Hindawi Publishing Corporation Advances in Volume 2016, Article ID 3424565, 15 pages http://dx.doi.org/10.1155/2016/3424565

Review Article Physical Environment of Accreting Neutron

J. Wang

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

Correspondence should be addressed to J. Wang; [email protected]

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

Academic Editor: Elmetwally Elabbasy

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

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

1. Introduction to Accretion-Powered released by matter of mass 𝑚 falling into its deep gravitational Neutron potential well amounts to 𝐺𝑀𝑚 Δ𝐸 = , Since the discovery for the first extrasolar X-ray source, acc 𝑅 (1) Scorpius X-1 in 1962 [1], a new field of astronomy—accreting 𝐺 compact objects in the —has arisen, which offers where is the gravitational constant. This is up to ∼ 20 −1 unique insight into the physics at extreme conditions. So 10 ergs g of accreted matter, that is, about a tenth of its ∼0.1𝑐2 𝑐 far,moreandmorebrightgalacticX-raysourceshavebeen rest-mass energy ( , with the speed of light ), which discovered, which show a clear concentration towards both makes accretion as an ideal source of power. Since each unit and galactic plane. They are highly variable of accreted mass releases an amount of gravitational potential energy 𝐺𝑀/𝑅 when it reaches the NS surface, the on timescales of seconds or less and display typical source 34 38 −1 generated by the accretion process is given by luminosity of 10 –10 ergs s , in the energy band of 1– 𝐺𝑀𝑚̇ 10 keV. It was suggested that the strong galactic X-ray sources 𝐿 = , acc (2) are NSs accreting material from their companions in close 𝑅 binary systems [2]. The detection of coherent and regular where 𝑚̇ is the accretion rate. Accordingly, to generate 37 −1 pulsations from the accreting X-ray sources Centaurus X-3 atypicalluminosityofabout10 ergss ,itrequiresan 17 −1 −9 −1 [3, 4] and Hercules X-1 [5] by Uhuru in 1971 provided the first accretion rate of ∼10 gs ∼10 𝑀⊙ yr . evidence that the compact objects in many of these sources During the accretion process, radiation passes through are indeed accreting neutron stars. Since then, accretion onto accretion flow and influences its dynamics in the case of rotatingNSshavebeennicelyconfirmedasthestandard a sufficiently large luminosity. When the outward pressure picture of strongest galactic X-ray sources, and the X-ray of radiation exceeds the inward gravitational attraction, pulsation periods are the spin periods of X-ray . the infalling flow will be halted, which implies a critical Accretion remains the only viable source of power to the luminosity, [9]: 𝑀 binary X-ray pulsars as a whole [6–8]. For a NS of mass 𝐿 ≈ 1.3 × 1038 −1, 6 Edd ergs s (3) 𝑀∼𝑀⊙,andofradius𝑅=10 cm, the quantity of energy 𝑀⊙ 2 Advances in Astronomy which, for spherically symmetric accretion, corresponds to a driven by radiation [25, 26]. The material leaves surface limit on a steady accretion rate, the Eddington accretion rate: uniformly in all directions. Some of material will be captured gravitationally by the NS. This mechanism is called stellar 𝑀̇ ≈1018𝑅 −1 ≈ 1.5 × 10−8𝑀 𝑅 −1. Edd 6 gs ⊙ 6 yr (4) wind accretion, and the corresponding systems are wind-fed Lower and higher accretion rates than this critical value are systems [7]. called the “sub-” and “super-”Eddington accretion, respec- In the course of its evolution, the optical companion star tively. mayincreaseinradiustoapointwherethegravitationalpull Phenomenologically, over 95% of the NS X-ray binaries of the companion can remove the outer layers of its envelope, fall into two distinct categories, that is, low-mass X-ray binary which is the RLOF phase [9, 27]. In addition to the swelling of (LMXB) and high-mass X-ray binary (HMXB) [10, 11]. Wind- the companion, this situation can also come about, due to, for fed system, in which the companion of NS loses mass in example, the decrease of binary separation as a consequence the form of a , requires a massive companion of magnetically coupled wind mass loss or gravitational radi- ation. Some of material begins to flow over to the with mass ≥10𝑀⊙ to drive the intense stellar wind and thus appears to be high-mass system, such as Centaurus X-3. NSs of NS through the inner Lagrangian point, without losing in HMXBs display relatively hard X-ray spectra with a peak energy. Because of the large , the matter energy larger than 15 keV [12] and tend to manifest as regular enters an at some distance from the NS. Subsequent 3 X-ray pulsars, with spin periods of 1–10 sandmagneticfields batches of outflowing matter from companion have nearly the 11 13 same initial conditions and hence fall on the same orbit. As a of 10 –10 G [13]. The companions in HMXBs are bright result, a ring of increasing density is formed by the gas near and luminous early-type Be or OB supergiant stars, which the NS. Because of the mutual collisions of individual density have larger than 10 solar masses and are short lived or cells, the angular momentum in the ring is and belong to the youngest in the galaxy, 5 7 redistributed. The ring spreads out into a disk in which the with ages of ∼10 –10 yrs [14–16]. They are distributed close matter rotates differentially. Gradually, the motion turns out to the galactic plane. On the other hand, X-ray binaries, into steady-state disk accretion on a time scale determined by with companion mass, ≤𝑀⊙,arecategorizedaslow-mass the . Some optical observations suggested that mass X-ray binaries (LMXBs), in which mass transfer takes place transfer by ROLF is taking place or perhaps mixed with that by Roche-lone overflow. This transfer is driven either by by stellar wind accretion in some systems. losing angular momentum due to gravitational radiation (for systems of very small masses and orbital separations) and 2.2. Accretion Regimes. Because of the different angular magnetic braking (for systems of orbital periods 𝑃 ≤ orb momentum carried by the accreted material and the motion 2 days)orbytheevolutionofthecompanionstar(for oftheNSrelativetothesoundspeed𝑐𝑠 in the medium, for systems of 𝑃 ≥2days).MostNSsinLMXBsareX- orb accretion onto a NS, several modes are possible [28]. ray busters [17], with relatively soft X-ray spectra of <10 keV in exponential fittings [11]. In LMXBs, the NSs possess 8 9 2.2.1. Spherically Symmetric Accretion [29]. The NS hardly relatively weak surface magnetic fields of 10 - Gandshort moves relative to the medium in its vicinity; that is, V∞ ≪𝑐𝑠. spinperiodsofafewmilliseconds[18],whilethecompanions The gas, of uniform density and pressure, is at rest and does of NSs are late-type main-sequence stars, white dwarfs, or notpossesssignificantangularmomentumatinfinityand stars with F-G type spectra [19, 20]. In space the falls freely, with spherically symmetrical and steady motion, LMXBs are concentrated towards the galactic center and towards the NS. have a fairly wide distribution around the galactic plane, which characterize a relatively old population, with ages of 2.2.2. Cylindrical Accretion [30]. Although the angular (0.5 1.5) × 1010 – yrs. A few of strong galactic X-ray sources, momentum is small, the NS velocity is comparable with, or for example, Hercules X-1, are assigned to be intermediate- larger than, the sound speed in the accreted matter; that is, mass X-ray binaries [21, 22], in which the companions of NSs V∞ ≳𝑐𝑠. In this case, a conical shock form will present as the 𝑀 have masses of 1-2 ⊙. Intermediate-mass systems are rare, result of pressure effects. since mass transfer via Roche lobe is unstable and would lead ∼ 3 5 to a very quick ( 10 –10 yrs) evolution of the system, while, 2.2.3. Disk Accretion [6, 31]. In a , if the material in the case of stellar wind accretion, the mass accretion rate supplied by the companion of the NS has a large angular is very low and the system would be very dim and hardly momentum due to the orbital motion, we must take both the detectable [23]. gravitational force and centrifugal forces into consideration. As a result, an ultimately forms around a NS in 2. Physical Processes in Accreting the binary plane. Neutron Stars In a number of astrophysical cases [32], the motion of accreted material in the vicinity of a compact object presents 2.1. Mass Transfer in Binary Systems. The mass transfer from two-stream accretion, that is, a quasi-spherically symmetric the companion to NS is either due to a stellar wind or to Roche flux of matter in addition to the accretion disk [33]. lobe overflow (RLOF), as described above [7, 24]. A massive companion star, at some evolutionary phase, 2.3. Accretion from a Stellar Wind. Mass loss in the case of ejects much of its mass in the form of intense stellar wind OB supergiant stars is driven by [34]. The Advances in Astronomy 3

