds The neutron stars here are observedas X-ray pulsars while disk accretion dominates the mass-transfer mechanism in NSLMXBs which occurs when theouter layer of the companion flows into the Roche lobeof the NS along the inner Lagrangian point However thetransferred material cannot be accreted all the way ontoNS surface due to the strong magnetic field At a preferredradius where the magnetic pressure can balance the rampressure of the infalling flows the accreting plasma willhalt and interact with the magnetic field building up amagnetosphere which exerts an accretion torque and resultsin the spin period evolution of NS In NSHMXBs because ofthe disordered stellar wind some instabilities and turbulencelead to fluctuations in the mass accretion rate and give riseto a stochastic accretion with positive and negative angularmomentum which alternatively contributes to phases ofspin-up and spin-down For disk-fed LMXBs theNSs usuallyare spunupby the angularmomentumcarried by the accretedmatter during which the magnetic field is buried by theaccumulated plasma As a result the NS is spun up to afew milliseconds and the magnetic field decays to 108-9 Gremaining a MSP In several systems the torque reversal[99ndash102] and state changes [103 104] were observed andinvestigated because of sudden dynamical changes triggeredby a gradual variation of mass accretion rates [105]

In the inner region of an accretion disk the accretingplasma is endowed with two additional frequencies dueto frame-dragging effects arising from the rotation of theNS which contribute to a band of closed circular orbits atcertain radii Such a disk structure is expected to be thequasi-quantized structure The orbiting material moving onthese closure circular orbits is in a stable state With aslight perturbation the test particle will oscillate around theminimummanifesting as radial drift of the orbital frequencyIf the perturbation is strong enough to transfer sufficientangularmomentumoutwardly and drive the particles to leavethis state thematerial will continue to follow the helical trackand spirals in

Competing Interests

The author declares that he has no competing interests

Acknowledgments

This work is supported by the Fundamental Research Fundsfor the Central Universities (Grant no 161gpy49) at Sun Yat-Sen University


References

[1] R Giacconi H Gursky F R Paolini and B B Rossi ldquoEvidencefor X rays from sources outside the solar systemrdquo PhysicalReview Letters vol 9 no 11 pp 439ndash443 1962

[2] Y B Zeldovich and O H Guseynov ldquoCollapsed stars inbinariesrdquoThe Astrophysical Journal vol 144 p 840 1966

[3] RGiacconi E Kellogg PGorensteinHGursky andHTanan-baum ldquoAn X-ray scan of the galactic plane from UHURUrdquoTheAstrophysical Journal Letters vol 165 p L27 1971

[4] E Schreier R Levinson H Gursky E Kellogg H TananbaumandRGiacconi ldquoEvidence for the binary nature of centaurusX-3 fromUHURU X-ray observationsrdquoTheAstrophysical Journalvol 172 pp L79ndashL89 1972

[5] H Tananbaum H Gursky E M Kellogg R Levinson ESchreier and R Giacconi ldquoDiscovery of a periodic pulsatingbinary X-ray source in hercules from UHURUrdquoThe Astrophys-ical Journal Letters vol 174 p L143 1972

[6] J E Pringle and M J Rees ldquoAccretion disc models for compactX-ray sourcesrdquo Astronomy amp Astrophysics vol 21 p 1 1972

[7] K Davidson and J P Ostriker ldquoNeutron-star accretion in astellar wind model for a pulsed X-ray sourcerdquoTheAstrophysicalJournal vol 179 pp 585ndash598 1973

[8] F K Lamb C J Pethick and D Pines ldquoA model for compactX-ray sources accretion by rotating magnetic starsrdquoThe Astro-physical Journal vol 184 pp 271ndash290 1973

[9] J Frank A King and D J Raine Accretion Power in Astro-physics Cambridge University Press Cambridge UK 2002

[10] H V D Bradt and J E McClintock ldquoThe optical counterpartsof compact galactic X-ray sourcesrdquoAnnual Review of Astronomyand Astrophysics vol 21 pp 13ndash66 1983

[11] J van Paradijs ldquoGalactic populations of X-ray binariesrdquo inTiming Neutron Stars H Ogelman and E P J van den HeuvelEds p 191 Kluwer AcademicPlenum New York NY USA1989

[12] C Jones ldquoEnergy spectra of 43 galactic X-ray sources observedby UHURUrdquo The Astrophysical Journal vol 214 pp 856ndash8731977

[13] S L Shapiro and S A Teukolsky Black Holes White Dwarfsand Neutron Stars The Physics of Compact Objects John Wileyamp Sons New York NY USA 1983

[14] Q Z Liu J van Paradijs and E P J van den Heuvel ldquoA cata-logue of high-mass X-ray binariesrdquoAstronomy and AstrophysicsSupplement Series vol 147 no 1 pp 25ndash49 2000

[15] Q Z Liu J van Paradijs and E P J van den Heuvel ldquoHigh-mass X-ray binaries in the magellanic cloudsrdquo Astronomy andAstrophysics vol 442 no 3 pp 1135ndash1138 2005

[16] Q Z Liu J van Paradijs and E P J van denHeuvel ldquoCatalogueof high-mass X-ray binaries in the Galaxy (4th edition)rdquoAstronomy amp Astrophysics vol 455 no 3 pp 1165ndash1168 2006

[17] D Bhattacharya and E P J van den Heuvel ldquoFormation andevolution of binary and millisecond radio pulsarsrdquo PhysicsReports vol 203 no 1-2 pp 1ndash124 1991

[18] M van der Klis ldquoMillisecond oscillations in X-ray binariesrdquoAnnual Review of Astronomy and Astrophysics vol 38 no 1 pp717ndash760 2000

[19] J van Paradijs and J E McClintock ldquoOptical and ultravioletobservations of X-ray binariesrdquo in X-Ray Binaries W H GLewin J van Paradijs and E P J van den Heuvel Eds p 58Cambridge University Press Cambridge UK 1995

[20] Q Z Liu J vanParadijs andE P J van denHeuvel ldquoA catalogueof low-mass X-ray binariesrdquo Astronomy and Astrophysics vol368 no 3 pp 1021ndash1054 2001

[21] E P J van den Heuvel ldquoThe formation of compact objectsin binary systemsrdquo in Fundamental Problems in the Theory ofStellar Evolution D Sugimoto D Q Lamb andDN SchrammEds vol 93 of International Astronomical Union Symposia pp155ndash175 Reidel Dordrecht The Netherlands 1981

[22] P Podsiadlowski S Rappaport and E D Pfahl ldquoEvolutionarysequences for low- and intermediate-mass X-ray binariesrdquo TheAstrophysical Journal vol 565 no 2 I pp 1107ndash1133 2002

[23] E P J van den Heuvel ldquoModes of mass transfer and classes ofbinary X-ray sourcesrdquoTheAstrophysical Journal Letters vol 198part 2 pp L109ndashL112 1975

[24] A F Illarionov and R A Sunyaev ldquoWhy the number of galacticX-ray stars is so smallrdquo Astronomy amp Astrophysics vol 39 pp185ndash196 1975

[25] N I Shakura and R A Sunyaev ldquoBlack holes in binary systemsObservational appearancerdquo Astronomy amp Astrophysics vol 24pp 337ndash355 1973

[26] R-P Kudritzki and J Puls ldquoWinds from hot starsrdquo AnnualReview of Astronomy and Astrophysics vol 38 pp 613ndash6662000

[27] G J Savonije ldquoRoche-lobe overflow in X-ray binariesrdquo Astron-omy and Astrophysics vol 62 pp 317ndash338 1978

[28] H Bondi and F Hoyle ldquoOn the mechanism of accretion bystarsrdquo Monthly Notices of the Royal Astronomical Society vol104 no 5 pp 273ndash282 1944

[29] H Bondi ldquoOn spherically symmetrical accretionrdquo MonthlyNotices of the Royal Astronomical Society vol 112 no 2 pp 195ndash204 1952

[30] E A Spiegel ldquoThe gas dynamics of accretionrdquo in Interstellar GasDynamics H J Habing Ed IAU Symposium no 39 p 201 DReidel Publishing Company DordrechtTheNetherlands 1970

[31] N I Shakura ldquoDisk model of gas accretion on a relativistic starin a close binary systemrdquo Astronomicheskii Zhurnal vol 49 pp921ndash930 1972

[32] V M Lipunov G Borner and R S Wadhwa Astrophysics ofNeutron Stars Astronomy and Astrophysics Library SpringerBerlin Germany 1992

[33] V M Lipunov ldquoNonradial accretion onto magnetized neutronstarsrdquo Astronomicheskii Zhurnal vol 57 pp 1253ndash1256 1980

[34] J I Castor D C Abbott and R I Klein ldquoRadiation-drivenwinds in of starsrdquoThe Astrophysical Journal vol 195 part 1 pp157ndash174 1975

[35] L Stella N E White and R Rosner ldquoIntermittent stellar windaccretion and the long-term activity of Population I binarysystems containing an X-ray pulsarrdquo The Astrophysical Journalvol 308 pp 669ndash679 1986

[36] R F Elsner and F K Lamb ldquoAccretion by magnetic neutronstars Imdashmagnetospheric structure and stabilityrdquoTheAstrophys-ical Journal vol 215 pp 897ndash913 1977

[37] R E Davies and J E Pringle ldquoSpindown of neutron starsin close binary systemsmdashIIrdquo Monthly Notices of the RoyalAstronomical Society vol 196 no 2 pp 209ndash224 1981

[38] E Bozzo M Falanga and L Stella ldquoAre there magnetars inhigh-mass X-ray binaries The case of supergiant fast X-raytransientsrdquo The Astrophysical Journal vol 683 no 2 pp 1031ndash1044 2008

[39] D Lynden-Bell and J E Pringle ldquoThe evolution of viscous discsand the origin of the nebular variablesrdquo Monthly Notices of theRoyal Astronomical Society vol 168 no 3 pp 603ndash637 1974


[40] Y-M Wang ldquoSpin-reversed accretion as the cause of intermit-tent spindown in slow X-ray pulsarsrdquo Astronomy and Astro-physics vol 102 pp 36ndash44 1981

[41] P Ghosh and F K Lamb ldquoDisk accretion by magnetic neutronstarsrdquo The Astrophysical Journal Letters vol 223 pp L83ndashL871978

[42] L Maraschi R Traversini and A Treves ldquoA model for A0538ndash66 the fast flaring pulsarrdquo Monthly Notices of the RoyalAstronomical Society vol 204 no 4 pp 1179ndash1184 1983

[43] J Wang and H-K Chang ldquoRetrograde wind accretionmdashanalternative mechanism for long spin period of SFXTsrdquo Astron-omy amp Astrophysics vol 547 article A27 2012

[44] V Urpin U Geppert and D Konenkov ldquoMagnetic and spinevolution of neutron stars in close binariesrdquoMonthly Notices ofthe Royal Astronomical Society vol 295 no 4 pp 907ndash920 1998

[45] V Urpin D Konenkov and U Geppert ldquoEvolution of neutronstars in high-mass X-ray binariesrdquoMonthly Notices of the RoyalAstronomical Society vol 299 no 1 pp 73ndash77 1998

[46] F K Lamb ldquoAccretion by magnetic neutron starsrdquo in HighEnergy Transients S EWoosley Ed p 179 AIP New York NYUSA 1984

[47] T Matsuda M Inoue and K Sawada ldquoSpin-up and spin-downof an accreting compact objectrdquo Monthly Notices of the RoyalAstronomical Society vol 226 no 4 pp 785ndash811 1987

[48] TMatsuda N Sekino K Sawada et al ldquoOn the stability of windaccretionrdquo Astronomy and Astrophysics vol 248 no 1 pp 301ndash314 1991

[49] J S Benensohn D Q Lamb and R E Taam ldquoHydrody-namical studies of wind accretion onto compact objects two-dimensional calculationsrdquo Astrophysical Journal vol 478 no 2pp 723ndash733 1997

[50] E Shima T Matsuda U Anzer G Borner and H M J BoffinldquoNumerical computation of two dimensional wind accretion ofisothermal gasrdquoAstronomyampAstrophysics vol 337 pp 311ndash3201998

