Progenitor neutron stars of the lightest and heaviest millisecond pulsars
aa r X i v : . [ a s t r o - ph . S R ] D ec Astronomy & Astrophysics manuscript no. progens_arxiv c (cid:13)
ESO 2018August 24, 2018
Progenitor neutron stars of the lightest and heaviestmillisecond pulsars
M. Fortin , , , M. Bejger , P. Haensel , and J. L. Zdunik N. Copernicus Astronomical Center, Polish Academy of Sciences, Bartycka 18, PL-00-716 Warszawa, Poland LUTh, UMR 8102 du CNRS, Observatoire de Paris, F-92195 Meudon Cedex, France Istituto Nazionale di Fisica Nucleare - Sezione di Roma, P.le Aldo Moro 2, 00185 Roma, Italy [email protected], [email protected], [email protected], [email protected]
Received xxx Accepted xxx
ABSTRACT
Context.
The recent mass measurements of two binary millisecond pulsars, PSR J1614 − M = 1 . ± .
04 M ⊙ and M = 1 . ± .
14 M ⊙ , respectively, indicate a wide range of masses for such objectsand possibly also a broad spectrum of masses of neutron stars born in core-collapse supernovæ. Aims.
Starting from the zero-age main sequence binary stage, we aim at inferring the birth masses of PSR J1614 − Methods.
Using simulations for the evolution of binary stars, we reconstruct the evolutionary tracks leading to theformation of PSR J1614 − Results.
The estimated birth mass of the neutron stars PSR J0751+1807 and PSR J1614 − . ⊙ and as high as . ⊙ , respectively. These values depend weakly on the equation of state and the assumedmodel for the magnetic field and its accretion-induced decay. Conclusions.
The masses of progenitor neutron stars of recycled pulsars span a broad interval from . ⊙ to . ⊙ .Including the effect of a slow Roche-lobe detachment phase, which could be relevant for PSR J0751+1807, would makethe lower mass limit even lower. A realistic theory for core-collapse supernovæ should account for this wide range ofmass. Key words. dense matter – equation of state – stars: neutron – pulsars: general – accretion disks
1. Introduction
Millisecond radio pulsars (defined here as those with a spinperiod
P < ms) have several unique properties thatmake them very interesting objects to study, both obser-vationally and theoretically. They are the most rapid stel-lar rotators with a spin frequency f = 1 /P up to 716 Hz(Hessels et al. 2006). Their spin periods are extremely sta-ble with a typical period increase, owing to the spin angularmomentum loss associated with magneto-dipole radiation: ˙ P ∼ − s s − ∼ − s yr − . Consequently, their mag-netic field B , as estimated from the timing properties, arethree to four orders of magnitude weaker than those of nor-mal radio pulsars, for which B ≃ G.According to the current theory of neutron star (NS)evolution, millisecond (radio) pulsars (MSPs) originate in“radio-dead” pulsars via the accretion-caused spin-up inlow-mass X-ray binaries (LMXBs, see Alpar et al. 1982;Radhakrishnan & Srinivasan 1982). During this “recycling”process, the rotation frequency increases from an initialvalue . . Hz to a final ∼ Hz in ∼ − yr.The process is associated with the accretion of matter via Send offprint requests to : M. Fortin an accretion disk around the NS. Millisecond X-ray pul-sars become millisecond radio pulsars after the accretionprocess stops. This scenario has been corroborated by thedetection of millisecond X-ray pulsations in LMXBs, in-terpreted as the manifestation of rotating and accretingNSs (Wijnands & van der Klis 1998) and the observationsof three objects in transition from a state of accretion withX-ray emission to a rotation-powered state with radio emis-sion and/or vice versa: PSR J1023+0038 (Archibald et al.2009; Patruno et al. 2014), PSR J1824-2452I (Papitto et al.2013) and XSS J12270-4859 (Bassa et al. 2014).The difference of typically three to four orders ofmagnitude in the magnetic field strength of MSPsand normal pulsars is explained either by the “bury-ing” of the original magnetic field under a layer ofaccreted material (Bisnovatyi-Kogan & Komberg 1971;Taam & van den Heuvel 1986; Cumming et al. 2001)and/or by the Ohmic dissipation of electric currents in theaccretion-heated crust (Romani 1990; Geppert & Urpin1994).It is expected that the “recycling” process in LMXBs is aparticularly widespread mechanism in dense stellar systemssuch as globular clusters. This is in accordance with the
Article number, page 1 of 13 &A proofs: manuscript no. progens_arxiv peculiar structure of the MSP population (Lorimer 2008):out of a total of about
MSPs, ∼ are in binaries,whereas for other (non-millisecond) pulsars, this percentageis one order of magnitude lower ( ∼ %). Simultaneously,about half (117) of all MSPs are located in Galacticglobular clusters. Finally, some 50% of the globular clus-ter MSPs are found in binary systems (Manchester et al.2005).The widely accepted recycling mechanism in LMXBssuggests that rapid MSPs (say those with P < ms)are likely to be massive. Therefore, they are importantfor the observational determination of the maximum al-lowable mass for NSs. This upper bound is a crucial con-straint on the poorly known equation of state (EOS) atsupra-nuclear density. The precise measurement of the mass M = 1 . ± .
02 M ⊙ of PSR J1903+0327 (Freire et al.2011) and, even more so, of . ± .
04 M ⊙ for PSRJ1614 − M = 2 . ± .
04 M ⊙ (Antoniadis et al. 2013). Its properties are a relatively longspin period ( P = 39 ms) and a short orbital period( P orb = 2 . hr) for a recycled pulsar. Combined withthe low mass of its helium white-dwarf (WD) companion, M WD = 0 .
17 M ⊙ , the case of PSR J0348+0432 is challeng-ing for stellar evolution theory (see e.g. Antoniadis et al.2013), and its formation will not be addressed in the fol-lowing. On the other side of the mass spectrum, PSRJ0751+1807 only has a mass of . ± .
14 M ⊙ (Nice et al.2008; all measurements are given at σ confidence level).As of today, the masses of MSPs are therefore bracketedby . ± .
14 M ⊙ and . ± .
04 M ⊙ . We focus on PSRJ1614 − P and ˙ P for MSPs and are similar to each other. In this paper,by “progenitor NS” we denote the NS as it was born in asupernova explosion.The binary MSPs PSR J1614 − − − − ATNF Pulsar Catalogue out to have very different masses. This may be interesting inthe context of the (still incomplete) theory of the formationof NSs in core-collapse supernovæ.The article is composed as follows. In Sect. 2, the evolu-tionary scenarios for the formation of PSR J1614 − −
2. Evolutionary scenarios of formation of twobinaries
We begin by sketching the plausible evolutionary scenariosthat could have led to the present binaries containing PSRJ1614 − M will eventuallyproduce a MSP, while a less massive secondary of initialmass M will become a WD. The scenarios presented hereare but brief and approximate summaries based on exist-ing work. The main stages of the evolution leading to PSRJ1614 − − t RLDP , is much shorter than the timescale for
Article number, page 2 of 13ortin et al.: Progenitors of PSR J1614 − Table 1.