2 wind material is accelerated outwards from stellar surface to a 𝑀/̇ V(𝑟 )4𝜋𝑟 and V(𝑟 )=√2𝐺𝑀/𝑟 ,respectively. V V = mag mag mag mag final velocity ∞,whichisrelatedtotheescapevelocity esc 𝑀̇ 𝑟 𝑎 √2𝐺𝑀 /𝑅 𝑀 The accretion rate depends on acc and , according to [9] OB OB, for an OB companion with mass OB and 𝑅 V ≈3V 2 radius OB.Inmostcases, ∞ esc. In these circumstances, 𝑀̇ 𝑟 𝑀 2 ∼ acc ∼10−5 ( ) V−4 𝑎−2. the stellar wind is intense, characterized by a mass loss rate ̇ 2 𝑤,8 10 (8) ̇ −6 −5 −1 𝑀𝑤 (4𝑎 ) 𝑀⊙ 𝑀𝑤 ∼10 –10 𝑀⊙ yr ,andhighlysupersonic,withawind −1 velocity of V𝑤 ∼ 1000–2000 km s , which is much larger than V ∼ 200 −1 2𝑅 Then the magnetospheric radius is given by using (6) which the orbital velocity orb km s at OB. yields

2.3.1. Characteristic Radii. Inthewindaccretiontheory,three 𝑀 −5/7 𝑟 ∼1010 ( ) 𝑀̇ −2/7V8/7𝑎4/7𝜇4/7 . characteristic radii [7, 8, 35] can be defined as follows. mag 𝑤,−6 𝑤,8 10 33 cm (9) 𝑀⊙ Accretion Radius. Onlypartofstellarwindwithinacertain 𝑟 radius will be captured by the gravitational field of the NS, Corotation Radius. A corotation radius cor can be defined Ω =2𝜋/𝑃 whereas the wind material outside that radius will escape. when the spin angular velocity ( 𝑠 𝑠)ofNSisequal 𝑟 Ω = √𝐺𝑀/𝑟3 This radius, called the accretion radius acc [28], can be to the Keplerian angular velocity ( 𝑘 )ofmatter defined by noting that material will only be accreted if it being accreted: has a kinetic energy lower than the potential energy in the 1/3 gravitational potential well of NS; that is, 10 𝑀 2/3 𝑟 ∼10 ( ) 𝑃 , (10) cor 𝑀 𝑠,3 cm 2𝐺𝑀 𝑀 ⊙ 𝑟 = ∼1010 V−2 , acc V2 𝑀 𝑤,8 cm (5) 𝑤 ⊙ where 𝑃𝑠,3 isthespinperiodoftheneutronstar𝑃𝑠 in units of 3 −1 10 s. where V𝑤,8 isawindvelocityinunitsof1000kms (V𝑤,8 = V /(108 −1 𝑟 𝑤 cm s )). We consider now two cases: mag smaller than 𝑟 and 𝑟 larger than 𝑟 . 2.3.2. Accretion onto a Magnetized NS. The stellar wind flows acc mag acc approximately spherically symmetric outside the accretion 𝑟 <𝑟 Magnetospheric Radius.(i) mag acc: when accreting, the radius. When approaching the NS, according to the relative electromagnetic field of NS will hinder the from dimensions of three characteristic radii, different accretion falling all the way to its surface. The inflow will come to regimes can be identified. stop at certain distance, at which the magnetic pressure of NS can balance the ram pressure of accreting flow. Here (I) Magnetic Inhibition: Propeller. In a system in which the 𝑟 magnetospheric radius is larger than the accretion radius the forms at a magnetospheric radius mag 𝑟 >𝑟 [36, 37], which is defined by ( mag acc),thestellarwind,flowingaroundtheNS,may not experience a significant gravitational field and interacts 2 directly with the magnetosphere. In this case, very little 𝐵(𝑟 ) 2 mag =𝜌(𝑟 ) V (𝑟 ) , (6) material penetrates the magnetospheric radius and will be mag mag 8𝜋 accreted by NS [24]. Most of wind material is ejected by 𝐵(𝑟 )∼𝜇/𝑟3 𝑟 the rotation energy of the NS. This is called the propeller where mag mag is the magnetic field strength at mag mechanism. The NS in this situation behaves like a propeller 𝜌(𝑟 ) V(𝑟 ) and mag and mag arewinddensityandwindvelocity and spins down due to dissipation of rotational energy 𝑟 𝜇 at mag,respectively. is the magnetic moment of the NS. (ii) resulting from the interaction between the magnetosphere 𝑟 >𝑟 mag acc: here the inflows cannot experience a significant and stellar wind. The spin-down torque is expressed as [39, gravitational field. Assuming a nonmagnetized spherically 40] 𝜌(𝑟 ) symmetricwind[36],thewinddensityneartheNS mag 𝜌(𝑟 )≈𝑀̇ /4𝜋𝑎2V 𝑎 𝜅 𝜇2 given by mag 𝑤 𝑤 [37, 38], where is the 𝑇 =− 𝑡 , 𝑎≫𝑟 𝑚,pro 3 (11) orbital separation between two components and mag, 𝑟 is assumed. Therefore, the magnetospheric radius in this case mag 𝜌(𝑟 )V2 =𝐵(𝑟 )2/8𝜋 is given by setting mag 𝑤 mag ,whichyields where 𝜅𝑡 ≤1is a dimensionless parameter of order unity [37]. The equation governing the spin evolution can be written as 𝑟 ∼1010𝑀̇ −1/6V−1/6𝑎1/3𝜇1/3 , mag 𝑤,−6 𝑤,8 10 33 cm (7) [7, 41] ̇ ̇ −6 33 3 𝑑 1 where 𝑀𝑤,−6 = 𝑀𝑤/(10 𝑀⊙/yr), 𝜇33 =𝜇/(10 Gcm ), 2𝜋𝐼 =𝑇, 12 𝑑𝑡 𝑃 (12) and 𝑎10 isrelatedtotheorbitalseparationas𝑎∼10 𝑎10 cm 𝑠 12 2/3 1/3 ∼10𝑃 𝑀 cm. 𝑃𝑏,10 is the orbital period 𝑃𝑏 in units 𝑏,10 30 where 𝐼 is the moment of inertia of NS and 𝑇 is the total of 10 days, and 𝑀30 is the companion mass in units of 30 𝑟 <𝑟 torque imposed on NS. Therefore, the spin-down rate will be solar masses. When mag acc,thegravitationalfieldofNS dominates the falling of wind matter. The wind density and ̇ −6 −1 2 −1 ̇ 1/2 1/2 −1 𝑟 𝜌(𝑟 )= 𝑃𝑠,𝑚 ∼10 𝜅𝑡𝜇33𝐼45 𝑃𝑠,3𝑎10 𝑀𝑤,−6V𝑤,8 ss , (13) wind velocity at mag can therefore be taken as mag 4 Advances in Astronomy