[51] P Ghosh and F K Lamb ldquoPlasma physics of accreting neutronstarsrdquo in Neutron Stars Theory and Observation J Venturaand D Pines Eds vol 344 of NATO ASI Series pp 363ndash444Springer Dordrecht The Netherlands 1991

[52] R F Elsner and F K Lamb ldquoAccretion by magnetic neutronstars IImdashplasma entry into the magnetosphere via diffusionpolar cusps andmagnetic field reconnectionrdquoTheAstrophysicalJournal vol 278 pp 326ndash344 1984

[53] P Ghosh and F K Lamb ldquoAccretion by rotating magneticneutron stars IImdashradial and vertical structure of the transitionzone in disk accretionrdquo The Astrophysical Journal vol 232 pp259ndash276 1979

[54] P Ghosh C J Pethick and F K Lamb ldquoAccretion by rotatingmagnetic neutron stars Imdashflow of matter inside the magneto-sphere and its implications for spin-up and spin-down of thestarrdquoThe Astrophysical Journal vol 217 pp 578ndash596 1977

[55] R N Manchester G B Hobbs A Teoh and M HobbsldquoThe Australia telescope national facility pulsar cataloguerdquoAstronomical Journal vol 129 no 4 pp 1993ndash2006 2005

[56] M A Alpar A F Cheng M A Ruderman and J Shaham ldquoAnew class of radio pulsarsrdquo Nature vol 300 no 5894 pp 728ndash730 1982

[57] R E TaamandE P J van denHeuvel ldquoMagnetic field decay andthe origin of neutron star binariesrdquo The Astrophysical Journalvol 305 pp 235ndash245 1986

[58] D Bhattacharya and G Srinivasan ldquoThe magnetic fields ofneutron stars and their evolutionrdquo in X-Ray Binaries W H GLewin J van Paradijs and E P J van den Heuvel Eds p 495Cambridge University Press Cambridge UK 1995

[59] V Radhakrishnan and G Srinivasan ldquoOn the origin of therecently discovered ultra-rapid pulsarrdquo Current Science vol 51pp 1096ndash1099 1982

[60] C M Zhang and Y Kojima ldquoThe bottom magnetic field andmagnetosphere evolution of neutron star in low-mass X-raybinaryrdquo Monthly Notices of the Royal Astronomical Society vol366 no 1 pp 137ndash143 2006

[61] P Ghosh and F K Lamb ldquoAccretion by rotating magneticneutron stars IIImdashaccretion torques and period changes inpulsating X-ray sourcesrdquoTheAstrophysical Journal vol 234 pp296ndash316 1979

[62] Y-M Wang ldquoDisc accretion by magnetized neutron starsmdashareassessment of the torquerdquo Astronomy amp Astrophysics vol 183no 2 pp 257ndash264 1987

[63] A N Parmar N E White L Stella C Izzo and P FerrildquoThe transient 42 second X-ray pulsar EXO 2030+375 Imdashthediscovery and the luminosity dependence of the pulse periodvariationsrdquo The Astrophysical Journal vol 338 pp 359ndash3721989

[64] A Koenigl ldquoDisk accretion onto magnetic T Tauri starsrdquo TheAstrophysical Journal vol 370 pp L39ndashL43 1991

[65] Y-MWang ldquoOn the torque exerted by amagnetically threadedaccretion diskrdquoTheAstrophysical Journal Letters vol 449 no 2article L153 1995

[66] P Ghosh ldquoRotation of T Tauri stars accretion discs and stellardynamosrdquo Monthly Notices of the Royal Astronomical Societyvol 272 no 4 pp 763ndash771 1995

[67] P Ghosh and F K Lamb ldquoDiagnostics of disk-magnetosphereinteraction in neutron star binariesrdquo in X-Ray Binaries and theFormation of Binary and Millisecond Radio Pulsars E P J vandenHeuvel and S A Rappaport Eds p 487 Kluwer AcademicDordrecht Netherlands 1992

[68] K S Cheng and C M Zhang ldquoMagnetic field evolution ofaccreting neutron starsrdquo Astronomy amp Astrophysics vol 337 pp441ndash446 1998

[69] K S Cheng and C M Zhang ldquoEvolution of magnetic fieldand spin period in accreting neutron starsrdquo Astronomy ampAstrophysics vol 361 pp 1001ndash1004 2000

[70] E P J van den Heuvel and O Bitzaraki ldquoThe magnetic fieldstrength versus orbital period relation for binary radio pulsarswith low-mass companions evidence for neutron-star forma-tion by accretion-induced collapserdquoAstronomyampAstrophysicsvol 297 p L41 1995

[71] E P J van den Heuvel and O Bitzaraki ldquoEvolution of binarieswith neutron starsrdquo in The Lives of the Neutron Stars M AAlpar U Kiziloglu and J van Paradijs Eds p 421 KluwerAcademic Dordrecht The Netherlands 1995

[72] JWang CM Zhang YH Zhao Y Kojima X Y Yin and LMSong ldquoSpin period evolution of a recycled pulsar in an accretingbinaryrdquo Astronomy amp Astrophysics vol 526 article A88 2011

[73] JWang CM Zhang andH-K Chang ldquoTesting the accretion-induced field-decay and spin-up model for recycled pulsarsrdquoAstronomy amp Astrophysics vol 540 article A100 6 pages 2012

[74] R Wijnands and M van der Klis ldquoA millisecond pulsar in anX-ray binary systemrdquo Nature vol 394 no 6691 pp 344ndash3461998


[75] A M Archibald I H Stairs S M Ransom et al ldquoA radiopulsarX-ray binary linkrdquo Science vol 324 no 5933 pp 1411ndash1414 2009

[76] N Shibazaki T Murakami J Shaham and K Nomoto ldquoDoesmass accretion lead to field decay in neutron starsrdquoNature vol342 no 6250 pp 656ndash658 1989

[77] A van der Meer L Kaper M H van Kerkwijk M H MHeemskerk and E P J van den Heuvel ldquoDetermination of themass of the neutron star in SMC X-1 LMC X-4 and Cen X-3with VLTUVESrdquoAstronomy amp Astrophysics vol 473 no 2 pp523ndash538 2007

[78] H Quaintrell A J Norton T D C Ash et al ldquoThe mass ofthe neutron star in Vela X-1 and tidally induced non-radialoscillations in GPVelrdquoAstronomy and Astrophysics vol 401 pp313ndash324 2003

[79] G J Francischelli R A M J Wijers and G E BrownldquoThe evolution of relativistic binary progenitor systemsrdquo TheAstrophysical Journal vol 565 no 1 pp 471ndash481 2002

[80] A G Lyne M Burgay M Kramer et al ldquoA double-pulsarsystem a rare laboratory for relativistic gravity and plasmaphysicsrdquo Science vol 303 no 5661 pp 1153ndash1157 2004

[81] D R Lorimer ldquoBinary and millisecond pulsarsrdquo Living Reviewsin Relativity vol 11 article 8 2008

[82] C W Misner K S Thorne and J A Wheeler Gravitation WH Freeman San Francisco Calif USA 1973

[83] H Thirring ldquoUber die wirkung rotierender ferner massen inder Einsteinschen gravitationstheorierdquo Physikalische Zeitschriftvol 19 pp 33ndash39 1918

[84] J Lense and H Thirring ldquoOn the influence of the properrotation of a central body on the motion of the planets and themoon according to Einsteinrsquos theory of gravitationrdquo Zeitschriftfur Physik vol 19 pp 156ndash163 1918

[85] HThirring ldquoBerichtigung zu meiner arbeit ldquouber die wirkungrotierender massen in der Einsteinschen gravitationstheorierdquordquoPhysikalische Zeitschrift vol 22 p 29 1921

[86] BMashhoon FW Hehl andD STheiss ldquoOn the gravitationaleffects of rotating masses the thirring-lense papersrdquo GeneralRelativity and Gravitation vol 16 no 8 pp 711ndash750 1984

[87] A Einstein The Meaning of Relativity Princeton UniversityPress Princeton NJ USA 1950

[88] D Bini and R J Jantzen Reference Frames and Gravitomag-netism Edited by J-F Pascual-Sanchez L Floria A SanMigueland F Vicente World Scientific Singapore 2001

[89] B Mashhoon Reference Frames and Gravitomagnetism Editedby J-F Pascual-Sanchez L Floria A SanMiguel and F VicenteWorld Scientific Singapore 2001

[90] B Mashhoon F Gronwald and H I M Lichtenegger ldquoGravit-omagnetism and the clock effectrdquo in Gyros Clocks Interferom-eters Testing Relativistic Graviy in Space vol 562 of LectureNotes in Physics pp 83ndash108 Springer Berlin Germany 2001

[91] M L Ruggiero and A Tartaglia ldquoGravitomagnetic effectsrdquoNuovo Cimento vol 117 pp 743ndash768 2002

[92] S Weinberg Gravitation and Cosmology Principles and Appli-cations of the General Theory of Relativity Edited by S Wein-berg John Wiley amp Sons 1972

[93] B M Mirza ldquoGravitomagnetic resonance shift due to a slowlyrotating compact starrdquo International Journal of Modern PhysicsD vol 13 no 2 pp 327ndash333 2004

[94] J M Bardeen and R V Wagoner ldquoRelativistic disks I UniformrotationrdquoThe Astrophysical Journal vol 167 pp 359ndash423 1971

[95] J B Hartle ldquoSlowly rotating relativistic stars I Equations ofstructurerdquoThe Astrophysical Journal vol 150 p 1005 1967

[96] N Stergioulas ldquoRotating stars in relativityrdquo Living Reviews inRelativity vol 6 article 3 2003

[97] J J Wang and H-K Chang ldquoOrbital motion and quasi-quantized disk around rotating neutron starsrdquo InternationalJournal of Modern Physics D vol 23 no 6 Article ID 14500532014

[98] Z Stuchlık A Kotrlova andG Torok ldquoMulti-resonance orbitalmodel of high-frequency quasi-periodic oscillations possiblehigh-precision determination of black hole and neutron starspinrdquo Astronomy amp Astrophysics vol 552 article A10 2013

[99] F Nagase ldquoAccretion-powered X-ray pulsarsrdquo Publications ofthe Astronomical Society of Japan vol 41 no 1 pp 1ndash79 1989

[100] L Bildsten D Chakrabarty J Chiu et al ldquoObservations ofaccreting pulsarsrdquoThe Astrophysical Journal Supplement Seriesvol 113 no 2 pp 367ndash408 1997

[101] D Chakrabarty L Bildsten J M Grunsfeld et al ldquoTorquereversal and spin-down of the accretion-powered pulsar 4U1626ndash67rdquoAstrophysical Journal vol 474 no 1 pp 414ndash425 1997

[102] D K Galloway D Chakrabarty E H Morgan and R ARemillard ldquoDiscovery of a high-latitude accreting millisecondpulsar in an ultracompact binaryrdquo Astrophysical Journal vol576 no 2 pp L137ndashL140 2002

[103] R Narayan and I Yi ldquoAdvection-dominated accretion under-fed black holes and neutron starsrdquo The Astrophysical Journalvol 452 p 710 1995

[104] J Li and D TWickramasinghe ldquoOn spin-upspin-down torquereversals in disc accreting pulsarsrdquoMonthly Notices of the RoyalAstronomical Society vol 300 no 4 pp 1015ndash1022 1998

[105] R W Nelson L Bildsten D Chakrabarty et al ldquoOn the dra-matic spin-upspin-down torque reversals in accreting pulsarsrdquoThe Astrophysical Journal Letters vol 488 no 2 p L117 1997

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


FluidsJournal of

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Journal of



Advances in Condensed Matter Physics

OpticsInternational Journal of



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Biophysics


ThermodynamicsJournal of


which for spherically symmetric accretion corresponds to alimit on a steady accretion rate the Eddington accretion rate

Edd asymp 10181198776g sminus1 asymp 15 times 10minus8119872

⊙1198776yrminus1 (4)

Lower and higher accretion rates than this critical value arecalled the ldquosub-rdquo and ldquosuper-rdquoEddington accretion respec-tively