Measured parameters of the binary pulsars PSR J1614 − B is the canonical value of the magnetic field obtained using the dipole formula Eq. (3) for an ortogonal rotator andthe “canonical” NS radius and the moment of inertia: R = 10 km and I = 10 g cm . PSR M PSR
P f ˙ P P orb e M WD B (M ⊙ ) (ms) (Hz) ( − ) (M ⊙ ) ( G)J1614 − . ± . . × − . ± . . ± . . × − . ± . Fig. 1. (Colour online) Main stages of the evolution of the bi-nary system leading to the currently observed PSR J1614 − . M ⊙ WD (detailed description in Sect. 2.1). A - main sequence stage; B - common envelope stage (sec-ondary inside the primary); C - the primary becoming a Hestar; D - core-collapse supernova explosion of the primary; E -intermediate-mass X-ray binary stage, with a very strong massloss; F - LMXB, NS recycling stage via accretion disk, secondarymass loss via RLO; G - current state, wide binary MSP+WD. transmitting the effect of braking to the NS t torque , thenthe effect of the RLDP spin-down is negligible comparedto the spin-up during the LMXB stage or the intermediate-mass X-ray binary (IMXB) stage. This was shown to be thecase for PSR J1614 − t RLDP ≪ t torque )RLDP. However, PSR J0751+1807 has a He WD compan-ion, and it is expected that the LMXB stage terminatesthere by a slow RLDP, so that the accreted mass that wecalculate when neglecting the RLDP effect is an underesti-mate. An example of a slow RLDP is illustrated in Fig. 7of Tauris et al. (2012), but it refers to a binary MSP that isquite different from PSR J0751+1807. It has P = 5 . ms,instead of 3.48 ms for PSR J0751+1807. Incidentally, in thisexample the NS rotation period just before the RLDP co-incides with the present period of PSR J0751+1807 (whichwe get by construction at the end of our LMXB stage).The expected effect of including the RLDP braking on theaccreted mass required to reproduce the present period ofPSR J0751+1807 will be discussed in Sect. 10.Predicted masses, timescales, and orbital periods refer-ring to each stage, collected in Table 2, are but approximateestimates. We will stress the differences in the evolutionaryscenarios, conditioned by the present parameters of the pul-sars and their white dwarf companions. Fig. 2. (Colour online) Main stages of the evolution of the bi-nary system leading to the currently observed PSR J0751+1807,accompanied by a . M ⊙ WD (detailed description inSect. 2.2). A - main sequence stage; B - common envelope stage(secondary inside the primary); C - the primary becoming a Hestar; D - core-collapse supernova explosion of the primary; E - Roche lobe filling of the secondary and its strong mass loss,LMXB and recycling stage; F - current state: compact binaryMSP+WD. − Binary parameters and their evolution after the primarySN explosion are taken from Case A of Tauris et al. (2011)and Sect. 3.3 of Lin et al. (2011), with some modificationsfor the sake of consistency between our scenario for thespin-up of the NS during the LMXB stage and the presentpulsar parameters. In what follows we also use the reviewsof de Loore & Doom (1993) and Tauris & van den Heuvel(2006). In view of the uncertainties in the evolution models,we restrict ourselves to giving only approximate values ofmasses and timescales.
A: main sequence (MS)
At the zero-age main se-quence (ZAMS), M = 25 M ⊙ and M = 4 . ⊙ ,and the orbital period is ∼ yr. After some × yr(de Loore & Doom 1993), the primary becomes a red giant(RG) and its envelope absorbs the main sequence secondary.The binary then enters the common envelope stage. B and C: common envelope (CE)
The secondarystar spirals within the primary, transferring its angular mo-mentum to a weakly-bound envelope. As a consequence, theenvelope is shed away on a timescale of ∼ yr. The ac-cretion onto the secondary within the CE can be neglected.What remains out of the primary is a helium star with amass M He , = 7 M ⊙ , in a binary with a MS secondary.For a mass M He , > ⊙ , the star would have collapsedinto a black hole instead of a NS (Tauris & van den Heuvel2006). As an outcome of the CE stage,
18 M ⊙ has beenejected from the binary. Frictional dissipation of kinetic en- Article number, page 3 of 13 &A proofs: manuscript no. progens_arxiv
Table 2.
Summary of evolutionary stages that led from the two ZAMS binaries to the present millisecond pulsar - WD binaries,for PSR J1614 − Pulsar ZAMS 1st RG CE SNIb/c 2nd RG IMXB/LMXB todayPSR J1614 −
25 M ⊙ × yr ⊙ . ⊙ .
97 M ⊙ secondary . ⊙ . ⊙ . ⊙ × yr × yr .
50 M ⊙ P orb yr d d . dPSR J0751+1807 primary
15 M ⊙ yr ⊙ . ⊙ .
26 M ⊙ secondary . ⊙ . ⊙ . ⊙ × yr yr .
12 M ⊙ P orb yr d d . hergy causes the binary to shrink and shortens its orbitalperiod to ∼ d. D: Supernova (SN)
The outcome of SNIb/c deter-mines the initial state for the IMXB/LMXB evolutionstages (Tauris et al. 2011; Lin et al. 2011). We assume M NS ≈ . ⊙ , higher than NS masses in Tauris et al.(2011) and Lin et al. (2011) in order to be consistent withour NS spin-up scenario during the LMXB stage. As a re-sult, in our scenario the helium star of M He , = 7 M ⊙ col-lapses into a massive NS of M NS = 1 . ⊙ , while as muchas ⊙ is ejected in a SNIb/c explosion. The orbit becomesstrongly eccentric. The angular momentum loss during thesupernova explosion decreases the orbital period to ∼ d.A massive radio pulsar is born in the centre of the super-nova. After a few tens of Myr of magnetic-dipole rotationbraking, the pulsar period increases to a few seconds. Whilekeeping its original surface magnetic field ∼ G, the NSrotates too slowly to generate radio emission and enters thepulsar graveyard. Then, × yr after the ZAMS stage,the secondary leaves the MS. E and F: Intermediate-mass and low-mass X-ray binary
We adapt the Case A scenario of Tauris et al.(2011), which is consistent with the high NS mass scenarioin Sect. 3.3 of Lin et al. 2011, with some additional com-ments. The binary enters the stage of intermediate-massX-ray binary, called thus because the donor star has aninitial mass of . ⊙ , which is substantially higher thanwhat is characteristic of a donor star in the initial stage ofLMXBs ( < ⊙ ). After filling its Roche lobe and startingto lose mass, the secondary becomes unstable on a ther-mal timescale of ∼ yr (Langer et al. 2000). The massloss via the RLO is . ⊙ , so that the mass of the sec-ondary decreases to . ⊙ (Fig. 5 of Tauris et al. 2011).The mass accreted by the NS is assumed to be negligi-ble, at most .