45 2 where 𝐼45 is the moment of inertia in units of 10 gcm . to the propeller mechanism and imposes a spin-down torque Accordingly, if the NS can spin down to a period of 1000 s on the NS, which is expressed by [32] under this torque, the spin-down timescale is given by 𝑀̇ V2 𝑇 =− 𝑤 . 𝜏 =𝑃/𝑃̇ 𝑐,pro (20) 𝑚,pro 𝑠 𝑠,pur Ω𝑠 (14) 3 −1 −1 −1 ̇ −1/2 −1/2 ∼10𝜅𝑡 𝜇33 𝐼45𝑃𝑠,3 𝑎10𝑀𝑤,−6V𝑤,8 yr. In addition, due to the high speed of stellar wind, the accumulation rate of accreted matter is higher than the In this regime, the spin period 𝑃𝑠 must satisfy ejection rate. Consequently, more and more wind clumps are deposited outside the magnetosphere [42]. This regime 𝑀 −2 is in connection with the spin-down evolution of young X- 𝑃𝑠 > 11.2 V𝑤,8 s. (15) 𝑀⊙ ray pulsars [24]. The minimum X-ray luminosity, at which accretion possibly occurs, is given by In addition, the mechanism of magnetic inhibition of accre- tion only occurs before the centrifugal barrier acts; that is, the 𝐿𝑥,𝑐 (min) 𝑟 >𝑟 condition cor mag must be satisfied, which gives 𝑀 −2/3 𝑃 −7/3 (21) ∼1043𝑅−1 ( ) 𝜇2 ( 𝑠 ) −1. 3 𝑀 −3 6 33 ergs s 𝑀⊙ 1 s 𝑃𝑠 > 6.4 × 10 V𝑤,8 s. (16) 𝑀⊙ An X-ray , with luminosity below this value, turns off 𝑟 =𝑟 By imposing mag acc, we get the minimum X-ray lumi- as a result of the centrifugal mechanism. Correspondingly, a nosity, threshold in wind parameters is given by a relation between spin period and orbital period [35], 𝑀 −3 𝐿 ( ) ∼1033 ( ) 𝑅−1𝜇2 V7 −1, 𝑥,𝑚 min 6 33 𝑤,8 ergs s (17) 𝑃𝑠,𝑐 (min) 𝑀⊙ −11/7 4/7 𝑀 𝑃 −3/7 (22) and the corresponding maximum orbital period, 2/7 6/7 orb ̇ 12/7 ∼( ) 𝑀30 𝜇33 ( ) 𝑀𝑤,−6V𝑤,8 s, 𝑀⊙ 1 day 𝑃 ,𝑚 (max) orb 𝑟 =𝑟 whichisobtainedbyusingthecondition mag cor,Kepler’s 9/2 𝑀 3/4 (18) third law, and a free-fall approximation for the wind material ∼103 ( ) 𝑀−1/2 𝜇−3/2𝑀̇ V−33/4 . OB,30 33 𝑤,−6 𝑤,8 days 𝑟 𝑀⊙ within acc. In the vicinity of the NS, the deposited wind matter is For very fast rotating X-ray pulsars (𝑃𝑠 ≪1s), the magnetic disordered because of the supersonic rotation, which causes pressure exerted by pulsar on wind material provides an addi- turbulent motion around the magnetosphere. The velocity 𝑟 tional mechanism for inhibiting accretion through acc [24]. of the turbulent wind at the magnetospheric boundary is 𝑟 The condition that magnetic pressure at acc counterbalances close to the sound speed [32]. Consequently, the accumulated the wind ram pressure gives the minimum X-ray luminosity matter can rotate either prograde or retrograde [43]. When 𝐿 ( ) 𝑟 𝑥,𝑟 min possible for accretion through acc to take place, flowing prograde, an accretion torque will transfer some angular momentum onto the NS and spin it up. The spin-up 𝑃 −4 𝑀 torque is 𝐿 ( ) ∼1043𝜇2 ( 𝑠 ) 𝑅−1V−1 −1. 𝑥,𝑟 min 33 0.1 𝑀 6 𝑤,8 ergs s (19) s ⊙ 𝑇 = 𝑀Ω̇ 𝑟2 . 𝑐,acc,𝑝 𝑠 mag (23) 𝑟 Once the mass capture rate at acc is higher than that of (19), the flow will continue to be accreted by the NS until Therefore, the total torque is written as reaching the centrifugal barrier. This scenario is the magnetic 𝑀̇ V2 inhibition of accretion like that of a radio pulsar [35]. 𝑇 =𝑇 +𝑇 = 𝑀Ω̇ 𝑟2 − 𝑤 . 𝑐,𝑝 𝑐,pro 𝑐,acc,𝑝 𝑠 mag (24) Ω𝑠 (II) Centrifugal Inhibition: Quasi-Spherical Capture. If the magnetospheric radius is smaller than the accretion radius, The NS then behaves as a prograde propeller. the wind material penetrates through the accretion radius Ifthewindflowsretrogradely,itimposesaninverted 𝑟 and halts at mag. The captured material cannot penetrate any torque and spins down the NS, which is further because of a super-Keplerian drag exerted by the mag- 𝑟 𝑇 =−𝑇 =−𝑀Ω̇ 𝑟2 . netic field of NS. The inflow in the region between acc and 𝑐,acc,𝑟 𝑐,acc,𝑝 𝑠 mag (25) 𝑟 mag falls approximately in a spherical configuration. The NS behaves like a supersonic rotator, that is, the linear velocity of Accordingly, the total torque imposed on the NS in this magnetosphere is much higher than the sound speed in wind scenario is clumps, which strongly shocks the flow at magnetospheric ̇ 2 𝑀V 2 boundaryandejectssomeofmaterialbeyondtheaccretion 𝑇 =𝑇 +𝑇 =− 𝑤 − 𝑀Ω̇ 𝑟 . (26) 𝑐,𝑟 𝑐,pro 𝑐,acc,𝑟 Ω 𝑠 mag radius, via propeller mechanism. This scenario corresponds 𝑠 Advances in Astronomy 5

The NS is then a retrograde propeller and can spin down toa atmosphere around the magnetosphere. The time-averaged 𝑟 very long spin period [43]. Substituting (26) into (12), we can X-ray luminosity is governed by the wind parameters at acc, obtain the spin evolutionary law in the regime of retrograde 𝑀 3 𝑃 −4/3 propeller: 𝐿 ∼1033𝑅−1 ( ) 𝑀−2/3 ( orb ) 𝑥 6 𝑀 30 1 9/7 ⊙ day (30) ̇ −7 −1 ̇ −2 −2 3 𝑀 𝑃𝑠,𝑐,𝑟 ∼10 𝐼45 [𝑀𝑤,−6V𝑤,8𝑎10 𝑃𝑠,3 +( ) ̇ −4 −1 𝑀⊙ ⋅ 𝑀𝑤,−6V𝑤,8 ergs s , (27) 5/7 assuming all gravitational potential energy is converted into ⋅ 𝑀̇ 𝑎−10/7V−20/7𝜇4/7𝑃 ] −1. 𝑤,−6 10 𝑤,8 33 𝑠,3 ss X-ray luminosity. When a gradient of density or velocity is present, it TheNScanspindownto1000sduringatimescaleof becomes possible to accrete angular momentum and the flows become unstable, instead of a steady state [47]. The 𝜏 ∼104𝐼 [𝑀̇ V−2 𝑎−2𝑃2 shock cone oscillates from one side of the accretor to the 𝑐,𝑟 45 𝑤,−6 𝑤,8 10 𝑠,3 other side, allowing the appearance of transient accretion disks. This instability, generally known as flip-flop oscillation −1 (28) 𝑀 9/7 [48], produces fluctuations in the accretion rate and gives +( ) 𝑀̇ 5/7 𝑎−10/7V−20/7𝜇4/7] . 𝑀 𝑤,−6 10 𝑤,8 33 yr rise to stochastic accretion of positive and negative angular ⊙ momentum, leading to the suggestion that it is the source of Therefore, the retrograde propeller can be an alternative thevariationsseeninthepulseevolutionofsomesupergiant mechanism for NSs with very long spin periods, such as X-ray binaries [49, 50]. If the accretion flow is stable, the supergiant fast X-ray transients [43]. amount of angular momentum transferred to NS is negligible. If the stellar wind is not strong enough, the accreting The more unstable the accretion flow, the higher the transfer matter is approximately in radial free fall as it approaches rate of angular momentum [48]. A secular evolution can the magnetospheric boundary. The captured material cannot be described by sudden jumps between states with counter- accumulate near the magnetosphere due to the supersonic rotating quasi-Keplerian atmosphere around the magneto- rotation of NS. Consequently, the accretion torque contains sphere, which imposes an inverted accretion torque on NS onlythespin-downtorqueimposedbythepropellermecha- expressed as (25). In addition, the reversal rotation between nism,andthedetailsarediscussedbyUrpinetal.[44,45]. the adiabatic atmosphere and the magnetosphere may also In the retrograde propeller phase, both interaction with cause a flip-flop behavior of some clumps in the atmosphere, magnetic fields by the propeller mechanism and inverted ejecting a part of material and leading to the loss of thermal accretion contribute to the luminosity, which results in energy, which imposes a torque, with the same form as (11) [32, 37]. Accordingly, the total spin-down torque in the 𝐺𝑀𝑀̇ 𝑀 2 regime of 𝑟 >𝑟 and 𝑟 >𝑟 is 𝐿 = 𝑀̇ V2 + ∼1032 [10 ( ) acc mag cor mag 𝑐,𝑟 𝑤 𝑟 𝑀 mag ⊙ 2 𝜇 ̇ 2 𝑇𝑎 =−𝜅𝑡 − 𝑀Ω𝑠𝑟 . (31) 𝑀 26/7 𝑟3 mag ̇ −2 −2 mag ⋅ 𝑀𝑤,−6V𝑤,8𝑎10 +( ) (29) 𝑀⊙ Substituting it into (12), we obtain the spin evolution [43]: 9/7 −36/7 −18/7 −4/7 −1 ⋅ 𝑀̇ V 𝑎 𝜇 ] ergs s . 𝑀 15/7 𝑤,−6 𝑤,8 10 33 𝑃̇ ∼10−6𝐼−1 [𝜅 𝜇2/7𝑃2 ( ) 𝑠,acc 45 𝑡 33 𝑠,3 𝑀⊙