Phenomenologically over 95 of the NS X-ray binariesfall into two distinct categories that is low-mass X-ray binary(LMXB) and high-mass X-ray binary (HMXB) [10 11]Wind-fed system in which the companion of NS loses mass inthe form of a stellar wind requires a massive companionwith mass ge10119872

⊙to drive the intense stellar wind and thus

appears to be high-mass system such as Centaurus X-3 NSsin HMXBs display relatively hard X-ray spectra with a peakenergy larger than 15 keV [12] and tend to manifest as regularX-ray pulsars with spin periods of 1ndash103s andmagnetic fieldsof 1011ndash1013 G [13] The companions in HMXBs are brightand luminous early-type Be or OB supergiant stars whichhave masses larger than 10 solar masses and are short livedand belong to the youngest stellar population in the galaxywith ages of sim105ndash107yrs [14ndash16] They are distributed closeto the galactic plane On the other hand X-ray binarieswith companion mass le 119872

⊙ are categorized as low-mass

X-ray binaries (LMXBs) in which mass transfer takes placeby Roche-lone overflow This transfer is driven either bylosing angular momentum due to gravitational radiation (forsystems of very small masses and orbital separations) andmagnetic braking (for systems of orbital periods 119875orb le

2 days) or by the evolution of the companion star (forsystems of 119875orb ge 2 days) Most NSs in LMXBs are X-ray busters [17] with relatively soft X-ray spectra of lt10 keVin exponential fittings [11] In LMXBs the NSs possessrelatively weak surface magnetic fields of 108-9G and shortspin periods of a fewmilliseconds [18] while the companionsof NSs are late-type main-sequence stars white dwarfs orsubgiant stars with F-G type spectra [19 20] In space theLMXBs are concentrated towards the galactic center andhave a fairly wide distribution around the galactic planewhich characterize a relatively old population with ages of(05ndash15) times 1010 yrs A few of strong galactic X-ray sourcesfor example Hercules X-1 are assigned to be intermediate-mass X-ray binaries [21 22] in which the companions of NSshave masses of 1-2119872

⊙ Intermediate-mass systems are rare

since mass transfer via Roche lobe is unstable and would leadto a very quick (sim103ndash105 yrs) evolution of the system whilein the case of stellar wind accretion the mass accretion rateis very low and the system would be very dim and hardlydetectable [23]

2 Physical Processes in AccretingNeutron Stars

21 Mass Transfer in Binary Systems Themass transfer fromthe companion toNS is either due to a stellarwind or toRochelobe overflow (RLOF) as described above [7 24]

A massive companion star at some evolutionary phaseejects much of its mass in the form of intense stellar wind

driven by radiation [25 26]The wind material leaves surfaceuniformly in all directions Some of material will be capturedgravitationally by the NS This mechanism is called stellarwind accretion and the corresponding systems are wind-fedsystems [7]

In the course of its evolution the optical companion starmay increase in radius to a point where the gravitational pullof the companion can remove the outer layers of its envelopewhich is the RLOF phase [9 27] In addition to the swelling ofthe companion this situation can also come about due to forexample the decrease of binary separation as a consequenceof magnetically coupled wind mass loss or gravitational radi-ation Some of material begins to flow over to the Roche lobeof NS through the inner Lagrangian point without losingenergy Because of the large angular momentum the matterenters an orbit at some distance from the NS Subsequentbatches of outflowingmatter from companion have nearly thesame initial conditions and hence fall on the same orbit As aresult a ring of increasing density is formed by the gas nearthe NS Because of the mutual collisions of individual densityor turbulence cells the angular momentum in the ring isredistributed The ring spreads out into a disk in which thematter rotates differentially Gradually the motion turns outinto steady-state disk accretion on a time scale determined bythe viscosity Some optical observations suggested that masstransfer by ROLF is taking place or perhaps mixed with thatby stellar wind accretion in some systems

22 Accretion Regimes Because of the different angularmomentum carried by the accreted material and the motionof the NS relative to the sound speed 119888

119904in the medium for

accretion onto a NS several modes are possible [28]

221 Spherically Symmetric Accretion [29] The NS hardlymoves relative to the medium in its vicinity that is V

infin≪ 119888119904

The gas of uniform density and pressure is at rest and doesnot possess significant angular momentum at infinity andfalls freely with spherically symmetrical and steady motiontowards the NS

222 Cylindrical Accretion [30] Although the angularmomentum is small the NS velocity is comparable with orlarger than the sound speed in the accreted matter that isVinfin≳ 119888119904 In this case a conical shock form will present as the

result of pressure effects

223 DiskAccretion [6 31] In a binary system if thematerialsupplied by the companion of the NS has a large angularmomentum due to the orbital motion we must take both thegravitational force and centrifugal forces into considerationAs a result an accretion disk ultimately forms around a NS inthe binary plane

In a number of astrophysical cases [32] the motion ofaccreted material in the vicinity of a compact object presentstwo-stream accretion that is a quasi-spherically symmetricflux of matter in addition to the accretion disk [33]

23 Accretion from a Stellar Wind Mass loss in the case ofOB supergiant stars is driven by radiation pressure [34] The












119903acc =2119866119872

V2119908

sim 1010 119872

119872⊙

Vminus21199088

cm (5)

where V1199088


=




119861 (119903mag)2

8120587= 120588 (119903mag) V (119903mag)

2

(6)

where 119861(119903mag) sim 1205831199033




2

119908= 119861(119903mag)




11988613

1012058313

33cm (7)

where 119908minus6

= 119908(10minus6119872⊙yr) 120583

33= 120583(10


1211988610cm

sim 101211987523

1198871011987213

30cm 119875


119887in units

of 10 days and 11987230


V(119903mag)41205871199032


119908

sim1199032

acc(41198862)

sim 10minus5(119872

119872⊙

)

2


10 (8)


119903mag sim 1010(119872

119872⊙

)

minus57

minus27

119908minus6V87119908811988647

1012058347

33cm (9)


119904= 2120587119875



being accreted

119903cor sim 1010(119872

119872⊙

)

13

11987523

1199043cm (10)

where 1198751199043


103 s



119879119898pro = minus

1205811199051205832

1199033mag (11)



2120587119868119889

119889119905

1

119875119904

= 119879 (12)


119904119898

sim 10minus612058111990512058333119868minus1

451198752

1199043119886minus1

1012

119908minus6V121199088

s sminus1 (13)


where 11986845


120591119898pro = 119875119904119904pur

sim 103120581minus1

119905120583minus1

3311986845119875minus1

119904311988610minus12


yr(14)


119875119904gt 112

119872

119872⊙

Vminus21199088

s (15)


119875119904gt 64 times 10

3 119872

119872⊙

Vminus31199088

s (16)


119871119909119898 (min) sim 1033 ( 119872

119872⊙

)

minus3

119877minus1

61205832

33V71199088

ergs sminus1 (17)


119875orb119898 (max)

sim 103(119872

119872⊙

)

92

119872minus12

OB30120583minus32

3334

119908minus6Vminus3341199088

days(18)



119871119909119903(min) sim 10431205832

33(119875119904

01 s)

minus4119872

119872⊙

119877minus1

6Vminus11199088

ergs sminus1 (19)




119879119888pro = minus

V2119908

Ω119904

(20)


119871119909119888 (min)


6(119872

119872⊙

)

minus23

1205832

33(119875119904

1 s)

minus73

ergs sminus1(21)


119875119904119888(min)

sim (119872

119872⊙

)

minus117

11987227

3012058367

33(119875orb1 day

)

47

minus37

119908minus6V1271199088

s(22)



119879119888acc119901 = Ω

1199041199032

mag (23)


119879119888119901= 119879119888pro + 119879119888acc119901 = Ω

1199041199032

mag minusV2119908

Ω119904

(24)




1199041199032

mag (25)


119879119888119903= 119879119888pro + 119879119888acc119903 = minus

V2119908

Ω119904

minus Ω1199041199032

mag (26)



119904119888119903


45[119908minus6


101198753

1199043+ (

119872

119872⊙

)

97

sdot 57

119908minus6119886minus107

10Vminus2071199088

12058347

331198751199043] s sminus1

(27)


120591119888119903sim 10411986845[119908minus6


101198752

1199043

+ (119872

119872⊙

)

97

57

119908minus6119886minus107

10Vminus2071199088

12058347

33]

minus1

yr

(28)




119871119888119903= V2

119908+119866119872

119903magsim 1032[10(

119872

119872⊙

)

2

sdot 119908minus6


10+ (

119872

119872⊙

)

267

sdot 97

119908minus6Vminus3671199088

119886minus187

10120583minus47

33] ergs sminus1

(29)




119871119909sim 1033119877minus1

6(119872

119872⊙

)

3

119872minus23

30(119875orb1 day

)

minus43

sdot 119908minus6

Vminus41199088

ergs sminus1

(30)



119879119886= minus120581119905

1205832

1199033magminus Ω

1199041199032

mag (31)


119904acc sim 10

minus6119868minus1

45[12058111990512058327

331198752

1199043(119872

119872⊙

)

157

sdot 67

119908minus6Vminus2471199088

119886minus127

10+ 12058387

331198751199043(119872

119872⊙

)

minus147

sdot 37

119908minus6V1671199088

11988687

10] s sminus1

(32)


120591119904acc

sim 10311986845[12058111990512058327

331198751199043(119872

119872⊙

)

157

67

119908minus6Vminus2471199088

119886minus127

10

+ 12058387

33(119872

119872⊙

)

minus147

37

119908minus6V1671199088

11988687

10]

minus1

yr

(33)



119871119909acc

sim 1031(119872

119872⊙

)

267

97

119908minus6Vminus3671199088

119886minus187

10120583minus47

33ergs sminus1

(34)



119870(119903) = (119866119872119903

3)12




V119891119891equiv (

2119866119872

119903)

12

= 16 times 109119903minus12

8(119872

119872⊙

)

12

cm sminus1

120588119891119891equiv

(V11989111989141205871199032)


817(119872

119872⊙

)

minus12

g cmminus3

(35)





then given by

119861 (119877119860)2

8120587= 120588119891119891(119877119860)

V119891119891(119877119860)2 (36)


radius reads

119877119860sim 108minus27

1712058347

30(119872

119872⊙

)

minus17

cm


3712058347

30(119872

119872⊙

)

minus17

119877minus27

6cm

(37)

where 12058330








(1198612

119905

8120587)

in+ 119875in = (

1198612

119905

8120587)

out+ 119875out (38)



(1198612

119905

8120587)

in+ 119875in + (120588V

2

119899)in

= (1198612

119905

8120587)

out+ 119875out + (120588V

2

119899)out

(39)







0and



Boundary

Outertransition

Transition

zonelayer

Magnetosphericflow

regionUnperturbed

disk flow

2h

rA

rsr0

120575



0to a






119904between 119903





8

9

10

11

12

13

14

15

16

log[B

(G)]


log[P(s)]

Spin-up line

Death lin

e





119904sim 01ndash10 s and 119875











119891by [60]

119861 (119905) =

119861119891



02119872⊙ and 119862 = 1 + radic1 minus 119909

2

0sim 2 with 1199092

0= (1198611198911198610)47 The



119860with a


119861119891= 132 times 10

8(

18

)

12

119872

119872⊙

14

119877minus54

6120601minus74 G (41)

where 18= 10




minus5[(

119872

119872⊙

)

minus37

119877127

6119868minus1

45]

times 11986127

12(11987511990411987137

37)2

119899 (120596119904) s yrminus1

(42)

where 11986112




120596119904equiv

Ω119904

Ω119870(119877119860)= 135 (

119872

119872⊙

)

minus27

119877157

611986167

12119875minus1

119904119871minus37

37 (43)


119899 (120596119904) = 14 times (

1 minus 120596119904120596119888

1 minus 120596119904

) (44)


Spin-up line

Death lin

e


log[B0] = 137

log[B0] = 127

log[B0] = 117

8

9

10

11

12

13

14

15

16lo

g[B

(G)]


log[P(s)]

(a)

Spin-up line

Death lin

e


8

9

10

11

12

13

14

15

16

log[B

(G)]


log[P(s)]

P0 = 100 sP0 = 10 sP0 = 1 s

(b)