01 M ⊙ (Tauris et al. 2011; Lin et al. 2011).Then the system enters the LMXB stage associated with aspin-up (recycling) of a dead pulsar via an disk accretionfrom its companion (donor) star. The RLO is initiated at P orb = 2 . d, and the final period P orb = 9 d (Tauris et al.2011). The widening of the orbit results from the mass lossfrom the system, and the magnetic braking is small. Accre-tion onto the NS induces the dissipation of its magnetic fieldto its current value ∼ G inferred from the measured P and ˙ P . During the ∼ × yr of the LMXB stage, the NSis spun-up to 317 Hz by accreting matter from the accretiondisk. The NS spin-up is not considered in Lin et al. (2011),where only NS mass and P orb are studied. The LMXB stageends after the mass loss of the secondary has stopped, leav- ing a . ⊙ WD. The binary orbit is circularized duringthe LMXB stage owing to the tidal dissipation, and theorbit eccentricity goes down to e ∼ − . As stressed inTauris et al. (2011), the proposed evolutionary scenario “isonly qualitative”. Both Tauris et al. (2011) and Lin et al.(2011) report that the initial mass of the recycled pulsaris higher than . ± . ⊙ , while we obtain ∼ . ⊙ (see Sect. 7). As a result of the IMXB/LMXB stage, dom-inated by the mass loss from the system with a negligibleeffect of the magnetic braking, we obtain a wide binary with P orb = 8 . d, composed of a WD of . ⊙ and a MSP of .
97 M ⊙ and f = 317 Hz . We assume that during the first 15 Myr, the binaryevolution follows the one summarized in Fig. 16.12 ofTauris & van den Heuvel (2006). We slightly deviate fromthis evolutionary track by assuming that the supernova ex-plosion of the primary produces a . ⊙ NS. This massis . ⊙ lower than in Tauris & van den Heuvel (2006)and should therefore result in a slightly higher eccentricityof the post-SN binary. We also assume that, because of aweaker angular momentum loss, the orbital period after theprimary SNIb/c explosion is initially 3 d (1 d longer than inTauris & van den Heuvel 2006). Together with the NS massof M = 1 . ⊙ , this is the starting point of an evolutionarytrack (converging LMXB in Fig. 2 E) that we select froma large set calculated by Istrate et al. (2014). According tothese results, magnetic braking operates at all times be-tween the ZAMS and the final state MSP+WD. Secondly,wind mass loss from a donor star is much less than the lossvia the RLO mechanism (Istrate et al. 2014). We adjust thespin-up duration and the mean accretion rate during theLMXB stage to reproduce the parameters of the presentNS+WD binary. All these changes, which should be takenwith a grain of salt, result from the lack of complete de-tailed evolutionary calculations for the currently observedbinary with PSR J0751+1807. A: Main sequence
At the ZAMS M = 15 M ⊙ and M = 1 . ⊙ , and the orbital period is ∼ yr. In ∼ yr,the primary becomes a RG and absorbs the MS secondary,and a brief ( ∼ yr) CE stage follows. B and C: Common envelope
The secondary starspirals towards the centre of the primary, transferring an-gular momentum to the envelope of the primary. As aconsequence, the envelope is shed away on a timescale of
Article number, page 4 of 13ortin et al.: Progenitors of PSR J1614 − ∼ yr. The envelope of
10 M ⊙ is ejected from the binary,kinetic energy is frictionally dissipated, the orbit shrinks,and the orbital period is reduced to d. The rest of theprimary reduces to a helium star with M He , = 5 M ⊙ in abinary with a MS secondary. D: Primary supernova explosion and Roche-lobeoverflow by the secondary
The evolved core of thehelium-star primary collapses into a light NS: M NS =1 . ⊙ , with most of the mass of the primary beingejected in a SNIb/c explosion. A low-mass radio pul-sar is born at the SNIb/c centre, and the orbital pe-riod increases to 3 d (by construction, 1 d longer than inTauris & van den Heuvel 2006. This can be easily obtainedby tuning the orbital angular momentum loss via mass lossin SNIb/c used up to this point). After a few tens of Myr,the NS enters the pulsar graveyard. Then, × yr afterthe ZAMS stage, the secondary fills its Roche lobe. Thisis because of a rapid orbital angular momentum loss asso-ciated with a very efficient magnetic braking. The orbitalperiod shortens by a factor of three (Istrate et al. 2014).Then the secondary overflows its Roche lobe, and the bi-nary enters the stage of the LMXB. E: Low-mass X-ray binary
In the following para-graph we rely on the modelling of Istrate et al. (2014) forthe evolution of the LMXB. During this phase. which lastsa few yr, the pulsar is spun-up by matter falling from anaccretion disk; however, we estimate that periods of intenseaccretion, during which the essential of the spin-up takesplace, occur on a time scale of yr. We adopt this valuein the following. The NS magnetic field is buried by the ac-creted matter, decreasing to a value ∼ G derived fromthe present ˙ P . The value of P orb = 3 d at the beginning ofmass transfer is below the bifurcation period P bif ∼ d, incontrast to Fig. 16.12 of Tauris & van den Heuvel (2006).The LMXB track is therefore of converging type, and P orb decreases in time (see Tauris & van den Heuvel 2006 andreferences therein). The orbital period shortens because ofan efficient angular momentum loss resulting from magneticbraking and gravitational wave radiation. In what followswe illustrate our case using an evolutionary track from alarge set of tracks calculated by Istrate et al. (2014). Tak-ing the LMXB model with initial P orb = 3 . d and adjustedmagnetic braking, one gets a compact NS+WD binary with P orb = 6 . h. The NS is spun-up to 287 Hz by accretionfrom the disk, and the WD mass at the end is .
16 M ⊙ ,which is quite close to the measured mass of WD. The ef-ficiency of accretion onto the NS is rather low (30%), sothat 70% of mass lost by the secondary leaves the binary.We assume that an appropriate small tuning of the LMXBstage can result in a decrease in the WD mass to a mea-sured value of .