In addition, energy is also released through the shock formed −14/7 𝑟 6/7 𝑀 near mag, and the corresponding luminosity is discussed by ̇ −24/7 −12/7 8/7 ⋅ 𝑀𝑤,−6V𝑤,8 𝑎10 +𝜇33 𝑃𝑠,3 ( ) (32) Bozzo et al. [38]. 𝑀⊙ 𝑟 >𝑟 (III) Direct Wind Accretion: Accretor.When acc mag and 𝑟 >𝑟 ⋅ 𝑀̇ 3/7 V16/7𝑎8/7] −1. cor mag, the captured material flows from the accretion 𝑤,−6 𝑤,8 10 ss radius and falls down directly toward the magnetosphere, where it is stopped by a collisionless shock. Because of a The spin-down timescale reads sub-Keplerian rotation at the magnetospheric boundary, the rotating NS cannot eject the wind material in this case, and 𝜏 𝑠,acc the inflows experience a significant gravitational field and accumulate near the magnetosphere. Some of them penetrate 𝑀 15/7 ∼103𝐼 [𝜅 𝜇2/7𝑃 ( ) 𝑀̇ 6/7 V−24/7𝑎−12/7 the NS magnetosphere, presumably in a sporadic fashion, by 45 𝑡 33 𝑠,3 𝑤,−6 𝑤,8 10 𝑀⊙ (33) means of Rayleigh-Taylor and Kelvin-Helmholtz instabilities [46].Alongtheopenfieldlinesabovethemagneticpoles −14/7 −1 𝑀 3/7 matter can also flow in relatively freely. Most accumulated +𝜇8/7 ( ) 𝑀̇ V16/7𝑎8/7] . 33 𝑀 𝑤,−6 𝑤,8 10 yr matter forms an extended adiabatic, tenuous, and rotating ⊙ 6 Advances in Astronomy

The corresponding X-ray luminosity in this scenario is stops the accreting plasma at a position where the pressure of field and the plasma become of the same order and leads to 𝐿 𝑥,acc the truncation and an inner boundary of accretion disk. 26/7 𝑀 9/7 (34) ∼1031 ( ) 𝑀̇ V−36/7𝑎−18/7𝜇−4/7 −1. 2.4.2. Magnetosphere. We consider a steady and radial free- 𝑀 𝑤,−6 𝑤,8 10 33 ergs s ⊙ fall accretion flow, with constant accretion rate 𝑀̇ ,approach- ing the magnetized NS. Thus the inward and 2.4. Accretion from an Accretion Disk mass density near the boundary are given by

2.4.1. Disk Formation. In some systems the companion 2𝐺𝑀 1/2 𝑀 1/2 V ≡( ) = 1.6 × 109𝑟−1/2 ( ) −1, evolves and fills its Roche lobe, and a RLOF occurs. A 𝑓𝑓 𝑟 8 𝑀 cm s consequence of RLOF is that the transferred material has a ⊙ rather high specific angular momentum, so that it cannot be 𝑀̇ accreted directly onto the mass-capturing NS. Note that the 𝜌𝑓𝑓 ≡ (V 4𝜋𝑟2) (35) matter must pass from the Roche lobe of the companion to 𝑓𝑓 that of the NS through the inner Lagrangian point. Within 𝑀 −1/2 the Roche lobe of the NS, the dynamics of high angular −10 −3/2 ̇ −3 = 4.9 × 10 𝑟8 𝑀17 ( ) gcm , momentum material will be controlled by the gravitational 𝑀⊙ field of the NS alone, which would give an elliptical orbit 8 lying in the binary plane. A continuous stream trying to where 𝑟8 is the distance 𝑟 in units of 10 cm. follow this orbit will therefore collide with itself, resulting The magnetic field of the NS begins to dominate theflow 2 in dissipation of energy via shocks, and finally settles down when the magnetic pressure 𝐵 /8𝜋 becomes comparable to 2 through postshock dissipation within a few orbital periods the pressure of accreting matter ∼𝜌𝑓𝑓 V𝑓𝑓 , which includes both into a ring of lowest energy for a given angular momentum, ram pressure and thermal pressure. The Alfven´ radius 𝑅𝐴 is that is, a circular ring. Since the gas has little opportunity to then given by riditselfoftheangularmomentumitcarried,wethusexpect 2 the gas to orbit the NS in the binary plane. Such a ring will 𝐵(𝑅𝐴) 2 =𝜌𝑓𝑓 (𝑅 )V𝑓𝑓 (𝑅𝐴) , (36) spread both inward and outward, as effectively dissipative 8𝜋 𝐴 processes, for example, collisions of gas elements, shocks, 3 and viscous dissipation, and convert some of the energy of for a dipolar field outside the NS 𝐵(𝑟) = 𝜇/𝑟 .SotheAlfven´ the bulk orbital motion into internal energy, which is partly radius reads radiatedandthereforelosttothegas.Theonlywayinwhich −1/7 the gas can meet this drain of energy is by sinking deeper 8 ̇ −2/7 4/7 𝑀 𝑅𝐴 ∼10𝑀17 𝜇30 ( ) cm into the gravitational potential of the NS, which in turn 𝑀⊙ requires it to lose angular momentum. The orbiting gas then (37) −1/7 redistributes its angular momentum on a timescale much 8 −2/7 4/7 𝑀 −2/7 ∼10𝐿37 𝜇30 ( ) 𝑅6 cm, longer than both the timescale over which it loses energy by 𝑀⊙ radiativecoolingandthedynamicaltimescale.Asaresult,the gas will lose as much energy as it can and spiral in towards where 𝜇30 is the magnetic moment of NS in units of 30 3 the NS through a series of approximately Keplerian circular 10 Gcm ,and𝐿37 is the accretion luminosity in units of 37 −1 in the orbital plane of binary, forming an accretion disk 10 ergs s . around the NS, with the gas in the disk orbiting at Keplerian Initially, the radial flow tends to sweep the field lines 3 1/2 angular velocity Ω𝐾(𝑟) = (𝐺𝑀/𝑟 ) . inward, due to the high conductivity of plasma. Since the To be accreted onto the NS, the material must somehow plasma is fully ionized and therefore highly conducting, we get rid of almost all its original angular momentum. The wouldexpectittomovealongthefieldlinescloseenough processes that cause the energy conversion exert torques on to the NS surface. However, it is the high conductivity of the inspiraling material, which transport angular momentum accreting plasma that complicates its dynamics, since it denies outward through the disk. Near the outer edge of the disk the plasma ready access to the surface [51]. The plasma tends some other process finally removes this angular momentum. to be frozen to the magnetic flux and be swept up, building It is likely that angular momentum is fed back into the up magnetic pressure, which halts the flow and truncates orbital motion of a binary system by tidal interaction between the inner disk. Finally, an empty magnetic cavity is created the outer edge of disk and companion star. An important and surrounded by the plasma that is piling up, driving consequence of this angular momentum transport is that the the boundary inward as the plasma pressure there keeps outer edge of disk will in general be at some radius exceeding on increasing. Such transition behavior may possibly occur the circularization radius given by Frank et al. [9], and it is at the very beginning of the accretion flow, but a steady obvious that the outer disk radius cannot exceed the Roche state is quickly reached. When the magnetic pressure of the lobe of the NS. Typically the maximum and minimum disk NS magnetic field can balance the ram pressure of inflows, radii differ by a factor of 2-3. However, the disk cannot extend the radial flow eventually halts, creating a restricted region all the way to NS surface, because of the magnetic field, which inside which the magnetic field dominates the motion of the Advances in Astronomy 7 plasma. This is the magnetosphere [36], whose formation isa consequence of the transition of the plasma flow. The boundary of the static magnetosphere is determined by the condition of static pressure balance between the inside andoutsideofthemagnetosphere,includingbothplasma 2 pressure 𝑃 and magnetic pressure 𝐵 /8𝜋;thatis, 2h