0= 5 times 10






119904)


value 120596119888 For 120596






119875eq = 2120587120596minus1

119888Ω119870119903mag = 189119861

67

9ms (45)












⊙) is


⊙ndash001119872

⊙






18084-3658 [74] and PSR J 1748-2446 [81]




7

8

9

10

11

12

13lo

g[B

(G)]


log[P(s)]

(a)

7

8

9

10

11

12

13

log[B

(G)]


log[ΔM(M⊙)]

(b)

minus3

minus2

minus1

0

log[P

(s)]


log[ΔM(M⊙)]

(c)

1

2

3

4

5

P (m

s)

04 06 08 1002

ΔMM⊙

(d)


0= 5 times 10






119892120583] = 120578120583] + ℎ120583]

10038161003816100381610038161003816ℎ120583]10038161003816100381610038161003816≪ 1 (46)


where ℎ = 120578120583]ℎ120583] = ℎ

120572



120583] = 8120587119866119879120583] in powers of ℎ120583]


◻ℎ120583] = minus16120587119866119879120583] (47)







ℎ120583] = 4119866int

119879120583] (1199020 minus

10038161003816100381610038161003816119902119894minus 1199021015840

119894

10038161003816100381610038161003816 1199021015840

119894)

1003816100381610038161003816119902119894 minus 1199021015840

119894

1003816100381610038161003816

11988931199021015840

119894 (48)





Spin-up line

Death lin

e



log[P(s)]

7

8

9

10

11

12

13

14

15

16

log[B

(G)]

M16 120596c = 09M17 120596c = 09M18 120596c = 09




1198610= 5 times 10

12 G and period 1198750= 100 s



119894119895(1199020 119902119894) [82] we canwrite down


ℎ00= 4Φ

ℎ0119894= minus2119860

119894

(49)



Φ(1199020 119902119894) = minus

119866119872

119903

119860119894(1199020 119902119894) = 119866

119869119895119902119896

1199033120576119895119896

119894

(50)


119894of the NS


120597Φ

1205971199020

+1

2nabla119894119860119895120575119894119895= 0 (51)



119894

as [90]


120597

1205971199020

(1

2119860119895) 120575119894119895

R119894= nabla119895119860119896120576119895119896

119894

(52)







nabla119894119892119895120575119894119895= minus4120587119866120588

119892

nabla119895119892119896= minus

120597

1205971199020

(1

2R119894)

nabla119894R119895120575119894119895= 0

nabla119895R119896=120597119892119894

1205971199020

minus 4120587119866119895119892119894

(53)




0+ nabla119894119895119892119895120575119894119895= 0



119865GEM119894 = 119876119892119892119894 + 119876R119895R119896120576119895119896

119894 (54)



119871= minus

119876119892

119898

119871=119876RR

2119898

(55)


119871= |R|



120596119871=10038161003816100381610038161003816R10038161003816100381610038161003816=4

5

1198661198721198772

1199033Ω119904 (56)



1198891199042= 1198902](119903120579)

1198891199052minus 1198902120582(119903120579)

1198891199032

minus 1198902120583(119903120579)

[11990321198891205792+ 1199032sin2120579 (119889120601 minus 120596 (119903 120579) 119889119905)2]

(57)



119892119905120601= 119892120601119905= 1199032sin21205791198902120583(119903120579)120596 (119903 120579) (58)


119889120601

119889119905=119892120601119905

119892119905119905=

minus119892120601119905

119892120601120601

=minus11990321198902120583(119903120579)

120596 (119903)

minus11990321198902120583(119903120579)= 120596 (119903) (59)


120596 (119903) =2

5

radic119866119872

11990331198772Ω2

119904 (60)



119897120596 (119903) = 119899120596119871 (61)


1199033=41198992

1198972

119866119872

Ω2119904

(62)


3 Final Remarks



Competing Interests


Acknowledgments



References

















































































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics














119903acc =2119866119872

V2119908

sim 1010 119872

119872⊙

Vminus21199088

cm (5)

where V1199088


=




119861 (119903mag)2

8120587= 120588 (119903mag) V (119903mag)

2

(6)

where 119861(119903mag) sim 1205831199033




2

119908= 119861(119903mag)




11988613

1012058313

33cm (7)

where 119908minus6

= 119908(10minus6119872⊙yr) 120583

33= 120583(10


1211988610cm

sim 101211987523

1198871011987213

30cm 119875


119887in units

of 10 days and 11987230


V(119903mag)41205871199032


119908

sim1199032

acc(41198862)

sim 10minus5(119872

119872⊙

)

2


10 (8)


119903mag sim 1010(119872

119872⊙

)

minus57

minus27

119908minus6V87119908811988647

1012058347

33cm (9)


119904= 2120587119875



being accreted

119903cor sim 1010(119872

119872⊙

)

13

11987523

1199043cm (10)

where 1198751199043


103 s



119879119898pro = minus

1205811199051205832

1199033mag (11)



2120587119868119889

119889119905

1

119875119904

= 119879 (12)


119904119898

sim 10minus612058111990512058333119868minus1

451198752

1199043119886minus1

1012

119908minus6V121199088

s sminus1 (13)


where 11986845


120591119898pro = 119875119904119904pur

sim 103120581minus1

119905120583minus1

3311986845119875minus1

119904311988610minus12


yr(14)


119875119904gt 112

119872

119872⊙

Vminus21199088

s (15)


119875119904gt 64 times 10

3 119872

119872⊙

Vminus31199088

s (16)


119871119909119898 (min) sim 1033 ( 119872

119872⊙

)

minus3

119877minus1

61205832

33V71199088

ergs sminus1 (17)


119875orb119898 (max)

sim 103(119872

119872⊙

)

92

119872minus12

OB30120583minus32

3334

119908minus6Vminus3341199088

days(18)



119871119909119903(min) sim 10431205832

33(119875119904

01 s)

minus4119872

119872⊙

119877minus1

6Vminus11199088

ergs sminus1 (19)




119879119888pro = minus

V2119908

Ω119904

(20)


119871119909119888 (min)


6(119872

119872⊙

)

minus23

1205832

33(119875119904

1 s)

minus73

ergs sminus1(21)


119875119904119888(min)

sim (119872

119872⊙

)

minus117

11987227

3012058367

33(119875orb1 day

)

47

minus37

119908minus6V1271199088

s(22)



119879119888acc119901 = Ω

1199041199032

mag (23)


119879119888119901= 119879119888pro + 119879119888acc119901 = Ω

1199041199032

mag minusV2119908

Ω119904

(24)




1199041199032

mag (25)


119879119888119903= 119879119888pro + 119879119888acc119903 = minus

V2119908

Ω119904

minus Ω1199041199032

mag (26)



119904119888119903


45[119908minus6


101198753

1199043+ (

119872

119872⊙

)

97

sdot 57

119908minus6119886minus107

10Vminus2071199088

12058347

331198751199043] s sminus1

(27)


120591119888119903sim 10411986845[119908minus6


101198752

1199043

+ (119872

119872⊙

)

97

57

119908minus6119886minus107

10Vminus2071199088

12058347

33]

minus1

yr

(28)




119871119888119903= V2

119908+119866119872

119903magsim 1032[10(

119872

119872⊙

)

2

sdot 119908minus6


10+ (

119872

119872⊙

)

267

sdot 97

119908minus6Vminus3671199088

119886minus187

10120583minus47

33] ergs sminus1

(29)




119871119909sim 1033119877minus1

6(119872

119872⊙

)

3

119872minus23

30(119875orb1 day

)

minus43

sdot 119908minus6

Vminus41199088

ergs sminus1

(30)



119879119886= minus120581119905

1205832

1199033magminus Ω

1199041199032

mag (31)


119904acc sim 10

minus6119868minus1

45[12058111990512058327

331198752

1199043(119872

119872⊙

)

157

sdot 67

119908minus6Vminus2471199088

119886minus127

10+ 12058387

331198751199043(119872

119872⊙

)

minus147

sdot 37

119908minus6V1671199088

11988687

10] s sminus1

(32)


120591119904acc

sim 10311986845[12058111990512058327

331198751199043(119872

119872⊙

)

157

67

119908minus6Vminus2471199088

119886minus127

10

+ 12058387

33(119872

119872⊙

)

minus147

37

119908minus6V1671199088

11988687

10]

minus1

yr

(33)



119871119909acc

sim 1031(119872

119872⊙

)

267

97

119908minus6Vminus3671199088

119886minus187

10120583minus47

33ergs sminus1

(34)



119870(119903) = (119866119872119903

3)12




V119891119891equiv (

2119866119872

119903)

12

= 16 times 109119903minus12

8(119872

119872⊙

)

12

cm sminus1

120588119891119891equiv

(V11989111989141205871199032)


817(119872

119872⊙

)

minus12

g cmminus3

(35)





then given by

119861 (119877119860)2

8120587= 120588119891119891(119877119860)

V119891119891(119877119860)2 (36)


radius reads

119877119860sim 108minus27

1712058347

30(119872

119872⊙

)

minus17

cm


3712058347

30(119872

119872⊙

)

minus17

119877minus27

6cm

(37)

where 12058330








(1198612

119905

8120587)

in+ 119875in = (

1198612

119905

8120587)

out+ 119875out (38)



(1198612

119905

8120587)

in+ 119875in + (120588V

2

119899)in

= (1198612

119905

8120587)

out+ 119875out + (120588V

2

119899)out

(39)







0and



Boundary

Outertransition

Transition

zonelayer

Magnetosphericflow

regionUnperturbed

disk flow

2h

rA

rsr0

120575



0to a






119904between 119903





8

9

10

11

12

13

14

15

16

log[B

(G)]


log[P(s)]

Spin-up line

Death lin

e





119904sim 01ndash10 s and 119875











119891by [60]

119861 (119905) =

119861119891



02119872⊙ and 119862 = 1 + radic1 minus 119909

2

0sim 2 with 1199092

0= (1198611198911198610)47 The



119860with a


119861119891= 132 times 10

8(

18

)

12

119872

119872⊙

14

119877minus54

6120601minus74 G (41)

where 18= 10




minus5[(

119872

119872⊙

)

minus37

119877127

6119868minus1

45]

times 11986127

12(11987511990411987137

37)2

119899 (120596119904) s yrminus1

(42)

where 11986112




120596119904equiv

Ω119904

Ω119870(119877119860)= 135 (

119872

119872⊙

)

minus27

119877157

611986167

12119875minus1

119904119871minus37

37 (43)


119899 (120596119904) = 14 times (

1 minus 120596119904120596119888

1 minus 120596119904

) (44)


Spin-up line

Death lin

e


log[B0] = 137

log[B0] = 127

log[B0] = 117

8

9

10

11

12

13

14

15

16lo

g[B

(G)]


log[P(s)]

(a)

Spin-up line

Death lin

e


8

9

10

11

12

13

14

15

16

log[B

(G)]


log[P(s)]

P0 = 100 sP0 = 10 sP0 = 1 s

(b)





0= 5 times 10






119904)


value 120596119888 For 120596






119875eq = 2120587120596minus1

119888Ω119870119903mag = 189119861

67

9ms (45)












⊙) is


⊙ndash001119872

⊙






18084-3658 [74] and PSR J 1748-2446 [81]




7

8

9

10

11

12

13lo

g[B

(G)]


log[P(s)]

(a)

7

8

9

10

11

12

13

log[B

(G)]


log[ΔM(M⊙)]

(b)

minus3

minus2

minus1

0

log[P

(s)]


log[ΔM(M⊙)]

(c)

1

2

3

4

5

P (m

s)

04 06 08 1002

ΔMM⊙

(d)


0= 5 times 10






119892120583] = 120578120583] + ℎ120583]

10038161003816100381610038161003816ℎ120583]10038161003816100381610038161003816≪ 1 (46)


where ℎ = 120578120583]ℎ120583] = ℎ

120572



120583] = 8120587119866119879120583] in powers of ℎ120583]


◻ℎ120583] = minus16120587119866119879120583] (47)







ℎ120583] = 4119866int

119879120583] (1199020 minus

10038161003816100381610038161003816119902119894minus 1199021015840

119894

10038161003816100381610038161003816 1199021015840

119894)