12 M ⊙ . Finally, after the accretion onto theNS has stopped, the pulsar activity restarts. F: Current state
The evolution of the binary at theLMXB stage is dominated by the angular momentum lossassociated with an efficient magnetic braking. As a result,the present MSP+WD binary is relativistic with an orbitalperiod of only 6.3 h. The orbit, which is highly eccentricafter the SNIb/c explosion, has been circularized by thetidal interactions of the NS with the secondary down to e ∼ − . We used t MS ∝ M − for low-mass stars as deduced fromTable 11.2 of de Loore & Doom (1993) We have assumed that both binary systems, hereafter re-ferred to as H (heavy) and L (light), resulted from the evolu-tion of binaries that originally consisted of two ZAMS stars.Therefore, the striking differences between today’s NS+WDbinaries are the imprint of initial conditions. NS(H) origi-nated in a
25 M ⊙ ZAMS primary star, to be compared witha
15 M ⊙ ZAMS progenitor for NS(L). Even more dramaticis the difference between the ZAMS masses of progenitorsof WDs: M (H) = 4 . ⊙ ≈ M (L) .The ZAMS masses of stars in the H binary are signifi-cantly higher than the masses of their counterparts in theL binary. Consequently, H-binary evolution timescales aresignificantly shorter than L-binary ones. The formation ofthe first RG star in the H binary requires half of the timeneeded for this in the L binary. The second RG in the Hbinary is formed after × yr, which is only ∼ of thetime needed for that in the L binary.As much as
18 M ⊙ is lost by the H binary during the CEstage, nearly double the mass loss by the L binary duringCE evolution.The H-binary orbital evolution during the IMXB-LMXB stage is dominated by the mass loss, the effect ofmagnetic braking being absent, and P orb increases from2 d after SNIb/c explosion to the current value of 8.7 d.The L binary only goes through the LMXB stage. In con-trast to the H case, L-binary orbital evolution has to bedominated by the angular momentum loss caused by themagnetic braking, and this allows P orb to decrease from3 d just after the SNIb/c, down to 6 h today, so that P orb (H) = 35 P orb (L) .
3. Spin-up by accretion
We now briefly summarize the model for the recycling pro-cess, i.e. the spin-up of a progenitor NS by accretion ofmatter from a thin disk leading to the formation of a mil-lisecond pulsar. We do not model the evolution of the binarysystem consisting of the NS and its companion, but only thespin-up of the NS due to accretion of matter. Our approachfollows Bejger et al. (2011) and is applied here to the twopulsars PSR J1614 − M B , obtained for an assumed EOS. Theincrease in total stellar angular momentum J is calculatedby taking into account the transfer of specific orbital angu-lar momentum of a particle accreted from a thin accretiondisk. This proceeds at a distance r from the centre of theNS. It results from the interaction of the disk with the NSmagnetic field and is obtained using the prescription for themagnetic torque by Kluźniak & Rappaport (2007).The spin evolution of an accreting NS results from theinterplay between the spin up due to the accretion of matterassociated with angular momentum transfer and the brak-ing due to the interaction between the NS magnetic fieldand the accretion disk. For details of the implementationand tests, we refer to Bejger et al. (2011). The evolution ofthe total angular momentum transferred to the NS of mass Article number, page 5 of 13 &A proofs: manuscript no. progens_arxiv M and radius R is described by d J d M B = l tot ≡ l ( r ) − l m = l ( r ) − µ r ˙ M − s r c r ! , (1)with the magnetic moment µ = BR . Here, l ( r ) is the spe-cific angular momentum of an infalling particle at the innerboundary of the disk r and l m (magnetic torque dividedby the accretion rate ) describes the interaction of the NSmagnetic field with the accretion disk. One can define threecharacteristic lengths of the problem: the magnetosphericradius r m = ( GM ) − / ˙ M − / µ / , the corotation radius r c = (cid:2) GM/ (4 π f ) (cid:3) / , and the location of the relativisticmarginally stable orbit r ms . Then, the inner boundary r is determined by the condition of vanishing viscous torqueleading to an algebraic equation containing these lengths: f ms ( r ) = (cid:18) r m r (cid:19) / s r c r − ! with f ms ( r ) = 2Ω r d l d r , (2)which is a dimensionless function describing the locationof the marginally stable orbit for rotating NSs in GeneralRelativity, defined by f ms ( r ms ) = 0 .The modeling of the accretion phase consists in adjust-ing the value of the mass M (or equivalently the baryonicmass M B0 ), the magnetic field B i of the progenitor NS,and the mean accretion rate ˙ M (accretion rate averagedover the whole accretion process) so that at the end of theaccretion process, the NS has its parameters, i.e. its mass M (baryonic mass M B ) and magnetic field B , coincidingwith those of a given millisecond pulsar. We also considerthat the post-accretion frequency is equal to its currentlyobserved value f . The validity of this assumption is dis-cussed in Sect. 9. For a given final configuration (i.e. M , P , and B ), a family of sets of three parameters: M (hencethe amount of accreted matter M acc = M B − M B0 ), B i and ˙ M (or equivalently the duration of the accretion phase τ acc = M acc / ˙ M ), is obtained. As a consequence, a choice of τ acc imposes M acc and B i .
4. Equations of state
The EOS of the dense cores of NSs is still poorly known.This is due to, on the one hand, a lack of knowledge ofstrong interaction in dense matter and, on the other hand,deficiencies in the available many-body theories of densematter. This uncertainty is reflected in a rather broad scat-ter of theoretically derived and EOS-dependent maximumallowable mass for NSs (see e.g. Haensel et al. 2007). For-tunately, the recent measurements of the mass of PSRJ1614 − M = 1 . ± .
04 M ⊙ (Demorest et al.2010) and PSR J0348+0432 as M = 2 . ± .
04 M ⊙ (Antoniadis et al. 2013) introduce a fairly strong constraint For simplicity we adopt the notation ˙ M for the accretion rateinstead of ˙ M B . The accretion rate is indeed described in termsof baryon mass M B , since this number is well defined for thebinary system, see Bejger et al. (2011). on M max . It implies that the (true) EOS is rather stiff. Toillustrate the remaining uncertainty, we considered two dif-ferent models for dense matter: DH (Douchin & Haensel 2001). It is a non-relativisticmodel for the simplest possible composition of matter:neutrons, protons, electrons, and muons in β equilib-rium. The energy density functional is based on theSLy4 effective nuclear interaction. The model describesboth the dense liquid core of the NS and its crustin a unified way. This EOS yields a maximum mass M max = 2 .