2 2 𝐵 𝐵 𝛿 rs ( 𝑡 ) +𝑃 =( 𝑡 ) +𝑃 , r in out (38) A r0 8𝜋 8𝜋 Transition Unperturbed in out Magnetospheric region disk flow flow 𝐵 Outer where 𝑡 is the tangential component of magnetic field, and transition the subscripts “in” and “out” denote “inside” and “outside” of Boundary zone the magnetospheric boundary. This pressure balance relation layer can be used to estimate the scale size of magnetosphere in Figure 1: Schematic representation of the transition zone [41], 𝑟0 astaticconfiguration mag, which depends on the particular which consists of an outer transition zone and a boundary layer. physical conditions near the magnetospheric boundary [36]. However, although the supersonic flow of plasma toward theboundaryishalted[8],someplasmacanenterthe magnetosphere via particle entry through the cusps, stress associated with the twisted field lines, the radiative diffusion of plasma across the magnetospheric boundary, and transport of energy associated with the effective viscous magnetic flux reconnection [52], as well as the Rayleigh- dissipation and resistive dissipation of currents generated by Taylor instability at the boundary. If there is a plasma flow the cross-field motion of the plasma [53], and the dissipative into the magnetosphere, its static structure is broken, and stresses consisting of the usual effective viscous stress and the the static pressure balance (38) is no longer valid. Instead, magnetic stress associated with the residual magnetospheric the momentum balance and continuity of magnetic field field. In the outer transition zone, the poloidal field is components normal to the magnetospheric boundary imply screened on a length-scale ∼𝑟 by the azimuthal currents generated by the radial cross-field drift of plasma. The 𝐵2 𝑡 2 azimuthal pitch of the magnetic field increases from 𝑟0 to a ( ) +𝑃 +(𝜌V𝑛) 8𝜋 in in 𝑟 in maximum at the corotation point cor and begins to decrease. 𝑟 𝑟 (39) Thus the field lines between cor and 𝑠 are swept backward 2 𝐵𝑡 2 and exert a spin-down torque on the NS. =( ) +𝑃 +(𝜌V𝑛) , 𝑟 8𝜋 out out The Keplerian motion ends at radius 0, and then the out angularvelocityoftheplasmaisreducedfromtheKeplerian 2 to corotate with NS, which resorts to the transportation of where 𝜌V𝑛 is the dynamic pressure of plasma in terms of its angular momentum through magnetic stresses [54]. Accord- velocity components V𝑛 normal to the boundary. Obviously, the scale of magnetosphere is in the order of the Alfven´ ingly, the accreted matter is released from the disk and starts 𝑟 ∼𝑅 flowing toward the NS along the magnetic field lines. The radius, mag 𝐴. transition of the accretion flow requires a region with some extent. This is the boundary layer, with thickness of 𝛿≡ 2.4.3. Transition Zone and Boundary Layer. The region where 𝑟0 −𝑅𝐴 ≪𝑟0, in which the angular momentum is conserved, the field lines thread the disk plasma forms a transition and the angular velocity of plasma continuously changes from zone between the unperturbed disk and the magnetosphere, Ω (𝑟 ) Ω 𝑟 𝑟 𝐾 0 to 𝑠 between 0 and cor.Becauseofasub-Keplerian in which the disk plasma across the field lines generates angular velocity, the close balance between centrifugal force currents. The currents confine the magnetic field inside a and gravitational force is broken, and the radial flow attains a screening radius 𝑟𝑠 ∼(10–100)𝑟 , which gives the extent cor much higher velocity, increasing inward continuously from of transition zone. Within this region, the accreted matter 𝑟 is forced to corotate with the NS, by means of transporting 0 where it must equal the slow radial drift characteristic angular momentum via stresses dominated by the magnetic of the outer transition zone due to continuity. Since the field. According to the angular velocity of accreting material, rising magnetic pressure gradient opposes the centrifugal the transition zone is divided into two parts (Figure 1), an support, the radial velocity passes through a maximum and is outer transition zone with Keplerian angular velocity, and reducedtozeroattheinneredgeoftheboundarylayer.The a boundary layer in which the plasma deviates from the boundary layer is basically an electromagnetic layer, in the Keplerian value significantly, which is separated at the radius sense that the dominant stresses are magnetic stress, and the 𝑟0. dominant dissipation is through electromagnetic processes, The structure of outer transition zone between 𝑟0 and which obey Maxwell’s equations. However, the mass flow also 𝑟𝑠 is very similar to that of a standard 𝛼-disk [25] at the plays an essential role in this layer, since the cross-field radial same radius, with three modifications, that is, the transport of flow generates the toroidal electric currents that screens the angular momentum between disk and NS due to the magnetic magnetic field of the NS. 8 Advances in Astronomy

16 compressed accreted matter causes the expansion of polar 15 zone in two directions, downward and equatorward [60]. Therefore, the magnetic flux in the polar zone is diluted, with 14 more and more plasma piling up onto the polar caps, which 13 subsequently diffuse outwards over the NS surface. This process expands the area of the polar caps, until occupying