1003816100381610038161003816119902119894 minus 1199021015840

119894

1003816100381610038161003816

11988931199021015840

119894 (48)





Spin-up line

Death lin

e



log[P(s)]

7

8

9

10

11

12

13

14

15

16

log[B

(G)]

M16 120596c = 09M17 120596c = 09M18 120596c = 09




1198610= 5 times 10

12 G and period 1198750= 100 s



119894119895(1199020 119902119894) [82] we canwrite down


ℎ00= 4Φ

ℎ0119894= minus2119860

119894

(49)



Φ(1199020 119902119894) = minus

119866119872

119903

119860119894(1199020 119902119894) = 119866

119869119895119902119896

1199033120576119895119896

119894

(50)


119894of the NS


120597Φ

1205971199020

+1

2nabla119894119860119895120575119894119895= 0 (51)



119894

as [90]


120597

1205971199020

(1

2119860119895) 120575119894119895

R119894= nabla119895119860119896120576119895119896

119894

(52)







nabla119894119892119895120575119894119895= minus4120587119866120588

119892

nabla119895119892119896= minus

120597

1205971199020

(1

2R119894)

nabla119894R119895120575119894119895= 0

nabla119895R119896=120597119892119894

1205971199020

minus 4120587119866119895119892119894

(53)




0+ nabla119894119895119892119895120575119894119895= 0



119865GEM119894 = 119876119892119892119894 + 119876R119895R119896120576119895119896

119894 (54)



119871= minus

119876119892

119898

119871=119876RR

2119898

(55)


119871= |R|



120596119871=10038161003816100381610038161003816R10038161003816100381610038161003816=4

5

1198661198721198772

1199033Ω119904 (56)



1198891199042= 1198902](119903120579)

1198891199052minus 1198902120582(119903120579)

1198891199032

minus 1198902120583(119903120579)

[11990321198891205792+ 1199032sin2120579 (119889120601 minus 120596 (119903 120579) 119889119905)2]

(57)



119892119905120601= 119892120601119905= 1199032sin21205791198902120583(119903120579)120596 (119903 120579) (58)


119889120601

119889119905=119892120601119905

119892119905119905=

minus119892120601119905

119892120601120601

=minus11990321198902120583(119903120579)

120596 (119903)

minus11990321198902120583(119903120579)= 120596 (119903) (59)


120596 (119903) =2

5

radic119866119872

11990331198772Ω2

119904 (60)



119897120596 (119903) = 119899120596119871 (61)


1199033=41198992

1198972

119866119872

Ω2119904

(62)


3 Final Remarks



Competing Interests


Acknowledgments



References

















































































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




where 11986845


120591119898pro = 119875119904119904pur

sim 103120581minus1

119905120583minus1

3311986845119875minus1

119904311988610minus12


yr(14)


119875119904gt 112

119872

119872⊙

Vminus21199088

s (15)


119875119904gt 64 times 10

3 119872

119872⊙

Vminus31199088

s (16)


119871119909119898 (min) sim 1033 ( 119872

119872⊙

)

minus3

119877minus1

61205832

33V71199088

ergs sminus1 (17)


119875orb119898 (max)

sim 103(119872

119872⊙

)

92

119872minus12

OB30120583minus32

3334

119908minus6Vminus3341199088

days(18)



119871119909119903(min) sim 10431205832

33(119875119904

01 s)

minus4119872

119872⊙

119877minus1

6Vminus11199088

ergs sminus1 (19)




119879119888pro = minus

V2119908

Ω119904

(20)


119871119909119888 (min)


6(119872

119872⊙

)

minus23

1205832

33(119875119904

1 s)

minus73

ergs sminus1(21)


119875119904119888(min)

sim (119872

119872⊙

)

minus117

11987227

3012058367

33(119875orb1 day

)

47

minus37

119908minus6V1271199088

s(22)



119879119888acc119901 = Ω

1199041199032

mag (23)


119879119888119901= 119879119888pro + 119879119888acc119901 = Ω

1199041199032

mag minusV2119908

Ω119904

(24)




1199041199032

mag (25)


119879119888119903= 119879119888pro + 119879119888acc119903 = minus

V2119908

Ω119904

minus Ω1199041199032

mag (26)



119904119888119903


45[119908minus6


101198753

1199043+ (

119872

119872⊙

)

97

sdot 57

119908minus6119886minus107

10Vminus2071199088

12058347

331198751199043] s sminus1

(27)


120591119888119903sim 10411986845[119908minus6


101198752

1199043

+ (119872

119872⊙

)

97

57

119908minus6119886minus107

10Vminus2071199088

12058347

33]

minus1

yr

(28)




119871119888119903= V2

119908+119866119872

119903magsim 1032[10(

119872

119872⊙

)

2

sdot 119908minus6


10+ (

119872

119872⊙

)

267

sdot 97

119908minus6Vminus3671199088

119886minus187

10120583minus47

33] ergs sminus1

(29)




119871119909sim 1033119877minus1

6(119872

119872⊙

)

3

119872minus23

30(119875orb1 day

)

minus43

sdot 119908minus6

Vminus41199088

ergs sminus1

(30)



119879119886= minus120581119905

1205832

1199033magminus Ω

1199041199032

mag (31)


119904acc sim 10

minus6119868minus1

45[12058111990512058327

331198752

1199043(119872

119872⊙

)

157

sdot 67

119908minus6Vminus2471199088

119886minus127

10+ 12058387

331198751199043(119872

119872⊙

)

minus147

sdot 37

119908minus6V1671199088

11988687

10] s sminus1

(32)


120591119904acc

sim 10311986845[12058111990512058327

331198751199043(119872

119872⊙

)

157

67

119908minus6Vminus2471199088

119886minus127

10

+ 12058387

33(119872

119872⊙

)

minus147

37

119908minus6V1671199088

11988687

10]

minus1

yr

(33)



119871119909acc

sim 1031(119872

119872⊙

)

267

97

119908minus6Vminus3671199088

119886minus187

10120583minus47

33ergs sminus1

(34)



119870(119903) = (119866119872119903

3)12




V119891119891equiv (

2119866119872

119903)

12

= 16 times 109119903minus12

8(119872

119872⊙

)

12

cm sminus1

120588119891119891equiv

(V11989111989141205871199032)


817(119872

119872⊙

)

minus12

g cmminus3

(35)





then given by

119861 (119877119860)2

8120587= 120588119891119891(119877119860)

V119891119891(119877119860)2 (36)


radius reads

119877119860sim 108minus27

1712058347

30(119872

119872⊙

)

minus17

cm


3712058347

30(119872

119872⊙

)

minus17

119877minus27

6cm

(37)

where 12058330








(1198612

119905

8120587)

in+ 119875in = (

1198612

119905

8120587)

out+ 119875out (38)



(1198612

119905

8120587)

in+ 119875in + (120588V

2

119899)in

= (1198612

119905

8120587)

out+ 119875out + (120588V

2

119899)out

(39)







0and



Boundary

Outertransition

Transition

zonelayer

Magnetosphericflow

regionUnperturbed

disk flow

2h

rA

rsr0

120575



0to a






119904between 119903





8

9

10

11

12

13

14

15

16

log[B

(G)]


log[P(s)]

Spin-up line

Death lin

e





119904sim 01ndash10 s and 119875











119891by [60]

119861 (119905) =

119861119891



02119872⊙ and 119862 = 1 + radic1 minus 119909

2

0sim 2 with 1199092

0= (1198611198911198610)47 The



119860with a


119861119891= 132 times 10

8(

18

)

12

119872

119872⊙

14

119877minus54

6120601minus74 G (41)

where 18= 10




minus5[(

119872

119872⊙

)

minus37

119877127

6119868minus1

45]

times 11986127

12(11987511990411987137

37)2

119899 (120596119904) s yrminus1

(42)

where 11986112




120596119904equiv

Ω119904

Ω119870(119877119860)= 135 (

119872

119872⊙

)

minus27

119877157

611986167

12119875minus1

119904119871minus37

37 (43)


119899 (120596119904) = 14 times (

1 minus 120596119904120596119888

1 minus 120596119904

) (44)


Spin-up line

Death lin

e


log[B0] = 137

log[B0] = 127

log[B0] = 117

8

9

10

11

12

13

14

15

16lo

g[B

(G)]


log[P(s)]

(a)

Spin-up line

Death lin

e


8

9

10

11

12

13

14

15

16

log[B

(G)]


log[P(s)]

P0 = 100 sP0 = 10 sP0 = 1 s

(b)





0= 5 times 10






119904)


value 120596119888 For 120596






119875eq = 2120587120596minus1

119888Ω119870119903mag = 189119861

67

9ms (45)












⊙) is


⊙ndash001119872

⊙






18084-3658 [74] and PSR J 1748-2446 [81]




7

8

9

10

11

12

13lo

g[B

(G)]


log[P(s)]

(a)

7

8

9

10

11

12

13

log[B

(G)]


log[ΔM(M⊙)]

(b)

minus3

minus2

minus1

0

log[P

(s)]


log[ΔM(M⊙)]

(c)

1

2

3

4

5

P (m

s)

04 06 08 1002

ΔMM⊙

(d)


0= 5 times 10






119892120583] = 120578120583] + ℎ120583]

10038161003816100381610038161003816ℎ120583]10038161003816100381610038161003816≪ 1 (46)


where ℎ = 120578120583]ℎ120583] = ℎ

120572



120583] = 8120587119866119879120583] in powers of ℎ120583]


◻ℎ120583] = minus16120587119866119879120583] (47)







ℎ120583] = 4119866int

119879120583] (1199020 minus

10038161003816100381610038161003816119902119894minus 1199021015840

119894

10038161003816100381610038161003816 1199021015840

119894)

1003816100381610038161003816119902119894 minus 1199021015840

119894

1003816100381610038161003816

11988931199021015840

119894 (48)





Spin-up line

Death lin

e



log[P(s)]

7

8

9

10

11

12

13

14

15

16

log[B

(G)]

M16 120596c = 09M17 120596c = 09M18 120596c = 09




1198610= 5 times 10

12 G and period 1198750= 100 s



119894119895(1199020 119902119894) [82] we canwrite down


ℎ00= 4Φ

ℎ0119894= minus2119860

119894

(49)



Φ(1199020 119902119894) = minus

119866119872

119903

119860119894(1199020 119902119894) = 119866

119869119895119902119896

1199033120576119895119896

119894

(50)


119894of the NS


120597Φ

1205971199020

+1

2nabla119894119860119895120575119894119895= 0 (51)



119894

as [90]


120597

1205971199020

(1

2119860119895) 120575119894119895

R119894= nabla119895119860119896120576119895119896

119894

(52)







nabla119894119892119895120575119894119895= minus4120587119866120588

119892

nabla119895119892119896= minus

120597

1205971199020

(1

2R119894)

nabla119894R119895120575119894119895= 0

nabla119895R119896=120597119892119894

1205971199020

minus 4120587119866119895119892119894

(53)




0+ nabla119894119895119892119895120575119894119895= 0



119865GEM119894 = 119876119892119892119894 + 119876R119895R119896120576119895119896

119894 (54)



119871= minus

119876119892

119898

119871=119876RR

2119898

(55)


119871= |R|



120596119871=10038161003816100381610038161003816R10038161003816100381610038161003816=4

5

1198661198721198772

1199033Ω119904 (56)



1198891199042= 1198902](119903120579)

1198891199052minus 1198902120582(119903120579)

1198891199032

minus 1198902120583(119903120579)

[11990321198891205792+ 1199032sin2120579 (119889120601 minus 120596 (119903 120579) 119889119905)2]

(57)



119892119905120601= 119892120601119905= 1199032sin21205791198902120583(119903120579)120596 (119903 120579) (58)


119889120601

119889119905=119892120601119905

119892119905119905=

minus119892120601119905

119892120601120601

=minus11990321198902120583(119903120579)

120596 (119903)

minus11990321198902120583(119903120579)= 120596 (119903) (59)


120596 (119903) =2

5

radic119866119872

11990331198772Ω2

119904 (60)



119897120596 (119903) = 119899120596119871 (61)


1199033=41198992

1198972

119866119872

Ω2119904

(62)