05 M ⊙ and a circumferential radius at max-imum mass R M max = 10 . km (for a non-rotating con-figuration). BM (Bednarek et al. 2012). It is a nonlinear relativisticmodel that allows for a softening owing to the appear-ance of hyperons at a density of ∼ × g cm − . Thenonlinear Lagrangian includes up to quartic terms in themeson fields. The meson fields σ, ω , and ρ are coupled tonucleons and hyperons, and hidden-strangeness mesonfields σ ⋆ and φ only couple to hyperons. The vector me-son φ generates high-density repulsion between hyper-ons. The EOS is calculated in the mean field approxima-tion. It yields M max = 2 .
03 M ⊙ and R M max = 10 . km(for a non-rotating configuration).Our models for baryonic matter do not include the ∆ resonance (first excited state of the nucleon) as a realconstituent of dense NS matter. In the vacuum, the ∆ is very different from hyperons. The instability of hyper-ons is due to the weak interactions, with a typical halfwidth to the rest-energy ratio for the lightest hyperon Λ : Γ Λ /m Λ c ≈ − . Hyperons in the vacuum are stablewith respect to the strong interaction processes. In con-trast, the ∆ is unstable because of the strong interactions,and Γ ∆ /m ∆ c ≈ . . In dense matter, the rest energy ofa baryon should be replaced by the self-energy (in-mediummass), when taking the interactions with other particlesinto account, as well as the exclusion principle blockingsome processes that were allowed in the vacuum. For exam-ple, when the ∆ is stabilized in dense matter, this meansa decrease in Γ ∆ from 120 MeV in the vacuum to zeroin dense matter. It has been argued that these medium-induced effects increase the threshold density for the ap-pearance of stable ∆ in dense matter above the maximumdensity reachable in NSs (e.g. Sawyer 1972; Nandy 1974;Glendenning 1985). In the relativistic mean field approxi-mation, the threshold density for the ∆ depends stronglyon the in-medium ∆ mass and the completely unknown ∆ coupling constant to the isovector-vector meson ρ . Recently,the question of a possible presence of ∆ in NSs was revivedin Drago et al. (2014a,b). Viewing the uncertainties in thein-medium effects on the ∆ at supra-nuclear densities, how-ever, we do not consider them in NS cores.
5. Magnetic field of a pulsar
Only an estimate of the magnetic field B of a given pulsarcan be obtained if its rotational period P and period deriva-tive ˙ P are known. Assuming that the pulsar is a rotatingmagnetic dipole and that its loss of rotational energy orig-inates in the emission of magneto-dipole radiation alone,one derives the following classical dipole formula (see e.g. Article number, page 6 of 13ortin et al.: Progenitors of PSR J1614 − Haensel et al. 2007): B = s c IP ˙ P π R α , (3)where α is the angle between the rotation and magneticaxes: < α ◦ . This formula describes a spinning dipolein a vacuum, meaning a dipole without plasma in the mag-netosphere. It breaks down for the case of aligned rotator, α = 0 ◦ . However, a more physically sound formula has beenderived by Spitkovsky (2006), approximating solutions ofthe force-free relativistic MHD equations in the magneto-sphere filled with plasma for both aligned and oblique ro-tators. For α ◦ , B = s c IP ˙ P π R
11 + sin α . (4)To test the dependence of the modelling on the estimateof the magnetic field of the pulsar during the recycling pro-cess, two different models are considered in the following,each of them corresponding to a given value of the param-eter β in the equation B = β q P ms ˙ P − √ I R G . (5)In the above formula, P ms is the period in ms, ˙ P − theperiod derivative in units of − s s − , I the moment ofinertia in units of g cm , and R the radius in cm.The dimensionless parameter β takes the following value: Model (a) : β = 1 . . This corresponds to taking thelowest possible value of the magnetic field for the mag-netic dipole obtained from the dipole formula (3) for anorthogonal rotator ( α = 90 ◦ ); Model (b) : β = 0 . derived using Spitkovsky’s for-mula (4) for an aligned rotator ( α = 0 ◦ ).The second model is consistent with the model of an ac-cretion disk presented in Kluźniak & Rappaport (2007), asused in our approach that assumes that the magnetic fieldand the rotation axes are aligned. The first model, whichis widely used in the literature, serves as a reference forcomparison with previous works (e.g. Bejger et al. 2011).With the above formula, one can calculate the value ofthe magnetic field of the pulsar at the end of the spin-upprocess consistently with the EOS, by taking the values of I and R for a NS rotating at the frequency and mass ofinterest. In the last column of Table 1, values B of thecanonical derived magnetic field are quoted, i.e., values ob-tained from Eq. (5) for the magnetic field model (a) andconsidering a NS with canonical values for the radius andthe moment of inertia: R = 10 km and I = 10 g cm .
6. Decay of NS magnetic field in LMXBs andIMXBs
Although observations do not give any evidence of mag-netic field decay during the radio pulsar phase, a substan-tial magnetic field decay (by some four orders of magni-tude) is expected to occur during the “recycling” process ina LMXB, leading to the formation of a millisecond pulsar
Table 3.