(G)] 12 the entire surface, and the magnetic flux is buried under the B accreted matter. Finally, the magnetosphere is compressed to 11 log[ the NS surface, and no field lines drag the plasma, remaining 10 Spin-up line an object with weak magnetic field. The minimum field is determined by the condition that the magnetospheric radius 9 ∼108 Death line equals the NS radius, which is about G. Meanwhile, the 8 angular momentum carried by the accreted matter increases −3.0 −2.5 −2.0 −1.5 −1.0 −0.5 0.0 0.5 1.0 1.5 to the spin angular momentum of the NS and spins it log[P(s)] up,leavingaMSPwithspinperiodofafewmilliseconds. The combined field decay and spin-up process is called the Normal pulsars recycling process of the NS. MSPs During the recycling process, the accretion-induced mag- Binaries netic field evolution can be obtained analytically for an initial Figure 2: Magnetic fields and spin periods of observed pulsars (data field 𝐵(𝑡 = 0)0 =𝐵 and final magnetic field 𝐵𝑓 by [60] taken from the ATNF pulsar catalogue). dots denote normal 𝐵 pulsars. Red dots represent MSPs. Green triangles are pulsars in 𝐵 (𝑡) = 𝑓 . 7/4 (40) binaries. The “spin-up line” represents the minimum spin period (1 − [𝐶 (−𝑦) − 1]2) to which a spin-up process may proceed in an Eddington-limited exp accretion, while the “death line” corresponds to a polar cap voltage Here, we define the parameters as follows, 𝑦 = (2/7)(Δ𝑀/ below which the pulsar activity is likely to switch off [17]. 𝑀 ) Δ𝑀 = 𝑀𝑡̇ 𝑀 ∼ cr , the accreted mass ,thecrustmass cr 2 2 4/7 0.2𝑀⊙,and𝐶=1+√1−𝑥0 ∼2with 𝑥0 =(𝐵𝑓/𝐵0) .The bottom magnetic field 𝐵𝑓 is defined by the magnetospheric 𝑟 (𝐵 )=𝑅 2.4.4. Accretion-Induced Field Decay and Spin-Up radius matching the NS radius; that is, mag 𝑓 .The 𝑟 𝑟 =𝜙𝑅 magnetospheric radius mag is taken as mag 𝐴 with a model dependent parameter 𝜙 of about 0.5 [9, 13, 61]. Using FormationoftheQuestion.Observationally, pulsars are usu- 𝑟 (𝐵 )=𝑅 the relation mag 𝑓 , we can obtain the bottom magnetic ally assigned to be of two distinct kinds, whose spin periods field, and surface magnetic fields are distributed almost bimodally, 𝑀̇ 1/2 𝑀 1/4 with a dichotomy, for magnetic fields and spin periods, which 𝐵 = 1.32 × 108 ( ) 𝑅−5/4𝜙−7/4 , 𝐵∼1011–13 𝐵∼108-9 𝑓 ̇ 6 G (41) are between Gand G, respectively, and 𝑀18 𝑀⊙ between 𝑃𝑠 ∼ 0.1–10 sand𝑃𝑠 ∼1–20 ms, respectively [55]. ̇ ̇ 18 −1 The large population of pulsars, with high surface magnetic where 𝑀18 = 𝑀/10 gs . For details see [60]. 11–13 fields of 𝐵∼10 Gandlongspinperiodsof𝑃𝑠 ∼ a In the meantime, the NS is spun up by the angular fewseconds,representstheoverwhelmingmajorityofnormal momentum carried by accreted matter, according to [61] pulsars. Most of them are isolated pulsars, with only very 𝑀 −3/7 −𝑃̇ = 5.8 × 10−5 [( ) 𝑅12/7𝐼−1] few sources in binary systems. On the other hand, there is 𝑠 𝑀 6 45 a smaller population of sources which display much weaker ⊙ (42) 8 9 𝐵∼10- 𝑃 ≤ 2 magnetic fields Gandshortspinperiods 𝑠 ×𝐵2/7 (𝑃 𝐿3/7) 𝑛(𝜔) −1, 20 ms, and most of this population is associated with binaries 12 𝑠 37 𝑠 syr (Figure 2). These two typical populations are connected with 12 where 𝐵12 is the magnetic field in units of 10 G, and the athinbridgeofpulsarsinbinaries.Theabnormalbehavior dimensionless torque 𝑛(𝜔𝑠) is a function of the fastness of the second group, particularly the fact that these are parameter [36], which is introduced in order to describe the mostly found in binaries, had led to extensive investigations relative impotance of and defined by the ratio concerning the formation of millisecond pulsars (MSPs). parameteroftheangularvelocities[36,54], The currently widely accepted idea is the accretion-induced −2/7 magnetic field decay and spin-up; that is to say the normal Ω𝑠 𝑀 15/7 6/7 −1 −3/7 𝜔𝑠 ≡ = 1.35 ( ) 𝑅6 𝐵12 𝑃𝑠 𝐿37 . (43) pulsars in binary systems have been “recycled” [56–58]. The Ω𝐾 (𝑅𝐴) 𝑀⊙ objectsformedbysucharecyclingprocessarecalledrecycled pulsars [17, 59]. Therefore, the dimensionless accretion torque on NS is given by [61] Accretion-Induced Field Decay and Spin-Up. The accreted 1−𝜔/𝜔 matter, in the transition zone, begins to be channeled onto 𝑛(𝜔) = 1.4 × ( 𝑠 𝑐 ), 𝑠 1−𝜔 (44) the polar patches of the NS by the field lines, where the 𝑠 Advances in Astronomy 9

16 16

15 15

14 14

13 13

(G)] 12

(G)] 12 B B 11 11 log[ log[ 10 Spin-up line 10 Spin-up line

9 9

8 Death line 8 Death line −3 −2 −1 012 −3 −2 −1 012 log[P(s)] log[P(s)]

Normal PSRs log[B0] =13.7 Normal PSRs P0 = 100 s MSPs log[B0] =12.7 MSPs P0 =10s Binaries log[B0] =11.7 Binaries P0 =1s (a) (b)

13 Figure 3: Magnetic field and spin period evolutionary tracks. In (a) tracks are plotted for different initial magnetic fields: 𝐵0 =5×10 G 12 11 17 −1 (solid line), 5×10 G(dashline),and5×10 G (dot line), at fixed initial spin period 𝑃0 = 100 s and accretion rate 10 gs .In(b)with 12 different initial periods, 𝑃0 =1s(dottedline),10 s(dashedline),and100 s (solid line), at fixed initial field 𝐵0 =5×10 G and accretion rate 17 −1 10 gs .

where 𝜔𝑐 ∼ 0.2–1 [62–66], with an original value of 0.35 of the transition from an X-ray binary to a radio pulsar PSR [61, 67]. For a slowly rotating star, 𝜔𝑠 ≪1,thetorqueis J 1023+0038 [75]. The NSs in LMXBs are the evolutionary 𝑛(𝜔 ) ≈ 1.4 𝑇 ≈ 1.4𝑇 𝑛(𝜔 ) 𝑠 ,sothat disk 0. For fast rotators, 𝑠 precursors to “recycled” MSPs [57]. It is evident that X-ray decreases with increasing 𝜔𝑠 and vanishes for the critical pulsars and recycled pulsars are correlated with both the value 𝜔𝑐.For𝜔𝑠 >𝜔𝑐, 𝑛(𝜔𝑠) becomes negative and increases duration of accretion phase and the total amount of accreted with the increasing 𝜔𝑠, until it reaches a maximum value, matter [57, 76]. When a small amount of mass (≲0.001𝑀⊙)is 𝜔 ∼ 0.95 typically, max . transferred, the spin period mildly changes, which may yield The NS cannot be spun up to infinity because of the a HMXB with a spin period of some seconds, such as Her X-1 critical fastness 𝜔𝑐, at which this torque vanishes and the NS is [77] and Vela X-1 [78]. If the NS accretes a small quantity of spun up to reach the shortest spin period. Then, the rotation mass from its companion, for example, ∼0.001𝑀⊙–0.01𝑀⊙, 𝑃 10 reaches an equilibrium spin period eq: a recycled pulsar with a mildly weak field𝐵∼10 ( G) and short spin period (𝑃𝑠 ∼50ms) will be formed [79], like PSR 𝑃 =2𝜋𝜔−1Ω 𝑟 = 1.89𝐵6/7 , eq 𝑐 𝐾 mag 9 ms (45) 1913+16 and PSR J 0737-3039 [80]. After accreting sufficient mass (Δ𝑀 ≳ 0.1𝑀⊙), the lower magnetic field and shorter 9 𝑃 ≲20 where 𝐵9 is 𝐵 in units of 10 G, and the second relation is spin period ( 𝑠 ms)ofaMSPform,forexample,SAXJ obtained by taking 𝑀 = 1.4𝑀⊙ and 𝑅6 =1and an Eddington 1808.4-3658 [74] and PSR J 1748-2446 [81]. limit accretion rate [17]. This line in field-period diagram is referred to as the “spin-up line”. 2.4.5. Rotation Effects and Quasi-Quantized Disk. The orbital Attheendoftheaccretionphase(withanaccretedmass motion of an accreting flow controlled by the potential of ≳0.2𝑀⊙), the magnetic field and spin period of recycled a rotating NS is different from that in a flat space-time, pulsars arrive at bottom values, which cluster in a range of 𝐵=108-9 𝑃 <20 because of the rotation effects and the strong of the Gand 𝑠 ms [68, 69], and the NSs remain to NS.Firstly,stellaroblatenessarisesfromtherotationandthus be MSPs [70, 71]. The minimum spin period and bottom field a quadruple term in the gravitational potential appears [82]. are independent of the initial values of magnetic field and spin Secondly, a rotating massive object will impose a rotational period;seeFigure3[72].Thebottomfieldsmildlyvarywith frame-dragging effect on the local inertial frame, which is the accretion rates and the accreted mass (Figure 4), while the known as gravitoelectromagnetism [83–86]. minimum period is insensitive to them (Figure 5) [72, 73]. The direct evidence for this recycled idea has been found Gravitoelectromagnetism. Gravitoelectromagnetism (GEM) in LMXBs with an accreting millisecond X-ray pulsar, for is based on an analogy between Newton’s law of gravitation example, SAX J 1808.4-3658 [74], and in observing the link and Coulomb’s law of electricity. The Newtonian solution of between LMXBs and millisecond radio pulsars in the form the gravitational field can be alternatively interpreted as a 10 Advances in Astronomy