3 Final Remarks



Competing Interests


Acknowledgments



References

















































































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics





119904119888119903


45[119908minus6


101198753

1199043+ (

119872

119872⊙

)

97

sdot 57

119908minus6119886minus107

10Vminus2071199088

12058347

331198751199043] s sminus1

(27)


120591119888119903sim 10411986845[119908minus6


101198752

1199043

+ (119872

119872⊙

)

97

57

119908minus6119886minus107

10Vminus2071199088

12058347

33]

minus1

yr

(28)




119871119888119903= V2

119908+119866119872

119903magsim 1032[10(

119872

119872⊙

)

2

sdot 119908minus6


10+ (

119872

119872⊙

)

267

sdot 97

119908minus6Vminus3671199088

119886minus187

10120583minus47

33] ergs sminus1

(29)




119871119909sim 1033119877minus1

6(119872

119872⊙

)

3

119872minus23

30(119875orb1 day

)

minus43

sdot 119908minus6

Vminus41199088

ergs sminus1

(30)



119879119886= minus120581119905

1205832

1199033magminus Ω

1199041199032

mag (31)


119904acc sim 10

minus6119868minus1

45[12058111990512058327

331198752

1199043(119872

119872⊙

)

157

sdot 67

119908minus6Vminus2471199088

119886minus127

10+ 12058387

331198751199043(119872

119872⊙

)

minus147

sdot 37

119908minus6V1671199088

11988687

10] s sminus1

(32)


120591119904acc

sim 10311986845[12058111990512058327

331198751199043(119872

119872⊙

)

157

67

119908minus6Vminus2471199088

119886minus127

10

+ 12058387

33(119872

119872⊙

)

minus147

37

119908minus6V1671199088

11988687

10]

minus1

yr

(33)



119871119909acc

sim 1031(119872

119872⊙

)

267

97

119908minus6Vminus3671199088

119886minus187

10120583minus47

33ergs sminus1

(34)



119870(119903) = (119866119872119903

3)12




V119891119891equiv (

2119866119872

119903)

12

= 16 times 109119903minus12

8(119872

119872⊙

)

12

cm sminus1

120588119891119891equiv

(V11989111989141205871199032)


817(119872

119872⊙

)

minus12

g cmminus3

(35)





then given by

119861 (119877119860)2

8120587= 120588119891119891(119877119860)

V119891119891(119877119860)2 (36)


radius reads

119877119860sim 108minus27

1712058347

30(119872

119872⊙

)

minus17

cm


3712058347

30(119872

119872⊙

)

minus17

119877minus27

6cm

(37)

where 12058330








(1198612

119905

8120587)

in+ 119875in = (

1198612

119905

8120587)

out+ 119875out (38)



(1198612

119905

8120587)

in+ 119875in + (120588V

2

119899)in

= (1198612

119905

8120587)

out+ 119875out + (120588V

2

119899)out

(39)







0and



Boundary

Outertransition

Transition

zonelayer

Magnetosphericflow

regionUnperturbed

disk flow

2h

rA

rsr0

120575



0to a






119904between 119903





8

9

10

11

12

13

14

15

16

log[B

(G)]


log[P(s)]

Spin-up line

Death lin

e





119904sim 01ndash10 s and 119875











119891by [60]

119861 (119905) =

119861119891



02119872⊙ and 119862 = 1 + radic1 minus 119909

2

0sim 2 with 1199092

0= (1198611198911198610)47 The



119860with a


119861119891= 132 times 10

8(

18

)

12

119872

119872⊙

14

119877minus54

6120601minus74 G (41)

where 18= 10




minus5[(

119872

119872⊙

)

minus37

119877127

6119868minus1

45]

times 11986127

12(11987511990411987137

37)2

119899 (120596119904) s yrminus1

(42)

where 11986112




120596119904equiv

Ω119904

Ω119870(119877119860)= 135 (

119872

119872⊙

)

minus27

119877157

611986167

12119875minus1

119904119871minus37

37 (43)


119899 (120596119904) = 14 times (

1 minus 120596119904120596119888

1 minus 120596119904

) (44)


Spin-up line

Death lin

e


log[B0] = 137

log[B0] = 127

log[B0] = 117

8

9

10

11

12

13

14

15

16lo

g[B

(G)]


log[P(s)]

(a)

Spin-up line

Death lin

e


8

9

10

11

12

13

14

15

16

log[B

(G)]


log[P(s)]

P0 = 100 sP0 = 10 sP0 = 1 s

(b)





0= 5 times 10






119904)


value 120596119888 For 120596






119875eq = 2120587120596minus1

119888Ω119870119903mag = 189119861

67

9ms (45)












⊙) is


⊙ndash001119872

⊙






18084-3658 [74] and PSR J 1748-2446 [81]




7

8

9

10

11

12

13lo

g[B

(G)]


log[P(s)]

(a)

7

8

9

10

11

12

13

log[B

(G)]


log[ΔM(M⊙)]

(b)

minus3

minus2

minus1

0

log[P

(s)]


log[ΔM(M⊙)]

(c)

1

2

3

4

5

P (m

s)

04 06 08 1002

ΔMM⊙

(d)


0= 5 times 10






119892120583] = 120578120583] + ℎ120583]

10038161003816100381610038161003816ℎ120583]10038161003816100381610038161003816≪ 1 (46)


where ℎ = 120578120583]ℎ120583] = ℎ

120572



120583] = 8120587119866119879120583] in powers of ℎ120583]


◻ℎ120583] = minus16120587119866119879120583] (47)







ℎ120583] = 4119866int

119879120583] (1199020 minus

10038161003816100381610038161003816119902119894minus 1199021015840

119894

10038161003816100381610038161003816 1199021015840

119894)

1003816100381610038161003816119902119894 minus 1199021015840

119894

1003816100381610038161003816

11988931199021015840

119894 (48)





Spin-up line

Death lin

e



log[P(s)]

7

8

9

10

11

12

13

14

15

16

log[B

(G)]

M16 120596c = 09M17 120596c = 09M18 120596c = 09




1198610= 5 times 10

12 G and period 1198750= 100 s



119894119895(1199020 119902119894) [82] we canwrite down


ℎ00= 4Φ

ℎ0119894= minus2119860

119894

(49)



Φ(1199020 119902119894) = minus

119866119872

119903

119860119894(1199020 119902119894) = 119866

119869119895119902119896

1199033120576119895119896

119894

(50)


119894of the NS


120597Φ

1205971199020

+1

2nabla119894119860119895120575119894119895= 0 (51)



119894

as [90]


120597

1205971199020

(1

2119860119895) 120575119894119895

R119894= nabla119895119860119896120576119895119896

119894

(52)







nabla119894119892119895120575119894119895= minus4120587119866120588

119892

nabla119895119892119896= minus

120597

1205971199020

(1

2R119894)

nabla119894R119895120575119894119895= 0

nabla119895R119896=120597119892119894

1205971199020

minus 4120587119866119895119892119894

(53)




0+ nabla119894119895119892119895120575119894119895= 0



119865GEM119894 = 119876119892119892119894 + 119876R119895R119896120576119895119896

119894 (54)



119871= minus

119876119892

119898

119871=119876RR

2119898

(55)


119871= |R|



120596119871=10038161003816100381610038161003816R10038161003816100381610038161003816=4

5

1198661198721198772

1199033Ω119904 (56)



1198891199042= 1198902](119903120579)

1198891199052minus 1198902120582(119903120579)

1198891199032

minus 1198902120583(119903120579)

[11990321198891205792+ 1199032sin2120579 (119889120601 minus 120596 (119903 120579) 119889119905)2]

(57)



119892119905120601= 119892120601119905= 1199032sin21205791198902120583(119903120579)120596 (119903 120579) (58)


119889120601

119889119905=119892120601119905

119892119905119905=

minus119892120601119905

119892120601120601

=minus11990321198902120583(119903120579)

120596 (119903)

minus11990321198902120583(119903120579)= 120596 (119903) (59)


120596 (119903) =2

5

radic119866119872

11990331198772Ω2

119904 (60)



119897120596 (119903) = 119899120596119871 (61)


1199033=41198992

1198972

119866119872

Ω2119904

(62)


3 Final Remarks



Competing Interests


Acknowledgments



References

















































































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics





119871119909acc

sim 1031(119872

119872⊙

)

267

97

119908minus6Vminus3671199088

119886minus187

10120583minus47

33ergs sminus1

(34)



119870(119903) = (119866119872119903

3)12




V119891119891equiv (

2119866119872

119903)

12

= 16 times 109119903minus12

8(119872

119872⊙

)

12

cm sminus1

120588119891119891equiv

(V11989111989141205871199032)


817(119872

119872⊙

)

minus12

g cmminus3

(35)





then given by

119861 (119877119860)2

8120587= 120588119891119891(119877119860)

V119891119891(119877119860)2 (36)


radius reads

119877119860sim 108minus27

1712058347

30(119872

119872⊙

)

minus17

cm


3712058347

30(119872

119872⊙

)

minus17

119877minus27

6cm

(37)

where 12058330








(1198612

119905

8120587)

in+ 119875in = (

1198612

119905

8120587)

out+ 119875out (38)



(1198612

119905

8120587)

in+ 119875in + (120588V

2

119899)in

= (1198612

119905

8120587)

out+ 119875out + (120588V

2

119899)out

(39)







0and



Boundary

Outertransition

Transition

zonelayer

Magnetosphericflow

regionUnperturbed

disk flow

2h

rA

rsr0

120575



0to a






119904between 119903





8

9

10

11

12

13

14

15

16

log[B

(G)]


log[P(s)]

Spin-up line

Death lin

e





119904sim 01ndash10 s and 119875











119891by [60]

119861 (119905) =

119861119891



02119872⊙ and 119862 = 1 + radic1 minus 119909

2

0sim 2 with 1199092

0= (1198611198911198610)47 The



119860with a


119861119891= 132 times 10

8(

18

)

12

119872

119872⊙

14

119877minus54

6120601minus74 G (41)

where 18= 10




minus5[(

119872

119872⊙

)

minus37

119877127

6119868minus1

45]

times 11986127

12(11987511990411987137

37)2

119899 (120596119904) s yrminus1

(42)

where 11986112




120596119904equiv

Ω119904

Ω119870(119877119860)= 135 (

119872

119872⊙

)

minus27

119877157

611986167

12119875minus1

119904119871minus37

37 (43)


119899 (120596119904) = 14 times (

1 minus 120596119904120596119888

1 minus 120596119904

) (44)


Spin-up line

Death lin

e


log[B0] = 137

log[B0] = 127

log[B0] = 117

8

9

10

11

12

13

14

15

16lo

g[B

(G)]


log[P(s)]

(a)

Spin-up line

Death lin

e


8

9

10

11

12

13

14

15

16

log[B

(G)]


log[P(s)]

P0 = 100 sP0 = 10 sP0 = 1 s

(b)





0= 5 times 10






119904)


value 120596119888 For 120596






119875eq = 2120587120596minus1

119888Ω119870119903mag = 189119861

67

9ms (45)












⊙) is


⊙ndash001119872

⊙






18084-3658 [74] and PSR J 1748-2446 [81]




7

8

9

10

11

12

13lo

g[B

(G)]


log[P(s)]

(a)

7

8

9

10

11

12

13

log[B

(G)]


log[ΔM(M⊙)]

(b)

minus3

minus2

minus1

0

log[P

(s)]


log[ΔM(M⊙)]

(c)

1

2

3

4

5

P (m

s)

04 06 08 1002

ΔMM⊙

(d)


0= 5 times 10






119892120583] = 120578120583] + ℎ120583]

10038161003816100381610038161003816ℎ120583]10038161003816100381610038161003816≪ 1 (46)


where ℎ = 120578120583]ℎ120583] = ℎ

120572



120583] = 8120587119866119879120583] in powers of ℎ120583]


◻ℎ120583] = minus16120587119866119879120583] (47)







ℎ120583] = 4119866int

119879120583] (1199020 minus

10038161003816100381610038161003816119902119894minus 1199021015840

119894

10038161003816100381610038161003816 1199021015840

119894)

1003816100381610038161003816119902119894 minus 1199021015840

119894

1003816100381610038161003816

11988931199021015840

119894 (48)