PSR J1614 − f =317 Hz of the moment of inertia I (in units of g cm ), theequatorial radius R (in cm), and of the magnetic field B (in G). The last is obtained from Eq. (5) for the differentmodels of magnetic field, the two EOSs, and the extrema andcentral values of the measured mass interval at σ level. EOS Mass ( M ⊙ ) B model I R B DH 1.97 (a) 1.99 1.08 1.98DH 1.97 (b) 1.99 1.08 1.62BM 1.97 (a) 2.19 1.18 1.57BM 1.93 (a) 2.22 1.21 1.48BM 2.01 (a) 2.10 1.13 1.77(Taam & van den Heuvel 1986, for a review see Colpi et al.2001).The theoretical modelling of the accretion-induced de-cay of B is a challenging task (Zhang & Kojima 2006, for areview see Bejger et al. 2011). Considering current uncer-tainties in the modeling of the magnetic field decay accom-panying the spin-up phase of a given millisecond pulsar,we employ phenomenological models, based to some extenton observations of NSs in LMXBs. Taam & van den Heuvel(1986) analysed several LMXBs of different ages. They sug-gested a possible inverse correlation between B and thetotal amount of accreted material. This suggestion waslater confirmed in a study by van den Heuvel & Bitzaraki(1995). After analysing a subset of LMXBs, Shibazaki et al.(1989) proposed to approximate the inverse correlation be-tween B and accreted mass by a formula: B ( M acc ) = B i / (1 + M acc /m B ) . (6)The scaling constant m B controls the pace of dissipation of B with increasing M acc . The values m B = 10 − , − M ⊙ are both consistent with the observed or estimated P, B, and M acc of binary and isolated millisecond radio pulsars(Shibazaki et al. 1989; Francischelli et al. 2002). In the fol-lowing, we adopt the value m B = 10 − M ⊙ unless statedotherwise. The limitations and uncertainties of Eq. (6) havebeen reviewed in detail in Bejger et al. (2011). We useEq. (6) as our baseline description for magnetic field de-cay in LMXBs and IMXBs. However, we also tested otherphenomenological models for the magnetic field decay, suchas the ones introduced in Kiel et al. (2008), Osłowski et al.(2011), and Wijers (1997). Our results depend weakly onthe choice of a specific model (for details see Fortin 2012).The aim of our analysis is to obtain evolutionary tracksthat lead to the formation of a millisecond pulsar whosemass, rotational frequency, and magnetic field are equal tothe currently observed values of PSR J1614 −
7. PSR J1614 − ˙ M and theprogenitor NS mass Five different setups are employed in modelling the accre-tion phase leading to the PSR J1614 − Article number, page 7 of 13 &A proofs: manuscript no. progens_arxiv
Fig. 3. (Colour online) Example of spin-up tracks of accretingNSs leading to the final configuration of PSR J1614 − M ⊙ / yr ), the initialmagnetic field (in G), and the duration of the accretion phaseare indicated. and central values of the σ mass interval are taken intoaccount. Their properties are indicated in Table 3.In Fig. 3 different spin-up tracks, i.e. the change in thespin frequency with the mass of the accreting NS, are shownfor the BM EOS, the model (a) for the magnetic field, andfor a set of initial (pre-accretion) masses, magnetic fields ofthe progenitor NS, and accretion rates. At the end of therecycling process, the final M , B, and f match the present-day parameters of PSR J1614 − years, which is muchshorter than the ones typical of the recycling process. Spin-up tracks 1, 2, and 3 in Fig. 3 are only plotted for illustrativepurposes since the duration of the accretion phase neededto reach PSR J1614 − ′ , which are indistinguishable from oneanother, are obtained for m B = 10 − and − M ⊙ , re-spectively. For a given set of final magnetic field, mass, andfrequency, models with m B = 10 − and − M ⊙ give equalvalues for the initial mass and the accretion rate. As a con-sequence of equation (6), initial magnetic fields are one or-der of magnitude larger for tracks with m B = 10 − M ⊙ than for the ones with m B = 10 − M ⊙ . Track 4, calculatedfor m B = 10 − M ⊙ , is obtained for an initial magnetic field B i ∼ G, consistent with the inferred magnetic field ofisolated radio pulsars B ∼ − G (Manchester et al.2005). For the value m B = 10 − M ⊙ used in Bejger et al.(2011, 2013) and shown by track ′ , B i ∼ G.In the following two quantities are used: the amount ofaccreted matter given by the relation M acc = M B − M B0 and the duration of the accretion phase: τ acc = M acc / ˙ M . To Fig. 4. (Colour online) Mass of accreted matter as a functionof the time needed to spin-up the progenitor NS to the observedproperties of PSR J1614 − M , B, and f ) for the fivemodels indicated in Table 3. The dot corresponds to a spin-uptrack proceeding at a rate equal to the Eddington accretion rate ˙ M Edd ∼ × − M ⊙ / yr ; the accretion timescale is then ∼ Myr.For tracks with τ acc . Myr, ˙ M is greater than ˙ M Edd . reach the mass of PSR J1614 − Fig. 5. (Colour online) Mean accretion rate versus duration ofthe accretion phase for DH and BM EOS needed to reach PSRJ1614 − − also longer for a low-mass progenitor than a high-mass one.Figures 4 and 5 show the relation between the accretiontime τ acc and the amount of accreted matter M acc and themean accretion rate needed to reach PSR J1614 − m B = 10 − and − M ⊙ are indistinguishable.The minimum amount of accreted mass necessary fora NS to become a millisecond pulsar is reached for a fi-nite value of τ acc , which corresponds to a minimum of thefunction M acc ( τ acc ) (see Figs. 4 and 7). For a final config-uration corresponding to PSR J1614-2230, the minimum isobtained for ˙ M min ∼ × − M ⊙ / yr and τ minacc ∼ − Myr. The existence of this minimum is a consequence of thefact that for accretion rates higher than ˙ M min , the mag-netic torque, or equivalently l m in Eq. (1), changes its sign,becoming positive (see Appendix). The value of τ minacc is sig-nificantly less than the timescales presented in Figs. 3 and6, and along all the tracks discussed in the following, l m isalways positive and thus counteracts l ( r ) . For an unreal-istically short duration of the recycling τ acc . Myr, themodel predicts that accretion proceeds at a higher rate thanthe Eddington rate ˙ M Edd ∼ × − M ⊙ / yr (Tauris et al.2012), as indicated in Fig. 4.Model (b) gives lower values for B than dipole model(a) at all stages of the recycling process. Therefore, themagnetic torque that opposes the spin up of the accretingNS is less for model (b), making the recycling process moreefficient. Thus less accreted matter is needed to reach thecurrent pulsar parameters. Moreover, since M acc is lowerfor model (b), so is the mean accretion rate in the recyclingprocess for given accretion time τ acc (see Fig. 5).Using evolutionary arguments one can constrain theminimal birth mass M , at the onset of accretion. For anaccretion phase lasting at most ≃ Myr (see Sect. 2 andTauris et al. 2011), the progenitor NS must have accretedless than ≃ .
06 M ⊙ . Therefore the progenitor NS was bornas massive independently of the EOS: M ≃ . ⊙ . Such aconfiguration is illustrated by the spin-up track 4 in Fig. 3.The value M acc ≃ .
06 M ⊙ should be considered as a lowerlimit, since we do not model the evolution of the binarysystem as in Tauris et al. (2011) or take the possible ejec-tion of matter from the magnetosphere or instabilities inthe accretion disk into account. Moreover, in our model thespin-up is assumed to be maximally efficient i.e., all angularmomentum from the accreted matter is transferred to theNS. If the efficiency of the accretion process is reduced by 50per cent (see discussion in Bejger et al. 2011), then our cal-culations show that the accretion of M acc ≃ .
11 M ⊙ is nec-essary to spin up the NS to PSR J1614 − r but fromthe magnetospheric radius r m and with no magnetic torque,unlike in Eq. (1). Finally, as shown in Bejger et al. (2011),if accretion proceeds from the marginally stable orbit r ms , M acc = 0 .