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12 12

11 11 (G)] (G)] B 10 B 10 log[ log[ 9 9

8 8

7 7 −3.5 −3.0 −2.5 −2.0 −1.5 −1.0 −0.5 0.0 0.5 −10 −8 −6 −4 −2 0 log[P(s)] log[ΔM(M⊙)] (a) (b)

0 5

4 −1

(s)] 3 P (ms) P

log[ −2 2

−3 1

−10 −8 −6 −4 −2 0 0.2 0.4 0.6 0.8 1.0

log[ΔM(M⊙)] ΔM/M⊙ (c) (d)

Figure 4: 𝐵 and 𝑃 evolution in the recycling process. (a) shows the joint evolution of 𝐵 and 𝑃. (b) and (c) are their evolution as a function 18 −1 17 −1 16 −1 of accreted mass Δ𝑀. The solid, dashed, and dotted lines are plotted for an accretion rate of 10 gs , 10 gs ,and10 gs ,respectively. 12 Initial values 𝐵0 =5×10 Gand𝑃0 =1s were taken. (d) is a zoom-in view of (c) in a linear scale for spin periods shorter than 5 ms.

gravitoelectric field. It is well known that a magnetic field is and keeping only the linear order terms, we obtain the field produced by the motion of electric charges, that is, electric equations (in this part, we set the light speed 𝑐=1)[82]: current. Accordingly, the rotating mass current would give ◻ℎ = −16𝜋𝐺𝑇 , rise to a gravitomagnetic field. In the framework of the 𝜇] 𝜇] (47) general theory of relativity, a non-Newtonian massive mass- 𝜇] ℎ =0 charge current can produce a gravitoelectromagnetic field where the Lorentz gauge condition ,] is imposed. In ◻𝐴] =4𝜋𝑗] [87–89]. analogy with Maxwell’s field equations ,wecan 𝐴] In the linear approximation of the gravitational field, the findthattheroleoftheelectromagneticvectorpotential ℎ Minkowski metric 𝜂𝜇] is perturbed due to the presence of a is played by the tensor potential 𝜇],whiletheroleofthe4- ] gravitating source with a perturbative term ℎ𝜇], and thus the current 𝑗 is played by the stress-energy tensor 𝑇𝜇]. Therefore, background reads the solution of (47) in terms of the retarded potential can be written as [90, 91] 󵄨 󵄨 󵄨 󵄨 𝑔 =𝜂 +ℎ , 󵄨ℎ 󵄨 ≪1. 󵄨 󸀠󵄨 󸀠 𝜇] 𝜇] 𝜇] 󵄨 𝜇]󵄨 (46) 𝑇𝜇] (𝑞0 − 󵄨𝑞𝑖 −𝑞𝑖 󵄨 ,𝑞𝑖 ) ℎ =4𝐺∫ 󵄨 󵄨 𝑑3𝑞󸀠, 𝜇] 󵄨 󸀠󵄨 𝑖 (48) 󵄨𝑞𝑖 −𝑞𝑖 󵄨 ℎ =ℎ −(1/2)𝜂 ℎ We define the trace-reversed amplitude 𝜇] 𝜇] 𝜇] , 𝑞 =(𝑞,𝑞) ℎ=𝜂𝜇]ℎ =ℎ𝛼 ℎ with coordinates 𝜇 0 𝑖 .(Notethatwechoose where 𝜇] 𝛼 is the trace of 𝜇]. Then, expanding Greeksubscriptsandindices(i.e.,𝜇, ], 𝛼, 𝛽) to describe the Einstein’s field equations 𝐺𝜇] =8𝜋𝐺𝑇𝜇] in powers of ℎ𝜇] the 4-dimension space-time components (0,1,2,3) for test Advances in Astronomy 11

16 In the approximation of weak field [82, 92] and slow rotation 𝑔 15 of the NS, 𝑖 contains mostly first-order corrections to flat space-time and denotes the Newtonian gravitational 14 Death line acceleration, whereas R𝑖 contains second-order corrections 13 andisrelatedtotherotationoftheNS. 12 Using (47), (51), and (52) and analogy with Maxwell’s (G)]

B equations, we get the gravitoelectromagnetic field equations 11 [90]: log[ 10 ∇𝑖𝑔𝑗𝛿𝑖𝑗 =−4𝜋𝐺𝜌𝑔, 9 Spin-up line 𝜕 1 8 ∇ 𝑔 =− ( R ), 𝑗 𝑘 𝜕𝑞 2 𝑖 7 0 −4 −3 −2 −1 012 (53) log[P(s)] ∇𝑖R𝑗𝛿𝑖𝑗 =0, ̇ Normal PSRs M16 𝜔c = 0.9 Ṁ 𝜔 = 0.9 MSPs 17 c 𝜕𝑔𝑖 ̇ ∇ R = −4𝜋𝐺𝑗 . Binaries M18 𝜔c = 0.9 𝑗 𝑘 𝑔𝑖 𝜕𝑞0 Figure 5: Magnetic field and spin period evolutionary tracks for 18 −1 17 −1 Here, 𝜌𝑔 is the mass density, and 𝑗𝑔 =𝜌𝑔× (velocity of the different accretion rates: 𝑀=10̇ gs (solid line), 𝑀=10̇ gs 16 −1 (dashed line), and 𝑀=10̇ gs (dotted line), at fixed initial field mass flow generating the gravitomagnetic field) denotes the 12 mass current density or mass flux. These equations include 𝐵0 =5×10 Gandperiod𝑃0 = 100 s. the conservation law for mass current 𝜕𝜌/𝜕𝑞0 +∇𝑖𝑗𝑔𝑗 𝛿𝑖𝑗 =0, as they should be. In analogy with the Lorentz force in an electromagnetic particles, while the subscripts and indices 𝑖, 𝑗, 𝑘 denote the 𝑚 𝑥 𝑦 𝑧 𝑟 𝜃 𝜙 field, the motion of a test particle in a gravitoelectromag- 3-dimension space coordinates , , (or , , ).) netic field is subject to a gravitoelectromagnetic force Neglecting all terms of high order and smaller terms that ℎ (𝑞 ,𝑞) 𝐹 =𝑄 𝑔 +𝑄 𝑞̇ R 𝜀𝑗𝑘, include the tensor potential 𝑖𝑗 0 𝑖 [82],wecanwritedown GEM𝑖 𝑔 𝑖 R 𝑗 𝑘 𝑖 (54) the tensor potential ℎ𝜇] [91]: where 𝑄𝑔 =−𝑚and 𝑄R =−2𝑚are the gravitoelectric charge and gravitomagnetic charge of the test particle [90], ℎ =4Φ, 00 respectively. Accordingly, the Larmor quantities in the gravi- (49) toelectromagnetic field would be ℎ0𝑖 =−2𝐴𝑖. 𝑄 𝑔⃗ 𝑎⃗ =− 𝑔 , Here, Φ and 𝐴𝑖 aretheNewtonianorgravitoelectricpotential 𝐿 𝑚 and the gravitomagnetic vector potential, respectively. They (55) canbeexpressedas ⃗ 𝑄RR 𝜔⃗ 𝐿 = . 𝐺𝑀 2𝑚 Φ(𝑞0,𝑞𝑖)=− , 𝑟 Therefore, the test particle has the translational acceleration (50) 𝑎⃗ = 𝑔⃗ 𝜔 =|R⃗ | 𝐽𝑗𝑞𝑘 𝐿 and the rotational frequency 𝐿 . 𝐴 (𝑞 ,𝑞)=𝐺 𝜀𝑗𝑘, Ω 𝑖 0 𝑖 𝑟3 𝑖 For a rotating NS with angular velocity 𝑠,thegravita- tional Larmor frequency of test particles can be written as [93] where 𝐴𝑖 isintermsoftheangularmomentum𝐽𝑖 of the NS. 2 󵄨 󵄨 4 𝐺𝑀𝑅 Accordingly, the Lorentz gauge condition reduces to 𝜔 = 󵄨R⃗ 󵄨 = Ω , (56) 𝐿 󵄨 󵄨 5 𝑟3 𝑠 𝜕Φ 1 which causes a split of the orbital motion of accreted particles + ∇𝑖𝐴𝑗𝛿𝑖𝑗 =0. (51) 𝜕𝑞0 2 and changes the circular orbital motion on the binary plane in the vertical direction of disk. Then we can define the gravitoelectromagnetic field, that is, the gravitoelectric field 𝑔𝑖 and the gravitomagnetic field R𝑖, Axisymmetrically Rotating Stars. The stationary and axisym- as [90] metric space-time metric arising from a rotating object, in the polar coordinates, has a form of [94] 𝜕 1 2 2](𝑟,𝜃) 2 2𝜆(𝑟,𝜃) 2 𝑔𝑖 =−∇𝑖Φ− ( 𝐴𝑗)𝛿𝑖𝑗 , 𝑑𝑠 =𝑒 𝑑𝑡 −𝑒 𝑑𝑟 𝜕𝑞0 2 (52) (57) 𝑗𝑘 2𝜇(𝑟,𝜃) 2 2 2 2 2 R𝑖 =∇𝑗𝐴𝑘𝜀𝑖 . −𝑒 [𝑟 𝑑𝜃 +𝑟 sin 𝜃(𝑑𝜙−𝜔(𝑟,) 𝜃 𝑑𝑡) ]. 12 Advances in Astronomy