Spin-up line

Death lin

e



log[P(s)]

7

8

9

10

11

12

13

14

15

16

log[B

(G)]

M16 120596c = 09M17 120596c = 09M18 120596c = 09




1198610= 5 times 10

12 G and period 1198750= 100 s



119894119895(1199020 119902119894) [82] we canwrite down


ℎ00= 4Φ

ℎ0119894= minus2119860

119894

(49)



Φ(1199020 119902119894) = minus

119866119872

119903

119860119894(1199020 119902119894) = 119866

119869119895119902119896

1199033120576119895119896

119894

(50)


119894of the NS


120597Φ

1205971199020

+1

2nabla119894119860119895120575119894119895= 0 (51)



119894

as [90]


120597

1205971199020

(1

2119860119895) 120575119894119895

R119894= nabla119895119860119896120576119895119896

119894

(52)







nabla119894119892119895120575119894119895= minus4120587119866120588

119892

nabla119895119892119896= minus

120597

1205971199020

(1

2R119894)

nabla119894R119895120575119894119895= 0

nabla119895R119896=120597119892119894

1205971199020

minus 4120587119866119895119892119894

(53)




0+ nabla119894119895119892119895120575119894119895= 0



119865GEM119894 = 119876119892119892119894 + 119876R119895R119896120576119895119896

119894 (54)



119871= minus

119876119892

119898

119871=119876RR

2119898

(55)


119871= |R|



120596119871=10038161003816100381610038161003816R10038161003816100381610038161003816=4

5

1198661198721198772

1199033Ω119904 (56)



1198891199042= 1198902](119903120579)

1198891199052minus 1198902120582(119903120579)

1198891199032

minus 1198902120583(119903120579)

[11990321198891205792+ 1199032sin2120579 (119889120601 minus 120596 (119903 120579) 119889119905)2]

(57)



119892119905120601= 119892120601119905= 1199032sin21205791198902120583(119903120579)120596 (119903 120579) (58)


119889120601

119889119905=119892120601119905

119892119905119905=

minus119892120601119905

119892120601120601

=minus11990321198902120583(119903120579)

120596 (119903)

minus11990321198902120583(119903120579)= 120596 (119903) (59)


120596 (119903) =2

5

radic119866119872

11990331198772Ω2

119904 (60)



119897120596 (119903) = 119899120596119871 (61)


1199033=41198992

1198972

119866119872

Ω2119904

(62)


3 Final Remarks



Competing Interests


Acknowledgments



References

















































































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics






(1198612

119905

8120587)

in+ 119875in = (

1198612

119905

8120587)

out+ 119875out (38)



(1198612

119905

8120587)

in+ 119875in + (120588V

2

119899)in

= (1198612

119905

8120587)

out+ 119875out + (120588V

2

119899)out

(39)







0and



Boundary

Outertransition

Transition

zonelayer

Magnetosphericflow

regionUnperturbed

disk flow

2h

rA

rsr0

120575



0to a






119904between 119903





8

9

10

11

12

13

14

15

16

log[B

(G)]


log[P(s)]

Spin-up line

Death lin

e





119904sim 01ndash10 s and 119875











119891by [60]

119861 (119905) =

119861119891



02119872⊙ and 119862 = 1 + radic1 minus 119909

2

0sim 2 with 1199092

0= (1198611198911198610)47 The



119860with a


119861119891= 132 times 10

8(

18

)

12

119872

119872⊙

14

119877minus54

6120601minus74 G (41)

where 18= 10




minus5[(

119872

119872⊙

)

minus37

119877127

6119868minus1

45]

times 11986127

12(11987511990411987137

37)2

119899 (120596119904) s yrminus1

(42)

where 11986112




120596119904equiv

Ω119904

Ω119870(119877119860)= 135 (

119872

119872⊙

)

minus27

119877157

611986167

12119875minus1

119904119871minus37

37 (43)


119899 (120596119904) = 14 times (

1 minus 120596119904120596119888

1 minus 120596119904

) (44)


Spin-up line

Death lin

e


log[B0] = 137

log[B0] = 127

log[B0] = 117

8

9

10

11

12

13

14

15

16lo

g[B

(G)]


log[P(s)]

(a)

Spin-up line

Death lin

e


8

9

10

11

12

13

14

15

16

log[B

(G)]


log[P(s)]

P0 = 100 sP0 = 10 sP0 = 1 s

(b)





0= 5 times 10






119904)


value 120596119888 For 120596






119875eq = 2120587120596minus1

119888Ω119870119903mag = 189119861

67

9ms (45)












⊙) is


⊙ndash001119872

⊙






18084-3658 [74] and PSR J 1748-2446 [81]




7

8

9

10

11

12

13lo

g[B

(G)]


log[P(s)]

(a)

7

8

9

10

11

12

13

log[B

(G)]


log[ΔM(M⊙)]

(b)

minus3

minus2

minus1

0

log[P

(s)]


log[ΔM(M⊙)]

(c)

1

2

3

4

5

P (m

s)

04 06 08 1002

ΔMM⊙

(d)


0= 5 times 10






119892120583] = 120578120583] + ℎ120583]

10038161003816100381610038161003816ℎ120583]10038161003816100381610038161003816≪ 1 (46)


where ℎ = 120578120583]ℎ120583] = ℎ

120572



120583] = 8120587119866119879120583] in powers of ℎ120583]


◻ℎ120583] = minus16120587119866119879120583] (47)







ℎ120583] = 4119866int

119879120583] (1199020 minus

10038161003816100381610038161003816119902119894minus 1199021015840

119894

10038161003816100381610038161003816 1199021015840

119894)

1003816100381610038161003816119902119894 minus 1199021015840

119894

1003816100381610038161003816

11988931199021015840

119894 (48)





Spin-up line

Death lin

e



log[P(s)]

7

8

9

10

11

12

13

14

15

16

log[B

(G)]

M16 120596c = 09M17 120596c = 09M18 120596c = 09




1198610= 5 times 10

12 G and period 1198750= 100 s



119894119895(1199020 119902119894) [82] we canwrite down


ℎ00= 4Φ

ℎ0119894= minus2119860

119894

(49)



Φ(1199020 119902119894) = minus

119866119872

119903

119860119894(1199020 119902119894) = 119866

119869119895119902119896

1199033120576119895119896

119894

(50)


119894of the NS


120597Φ

1205971199020

+1

2nabla119894119860119895120575119894119895= 0 (51)



119894

as [90]


120597

1205971199020

(1

2119860119895) 120575119894119895

R119894= nabla119895119860119896120576119895119896

119894

(52)







nabla119894119892119895120575119894119895= minus4120587119866120588

119892

nabla119895119892119896= minus

120597

1205971199020

(1

2R119894)

nabla119894R119895120575119894119895= 0

nabla119895R119896=120597119892119894

1205971199020

minus 4120587119866119895119892119894

(53)




0+ nabla119894119895119892119895120575119894119895= 0



119865GEM119894 = 119876119892119892119894 + 119876R119895R119896120576119895119896

119894 (54)



119871= minus

119876119892

119898

119871=119876RR

2119898

(55)


119871= |R|



120596119871=10038161003816100381610038161003816R10038161003816100381610038161003816=4

5

1198661198721198772

1199033Ω119904 (56)



1198891199042= 1198902](119903120579)

1198891199052minus 1198902120582(119903120579)

1198891199032

minus 1198902120583(119903120579)

[11990321198891205792+ 1199032sin2120579 (119889120601 minus 120596 (119903 120579) 119889119905)2]

(57)



119892119905120601= 119892120601119905= 1199032sin21205791198902120583(119903120579)120596 (119903 120579) (58)


119889120601

119889119905=119892120601119905

119892119905119905=

minus119892120601119905

119892120601120601

=minus11990321198902120583(119903120579)

120596 (119903)

minus11990321198902120583(119903120579)= 120596 (119903) (59)


120596 (119903) =2

5

radic119866119872

11990331198772Ω2

119904 (60)



119897120596 (119903) = 119899120596119871 (61)


1199033=41198992

1198972

119866119872

Ω2119904

(62)


3 Final Remarks



Competing Interests


Acknowledgments



References

















































































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




8

9

10

11

12

13

14

15

16

log[B

(G)]


log[P(s)]

Spin-up line

Death lin

e





119904sim 01ndash10 s and 119875











119891by [60]

119861 (119905) =

119861119891



02119872⊙ and 119862 = 1 + radic1 minus 119909

2

0sim 2 with 1199092

0= (1198611198911198610)47 The



119860with a


119861119891= 132 times 10

8(

18

)

12

119872

119872⊙

14

119877minus54

6120601minus74 G (41)

where 18= 10




minus5[(

119872

119872⊙

)

minus37

119877127

6119868minus1

45]

times 11986127

12(11987511990411987137

37)2

119899 (120596119904) s yrminus1

(42)

where 11986112




120596119904equiv

Ω119904

Ω119870(119877119860)= 135 (

119872

119872⊙

)

minus27

119877157

611986167

12119875minus1

119904119871minus37

37 (43)


119899 (120596119904) = 14 times (

1 minus 120596119904120596119888

1 minus 120596119904

) (44)


Spin-up line

Death lin

e


log[B0] = 137

log[B0] = 127

log[B0] = 117

8

9

10

11

12

13

14

15

16lo

g[B

(G)]


log[P(s)]

(a)

Spin-up line

Death lin

e


8

9

10

11

12

13

14

15

16

log[B

(G)]


log[P(s)]

P0 = 100 sP0 = 10 sP0 = 1 s

(b)





0= 5 times 10






119904)


value 120596119888 For 120596






119875eq = 2120587120596minus1

119888Ω119870119903mag = 189119861

67

9ms (45)












⊙) is


⊙ndash001119872

⊙






18084-3658 [74] and PSR J 1748-2446 [81]




7

8

9

10

11

12

13lo

g[B

(G)]


log[P(s)]

(a)

7

8

9

10

11

12

13

log[B

(G)]


log[ΔM(M⊙)]

(b)

minus3

minus2

minus1

0

log[P

(s)]


log[ΔM(M⊙)]

(c)

1

2

3

4

5

P (m

s)

04 06 08 1002

ΔMM⊙

(d)


0= 5 times 10






119892120583] = 120578120583] + ℎ120583]

10038161003816100381610038161003816ℎ120583]10038161003816100381610038161003816≪ 1 (46)


where ℎ = 120578120583]ℎ120583] = ℎ

120572



120583] = 8120587119866119879120583] in powers of ℎ120583]


◻ℎ120583] = minus16120587119866119879120583] (47)







ℎ120583] = 4119866int

119879120583] (1199020 minus

10038161003816100381610038161003816119902119894minus 1199021015840

119894

10038161003816100381610038161003816 1199021015840

119894)

1003816100381610038161003816119902119894 minus 1199021015840

119894

1003816100381610038161003816

11988931199021015840

119894 (48)





Spin-up line

Death lin

e



log[P(s)]

7

8

9

10

11

12

13

14

15

16

log[B

(G)]

M16 120596c = 09M17 120596c = 09M18 120596c = 09




1198610= 5 times 10

12 G and period 1198750= 100 s



119894119895(1199020 119902119894) [82] we canwrite down


ℎ00= 4Φ

ℎ0119894= minus2119860

119894

(49)



Φ(1199020 119902119894) = minus

119866119872

119903

119860119894(1199020 119902119894) = 119866

119869119895119902119896

1199033120576119895119896

119894

(50)


119894of the NS


120597Φ

1205971199020

+1

2nabla119894119860119895120575119894119895= 0 (51)



119894

as [90]


120597

1205971199020

(1

2119860119895) 120575119894119895

R119894= nabla119895119860119896120576119895119896

119894

(52)







nabla119894119892119895120575119894119895= minus4120587119866120588

119892

nabla119895119892119896= minus

120597

1205971199020

(1

2R119894)

nabla119894R119895120575119894119895= 0

nabla119895R119896=120597119892119894

1205971199020

minus 4120587119866119895119892119894

(53)




0+ nabla119894119895119892119895120575119894119895= 0



119865GEM119894 = 119876119892119892119894 + 119876R119895R119896120576119895119896

119894 (54)