076 M ⊙ . Therefore the mass M ≃ . ⊙ of theNS at birth is an upper limit.Assuming that the accretion time is well-constrained,one can also estimate the mean accretion rate using thecurrently observed parameters of PSR J1614 − ˙ M > (1 . − . × − M ⊙ / yr , the lower valuecorresponding to the DH EOS. Table 4.
PSR J0751+1807: Analogue of Table 3.
EOS Mass ( M ⊙ ) B model I R B DH 1.26 (a) 1.19 1.19 1.09DH 1.26 (b) 1.19 1.19 0.88BM 1.26 (a) 1.51 1.36 0.80BM 1.12 (a) 1.29 1.37 0.73BM 1.40 (a) 1.74 1.36 0.87
Fig. 6. (Colour online) Similar to Fig. 3, but for the pulsarJ0751+1807.
Modelling the evolution and dynamics of the binary sys-tem, Tauris et al. (2011) calculated that the NS accreted ∼ . ⊙ during 50 Myr, i.e. an averaged accretion rateof ˙ M ≃ × − M ⊙ /yr. The discrepancy between theseresults and ours stems from the fact that we do not modelthe evolution of the binary and of the donor star and, to alesser extent, from a different model for the spin-up phase.
8. PSR J0751+1807: lower bounds on ˙ M and M A similar approach was used for PSR J0751+1807. Proper-ties of the five models are presented in Table 4, and resultsare shown in Figs. 6, 7, and 8. As before, spin-up tracks1, 2, and 3 in Fig. 6 are plotted for illustrative purposes,and tracks 4 and ′ are obtained for m B = 10 − , − M ⊙ respectively. The figures are remarkably similar to the onesobtained for PSR J1614 − − m B = 10 − and − M ⊙ lies in the value of the initial magnetic field.For PSR J1614 − . , . , and .
01 M ⊙ in Figs. 4 and 5. This be-haviour is opposite to what is found for PSR J0751+1807(Figs. 7 and 8). The reason is the non-monotonic depen-dence of the moment of inertia I (and total angular momen- Article number, page 9 of 13 &A proofs: manuscript no. progens_arxiv
Fig. 7. (Colour online) Similar to Fig. 4, but for the pulsarJ0751+1807.
Fig. 8. (Colour online) Similar to Fig. 5, but for the pulsarJ0751+1807. tum of a star rotating at a fixed frequency) on the stellarmass M (Bejger 2013). For the BM EOS, for example, formasses lower than . ⊙ , I increases with M and a highermass corresponds to a larger total angular momentum. Asa consequence, for an equal transfer of angular momentumby accretion, a longer time is needed to spin up the starto a given frequency. The situation is opposite close to themaximum mass (for M > .
85 M ⊙ for BM model) where I decreases with M (see Tables 3 and 4).As Fig. 7 indicates, assuming that the accretion pro-ceeds at a rate lower than the Eddington rate for Gyr
Fig. 9. (Colour online) Mass M vs equatorial radius R eq dia-gram and spin-up tracks of the two pulsars. The colour code andthe numbers labelling the tracks correspond to the ones used inFigs. 3 and 6. For comparison relations between the mass andthe equatorial radius of equilibrium configurations are shown for f = 0 and 300 Hz. The large open-star symbol corresponds tothe PSR J1614 − (see Sect. 2), the progenitor NS of PSR J0751+1807 ac-quired ∼ . − .
10 M ⊙ , which implies that it was bornwith a very low mass, ∼ . − .
30 M ⊙ .The wide range of possible birth masses for PSRJ1614 − M vs.equatorial radius R eq diagram for the two pulsars. Duringthe recycling phase, the two NSs undergo a remarkably dif-ferent evolution in the M − R eq diagram: the equatorialradius of PSR J0751+1807 increases, while the one of PSRJ1614 − ∼ Hz is much more important for the struc-ture and oblateness of a star with the mass . ⊙ andradius km than for a much more compact NS with M close to the maximum mass.
9. The age of the MSPs and their true initial spinperiod
The present spin period of MSPs is expected to be longerthan the period at the end of the spin-up phase. The pul-sars’ spin-down age can be estimated with the formula τ PSR = P/ P . For the values given in Table 1, one ob-tains τ H PSR = 5 . Gyr and τ L PSR = 7 . Gyr. These valuesshould be treated as upper limits on τ PSR since the mea-sured ˙ P is larger than the true ˙ P because of the trans-verse motion of the binary system (Shklovskii 1970). Thiseffect turns out to be negligible for the low-mass pulsar(about ∼ ). It may be crucial for the high-mass pulsar Article number, page 10 of 13ortin et al.: Progenitors of PSR J1614 − since the true ˙ P could be more than one order of magni-tude smaller than the observed one (Bhalearo & Kulkarni2011). However this result is very sensitive to the uncertain-ties in the determination of the distance and proper motionof PSR J1614 − − P orb = 8 . d. It is far from being obvious for the L-binarywith P orb = 6 . h, but Bassa et al. (2006) found “a sur-prising lack of evidence for any heating”. Therefore, the ageof the MSPs in both the L and H binaries can be obtainedfrom modelling the cooling of their WD.Using their own observations of WD(L) Bassa et al.(2006) constrained its effective temperature T eff , radius,and the composition of its atmosphere. They found T eff ≈ K for the most likely pure He (or stronglyHe-dominated) atmosphere. Such a composition of the at-mosphere is a puzzle (see discussion in Bassa et al. 2006).We applied a cooling curve obtained for a .
15 M ⊙ He-coreWD with a He envelope calculated by Hansen & Phinney(1998) and obtained τ L WD ≈ . Gyr. We assume that thisis the age of PSR J0751+1807.The WD companion of PSR J1614 − . ⊙ , indicate thatthe WD(H) has a C/O core and an H atmosphere. Cool-ing sequences calculated by Chabrier et al. (2000) predictthat such a WD reaches the inferred absolute magnitude inthe R-band M R ≈ . obtained in Bhalearo & Kulkarni(2011) after τ H WD ≈ . Gyr. This is the age we assume forPSR J1614 − ˙ P and assuming a braking index n = 3 , which is consistent with the observed populationof MSPs (Kiziltan & Thorsett 2010), the initial period of apulsar at the beginning of the slowing down phase can be es-timated: P init = P obs h − ( n − τ WD ˙ P /P i / ( n − , where P obs is the currently observed period. Here, P init is shorterthan P obs by about for both pulsars. For the lightpulsar, the amount of accreted matter necessary to spin itup not to P obs but to P init , which is ∼ larger. As aconsequence, its progenitor NS could be born with a mass ∼ . ⊙ .