As a new feature of an axisymmetric structure, the nondiag- 3. Final Remarks onal elements appear: In NS X-ray binary systems, accretion is the only viable 2 2 2𝜇(𝑟,𝜃) 𝑔𝑡𝜙 =𝑔𝜙𝑡 =𝑟sin 𝜃𝑒 𝜔 (𝑟,) 𝜃 . (58) energy source that powers the X-ray emission. Accretion onto a NS is fed by mass transfer from the optical companion For a test particle at a great distance from NS in its equatorial star to the NS, via either a stellar wind or RLOF. Stellar 𝜃=𝜋/2 𝜙 plane ( ) falling inwards from rest, the coordinate at wind accretion always occurs in NS/HMXBs, with high- 𝜃=𝜋/2 obeys the equation [95]: mass OB supergiant companions or Be stars with radiation- 𝜙𝑡 2 2𝜇(𝑟,𝜃) driven stellar . The neutron stars here are observed 𝑑𝜙 𝑔 −𝑔𝜙𝑡 −𝑟 𝑒 𝜔 (𝑟) = = = =𝜔(𝑟) . as X-ray pulsars, while disk accretion dominates the mass- 𝑑𝑡 𝑔𝑡𝑡 𝑔 −𝑟2𝑒2𝜇(𝑟,𝜃) (59) 𝜙𝜙 transfer mechanism in NS/LMXBs, which occurs when the outer layer of the companion flows into the Roche lobe Therefore, 𝜔(𝑟) is referred to as the angular velocity of the local inertial frame, which can be expressed in terms of the of the NS along the inner Lagrangian point. However, the Keplerian velocity at radius 𝑟 [96]: transferred material cannot be accreted all the way onto NS surface, due to the strong magnetic field. At a preferred 2 𝐺𝑀 radius where the magnetic pressure can balance the ram 𝜔 (𝑟) = √ 𝑅2Ω2, (60) 5 𝑟3 𝑠 pressure of the infalling flows, the accreting plasma will haltandinteractwiththemagneticfield,buildingupa Consequently, the accreted matter experiences an increasing magnetosphere, which exerts an accretion torque and results drag in the direction of the rotation of NS and possesses a in the spin period evolution of NS. In NS/HMXBs, because of corresponding angular velocity 𝜔(𝑟). the disordered stellar wind, some instabilities and turbulence Closure Orbits and the Quasi-Quantized Disk. The accretion lead to fluctuations in the mass accretion rate and give rise flow in a turbulent Keplerian disk will flow helical trajectories, to a stochastic accretion with positive and negative angular that is, open circular orbits at each radius, and is endowed momentum, which alternatively contributes to phases of with two frequencies, that is, angular frequency in the spin-up and spin-down. For disk-fed LMXBs, the NSs usually direction of rotation and gravitational Larmor frequency in are spun up by the angular momentum carried by the accreted vertical direction of the disk. The former deviates the orbital matter, during which the magnetic field is buried by the motion from a circular orbit, and the latter leads to some accumulated plasma. As a result, the NS is spun up to a 8 9 vertical oscillation. If and only if the gravitational Larmor few milliseconds and the magnetic field decays to 10 - G, frequency and angular velocity of accreted plasma satisfy [97] remaining a MSP. In several systems, the torque reversal

𝑙𝜔 (𝑟) =𝑛𝜔𝐿, (61) [99–102] and state changes [103, 104] were observed and investigated because of sudden dynamical changes triggered where 𝑛 and 𝑙 are integers with 𝑛, 𝑙, ≥1 the vertical split by a gradual variation of mass accretion rates [105]. along disk and deformation in the direction of rotation of NS In the inner region of an accretion disk, the accreting can be harmonious [98], that is, the vertical oscillation and plasma is endowed with two additional frequencies due deformed orbital circular motion are syntonic. As a result, the to frame-dragging effects arising from the rotation of the orbital motion of accreted matter on separated and deformed NS, which contribute to a band of closed circular orbits at open circular orbits returns to a closed circular motion. The certain radii. Such a disk structure is expected to be the radii of closure circular orbits are give by quasi-quantized structure. The orbiting material, moving on 4𝑛2 𝐺𝑀 these closure circular orbits, is in a stable state. With a 𝑟3 = . 𝑙2 Ω2 (62) slight perturbation, the test particle will oscillate around the 𝑠 minimum, manifesting as radial drift of the orbital frequency. We call such a disk structure, that is, tenuous spiraling-in If the perturbation is strong enough to transfer sufficient gas with the appearance of a band of closure circular orbits, angular momentum outwardly and drive the particles to leave as a quasi-quantized disk. Moving on the closure circular this state, the material will continue to follow the helical track orbits, the accreted material is in a stable state, with the first and spirals in. derivative of angular momentum being larger than or equal to zero, which correspond to a minimum of the effective potential. With a slight perturbation, the test particle will Competing Interests oscillate around the minimum, manifesting as drift of the The author declares that he has no competing interests. orbital frequency. If the perturbation is strong enough to transfer sufficient angular momentum outwardly and drive the particles to leave this state, the material will continue to Acknowledgments follow the original helical track and spirals in. In a turbulent and viscous accretion disk, dissipative processes, for example, ThisworkissupportedbytheFundamentalResearchFunds viscosity, collisions of elements, and shocks, are responsible for the Central Universities (Grant no. 161gpy49) at Sun Yat- for the perturbation. Sen University. Advances in Astronomy 13

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