119871= minus

119876119892

119898

119871=119876RR

2119898

(55)


119871= |R|



120596119871=10038161003816100381610038161003816R10038161003816100381610038161003816=4

5

1198661198721198772

1199033Ω119904 (56)



1198891199042= 1198902](119903120579)

1198891199052minus 1198902120582(119903120579)

1198891199032

minus 1198902120583(119903120579)

[11990321198891205792+ 1199032sin2120579 (119889120601 minus 120596 (119903 120579) 119889119905)2]

(57)



119892119905120601= 119892120601119905= 1199032sin21205791198902120583(119903120579)120596 (119903 120579) (58)


119889120601

119889119905=119892120601119905

119892119905119905=

minus119892120601119905

119892120601120601

=minus11990321198902120583(119903120579)

120596 (119903)

minus11990321198902120583(119903120579)= 120596 (119903) (59)


120596 (119903) =2

5

radic119866119872

11990331198772Ω2

119904 (60)



119897120596 (119903) = 119899120596119871 (61)


1199033=41198992

1198972

119866119872

Ω2119904

(62)


3 Final Remarks



Competing Interests


Acknowledgments



References

















































































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




Spin-up line

Death lin

e


log[B0] = 137

log[B0] = 127

log[B0] = 117

8

9

10

11

12

13

14

15

16lo

g[B

(G)]


log[P(s)]

(a)

Spin-up line

Death lin

e


8

9

10

11

12

13

14

15

16

log[B

(G)]


log[P(s)]

P0 = 100 sP0 = 10 sP0 = 1 s

(b)





0= 5 times 10






119904)


value 120596119888 For 120596






119875eq = 2120587120596minus1

119888Ω119870119903mag = 189119861

67

9ms (45)












⊙) is


⊙ndash001119872

⊙






18084-3658 [74] and PSR J 1748-2446 [81]




7

8

9

10

11

12

13lo

g[B

(G)]


log[P(s)]

(a)

7

8

9

10

11

12

13

log[B

(G)]


log[ΔM(M⊙)]

(b)

minus3

minus2

minus1

0

log[P

(s)]


log[ΔM(M⊙)]

(c)

1

2

3

4

5

P (m

s)

04 06 08 1002

ΔMM⊙

(d)


0= 5 times 10






119892120583] = 120578120583] + ℎ120583]

10038161003816100381610038161003816ℎ120583]10038161003816100381610038161003816≪ 1 (46)


where ℎ = 120578120583]ℎ120583] = ℎ

120572



120583] = 8120587119866119879120583] in powers of ℎ120583]


◻ℎ120583] = minus16120587119866119879120583] (47)







ℎ120583] = 4119866int

119879120583] (1199020 minus

10038161003816100381610038161003816119902119894minus 1199021015840

119894

10038161003816100381610038161003816 1199021015840

119894)

1003816100381610038161003816119902119894 minus 1199021015840

119894

1003816100381610038161003816

11988931199021015840

119894 (48)





Spin-up line

Death lin

e



log[P(s)]

7

8

9

10

11

12

13

14

15

16

log[B

(G)]

M16 120596c = 09M17 120596c = 09M18 120596c = 09




1198610= 5 times 10

12 G and period 1198750= 100 s



119894119895(1199020 119902119894) [82] we canwrite down


ℎ00= 4Φ

ℎ0119894= minus2119860

119894

(49)



Φ(1199020 119902119894) = minus

119866119872

119903

119860119894(1199020 119902119894) = 119866

119869119895119902119896

1199033120576119895119896

119894

(50)


119894of the NS


120597Φ

1205971199020

+1

2nabla119894119860119895120575119894119895= 0 (51)



119894

as [90]


120597

1205971199020

(1

2119860119895) 120575119894119895

R119894= nabla119895119860119896120576119895119896

119894

(52)







nabla119894119892119895120575119894119895= minus4120587119866120588

119892

nabla119895119892119896= minus

120597

1205971199020

(1

2R119894)

nabla119894R119895120575119894119895= 0

nabla119895R119896=120597119892119894

1205971199020

minus 4120587119866119895119892119894

(53)




0+ nabla119894119895119892119895120575119894119895= 0



119865GEM119894 = 119876119892119892119894 + 119876R119895R119896120576119895119896

119894 (54)



119871= minus

119876119892

119898

119871=119876RR

2119898

(55)


119871= |R|



120596119871=10038161003816100381610038161003816R10038161003816100381610038161003816=4

5

1198661198721198772

1199033Ω119904 (56)



1198891199042= 1198902](119903120579)

1198891199052minus 1198902120582(119903120579)

1198891199032

minus 1198902120583(119903120579)

[11990321198891205792+ 1199032sin2120579 (119889120601 minus 120596 (119903 120579) 119889119905)2]

(57)



119892119905120601= 119892120601119905= 1199032sin21205791198902120583(119903120579)120596 (119903 120579) (58)


119889120601

119889119905=119892120601119905

119892119905119905=

minus119892120601119905

119892120601120601

=minus11990321198902120583(119903120579)

120596 (119903)

minus11990321198902120583(119903120579)= 120596 (119903) (59)


120596 (119903) =2

5

radic119866119872

11990331198772Ω2

119904 (60)



119897120596 (119903) = 119899120596119871 (61)


1199033=41198992

1198972

119866119872

Ω2119904

(62)


3 Final Remarks



Competing Interests


Acknowledgments



References

















































































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




7

8

9

10

11

12

13lo

g[B

(G)]


log[P(s)]

(a)

7

8

9

10

11

12

13

log[B

(G)]


log[ΔM(M⊙)]

(b)

minus3

minus2

minus1

0

log[P

(s)]


log[ΔM(M⊙)]

(c)

1

2

3

4

5

P (m

s)

04 06 08 1002

ΔMM⊙

(d)


0= 5 times 10






119892120583] = 120578120583] + ℎ120583]

10038161003816100381610038161003816ℎ120583]10038161003816100381610038161003816≪ 1 (46)


where ℎ = 120578120583]ℎ120583] = ℎ

120572



120583] = 8120587119866119879120583] in powers of ℎ120583]


◻ℎ120583] = minus16120587119866119879120583] (47)







ℎ120583] = 4119866int

119879120583] (1199020 minus

10038161003816100381610038161003816119902119894minus 1199021015840

119894

10038161003816100381610038161003816 1199021015840

119894)

1003816100381610038161003816119902119894 minus 1199021015840

119894

1003816100381610038161003816

11988931199021015840

119894 (48)





Spin-up line

Death lin

e



log[P(s)]

7

8

9

10

11

12

13

14

15

16

log[B

(G)]

M16 120596c = 09M17 120596c = 09M18 120596c = 09




1198610= 5 times 10

12 G and period 1198750= 100 s



119894119895(1199020 119902119894) [82] we canwrite down


ℎ00= 4Φ

ℎ0119894= minus2119860

119894

(49)



Φ(1199020 119902119894) = minus

119866119872

119903

119860119894(1199020 119902119894) = 119866

119869119895119902119896

1199033120576119895119896

119894

(50)


119894of the NS


120597Φ

1205971199020

+1

2nabla119894119860119895120575119894119895= 0 (51)



119894

as [90]


120597

1205971199020

(1

2119860119895) 120575119894119895

R119894= nabla119895119860119896120576119895119896

119894

(52)







nabla119894119892119895120575119894119895= minus4120587119866120588

119892

nabla119895119892119896= minus

120597

1205971199020

(1

2R119894)

nabla119894R119895120575119894119895= 0

nabla119895R119896=120597119892119894

1205971199020

minus 4120587119866119895119892119894

(53)




0+ nabla119894119895119892119895120575119894119895= 0



119865GEM119894 = 119876119892119892119894 + 119876R119895R119896120576119895119896

119894 (54)



119871= minus

119876119892

119898

119871=119876RR

2119898

(55)


119871= |R|



120596119871=10038161003816100381610038161003816R10038161003816100381610038161003816=4

5

1198661198721198772

1199033Ω119904 (56)



1198891199042= 1198902](119903120579)

1198891199052minus 1198902120582(119903120579)

1198891199032

minus 1198902120583(119903120579)

[11990321198891205792+ 1199032sin2120579 (119889120601 minus 120596 (119903 120579) 119889119905)2]

(57)



119892119905120601= 119892120601119905= 1199032sin21205791198902120583(119903120579)120596 (119903 120579) (58)


119889120601

119889119905=119892120601119905

119892119905119905=

minus119892120601119905

119892120601120601

=minus11990321198902120583(119903120579)

120596 (119903)

minus11990321198902120583(119903120579)= 120596 (119903) (59)


120596 (119903) =2

5

radic119866119872

11990331198772Ω2

119904 (60)



119897120596 (119903) = 119899120596119871 (61)


1199033=41198992

1198972

119866119872

Ω2119904

(62)


3 Final Remarks



Competing Interests


Acknowledgments



References

















































































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




Spin-up line

Death lin

e



log[P(s)]

7

8

9

10

11

12

13

14

15

16

log[B

(G)]

M16 120596c = 09M17 120596c = 09M18 120596c = 09




1198610= 5 times 10

12 G and period 1198750= 100 s



119894119895(1199020 119902119894) [82] we canwrite down


ℎ00= 4Φ

ℎ0119894= minus2119860

119894

(49)



Φ(1199020 119902119894) = minus

119866119872

119903

119860119894(1199020 119902119894) = 119866

119869119895119902119896

1199033120576119895119896

119894

(50)


119894of the NS


120597Φ

1205971199020

+1

2nabla119894119860119895120575119894119895= 0 (51)



119894

as [90]


120597

1205971199020

(1

2119860119895) 120575119894119895

R119894= nabla119895119860119896120576119895119896

119894

(52)







nabla119894119892119895120575119894119895= minus4120587119866120588

119892

nabla119895119892119896= minus

120597

1205971199020

(1

2R119894)

nabla119894R119895120575119894119895= 0

nabla119895R119896=120597119892119894

1205971199020

minus 4120587119866119895119892119894

(53)




0+ nabla119894119895119892119895120575119894119895= 0



119865GEM119894 = 119876119892119892119894 + 119876R119895R119896120576119895119896

119894 (54)



119871= minus

119876119892

119898

119871=119876RR

2119898

(55)


119871= |R|



120596119871=10038161003816100381610038161003816R10038161003816100381610038161003816=4

5

1198661198721198772

1199033Ω119904 (56)



1198891199042= 1198902](119903120579)

1198891199052minus 1198902120582(119903120579)

1198891199032

minus 1198902120583(119903120579)

[11990321198891205792+ 1199032sin2120579 (119889120601 minus 120596 (119903 120579) 119889119905)2]

(57)



119892119905120601= 119892120601119905= 1199032sin21205791198902120583(119903120579)120596 (119903 120579) (58)


119889120601

119889119905=119892120601119905

119892119905119905=

minus119892120601119905

119892120601120601

=minus11990321198902120583(119903120579)

120596 (119903)

minus11990321198902120583(119903120579)= 120596 (119903) (59)


120596 (119903) =2

5

radic119866119872

11990331198772Ω2

119904 (60)



119897120596 (119903) = 119899120596119871 (61)


1199033=41198992

1198972

119866119872

Ω2119904

(62)


3 Final Remarks



Competing Interests


Acknowledgments



References

















































































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics





119892119905120601= 119892120601119905= 1199032sin21205791198902120583(119903120579)120596 (119903 120579) (58)


119889120601

119889119905=119892120601119905

119892119905119905=

minus119892120601119905

119892120601120601

=minus11990321198902120583(119903120579)

120596 (119903)

minus11990321198902120583(119903120579)= 120596 (119903) (59)


120596 (119903) =2

5

radic119866119872

11990331198772Ω2

119904 (60)



119897120596 (119903) = 119899120596119871 (61)


1199033=41198992

1198972

119866119872

Ω2119904

(62)


3 Final Remarks



Competing Interests


Acknowledgments



References

















































































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




References

















































































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics












































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics








































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics








FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics



Documents

Review Article Physical Environment of Accreting Neutron …downloads.hindawi.com/journals/aa/2016/3424565.pdf · Review Article Physical Environment of Accreting Neutron Stars