10. Discussion and conclusions
We have presented the modelling of the accretion-inducedspin-up phase undergone by two millisecond pulsars: themassive PSR J1614 − − . ⊙ and therefore was very close to thecurrently measured value. This value is ∼ . ⊙ higherthan the NS birth mass obtained in Tauris et al. (2011),Tauris et al. (2012), and Lin et al. (2011), in which the ac-cretion of M acc = 0 . ⊙ is required, while for our spin-up models, ∼ .
06 M ⊙ is sufficient. This large discrepancypartly comes from differences in the modelling of the spin-up phase but mostly from the fact that we do not modelthe whole evolution of the binary system.Equation 14 in Tauris et al. (2012) yields a minimumvalue of M acc . This value is remarkably similar to our valueof M acc . Our calculations show that this estimate is in-dependent of the EOS and of the assumed model for themagnetic field and its accretion-induced decay. A reason-able estimate of the accreted mass could be obtained onthe basis of the current parameters of the pulsar ( P , ˙ P , M ). In our approach the evolution depends on the meanaccretion rate, which is not assumed to be the equilibriumone. However, there is a minimum value of the accretedmass M minacc needed to spin the star up to the observed ro-tational period. In our model, M minacc ≃ .
05 M ⊙ (for detailssee Appendix) and is obtained for a given τ minacc (see e.g.Figs. 4 and 7). The mean accretion rate is then given by ˙ M min = M minacc /τ minacc . All these values for the amount of ac-creted matter are actually lower limits since processes in theNS magnetosphere and in the accretion disk at the originof the ejection are not included.In our approach (Bejger et al. 2011), relativistic effectsare taken into account by introducing dimensionless func-tion f ms in Eq. (2). The role of GR for the radius r isnegligible, of the order of 1% (for details see Appendix inBejger et al. 2011). However, the specific angular momen-tum of a particle l ( r ) calculated in the framework of GRis larger than obtained in Newtonian theory. The increasein l due to GR is maximal at a marginally stable orbit (bya factor √ , see Kluźniak & Wagoner 1985), and in casesconsidered in this paper ( r > r ms ) is ∼ . As a con-sequence, disregarding GR effects results in less effectivespin-up; the mass needed to reach given frequency is there-fore higher by ∼ .The progenitor of PSR J0751+1807 was itself born witha very low mass, which as we estimate, could be as lowas .
05 M ⊙ . Considering that the pulsar spun down afterthe recycling, the mass of the progenitor NS is lowered to . ⊙ .The Roche lobe decoupling phase (RLDP), suggested re-cently by Tauris (2012) and Tauris et al. (2012) is relatedto some additional quasi-spherical accretion. Although wedid not model the RLDP, we estimated the additional M acc (RLDP) as follows: When assuming a slow RLDP witha timescale of ≃ Myr and a pre-RLDP accretion rate ˙ M ∼ − M ⊙ / yr in the case of PSR J0751+1807, theupper limit on M acc (RLDP) is 0.005 M ⊙ . Because of therapidly decreasing ˙ M and because some matter is ejectedduring the propeller phase, M acc (RLDP) is likely to besmaller. Considering that before the RLDP phase, a pul-sar should have been spun up to a higher frequency thanthe one observed now, by accreting more mass before thisphase, we can therefore conclude that the amount of mat-ter accreted during the RLDP is negligible compared to thetotal mass accreted by the NS during the recycling process. Article number, page 11 of 13 &A proofs: manuscript no. progens_arxiv
The wide range of NS birth masses, . ⊙ − . ⊙ derived from our simulations, agrees with recent modellingsby Ugliano et al. (2012) and Pejcha & Thompson (2014) ofsupernova explosions of a large set of massive stars progen-itors and metallicities in spherical symmetry. Acknowledgements.
We are grateful to Antonios Manousakis, JanuszZiółkowski, Hans-Thomas Janka, and an anonymous referee for read-ing the manuscript and for helpful remarks and suggestions. Wealso acknowledge helpful remarks of participants of the Comp-Star 2011 Workshop (Catania, Italy, 9-12 May, 2011). This workwas partially supported by the Polish NCN research grant no.2013/11/B/ST9/04528 and by the COST Action MP1304 ”NewComp-Star”.
Appendix: Approximate solution
The properties of the solutions of Eqs. (1) and (2) allowus to determine an approximate solution for our model. Itcontains only the parameters of the final configuration: B , f , M and a given mean accretion rate ˙ M and enables theproperties of the pre-accretion NS to be estimated. Thespin-up equation (1) can be factorized as d J d M b = p GM r c · λ ( f ) , (7)where λ ≡ l tot / √ GM r c is the ratio of total specific angularmomentum of an accreted particle to its value at the corota-tion radius. The main dependence on rotational frequencyis included in √ r c ∼ f − / .The ratios of the characteristic lengths describing thesystem ( r m , r c , r ) do not change significantly along a spin-up track, except for a small region of very slow rotation.Therefore, one can assume that λ ( f ) is constant along spin-up track as the Eq. (2) depends on r m /r and r c /r . Thisassumption holds with an accuracy of about 20%. However, λ is a rather sensitive function of the mean accretion rate.For example, for the set of evolutionary tracks with differentaccretion rates presented in Figs. 1 and 2, the value of λ changes by more than one order of magnitude.Neglecting the change in the NS moment of inertia I (which is a good assumption for configurations close to themaximum mass and/or for relatively small amount of ac-creted mass), we obtain the formula ∆ M ≃ J f l f , with l f = l ( r f ) − l m , (8)where J f is the angular momentum, l ( r f ) is specific angu-lar momentum of an accreted particle, and l m is magnetictorque divided by ˙ M , all three quantities corresponding tothe final (presently observed) state of the pulsar.Since the magnetic torque is proportional to / ˙ M = τ / ∆ M in the limit of a low accretion rate (large τ ),the angular momentum transferred to the star is l f = l ( r ) · (1 − A/ ˙ M ) = l ( r ) · (1 − Aτ / ∆ M ) , the depen-dence on τ is almost linear, as shown in Figs. 4 and 7: ∆ M ≃ J f l + A · τ, with A = µ r l ( r ) − s r c r ! . (9)A simple estimate of the amount of accreted matter neededto spin up a NS to a given configuration can be obtained Fig. 10. (Colour online) Mass of accreted matter as a functionof the time needed to spin-up the progenitor NS to the observedstate (i.e. M , B, and f ) for PSR J1614 − from Eq. (9). The radius r f at the inner boundary of thedisk can be calculated by solving the simple algebraic equa-tion Eq. (2) for values of f , B , and M corresponding theones of the final configuration. Then l ( r f ) can be deter-mined by the analytic formula given in Bejger et al. (2010).Examplary results based only on the present-day parame-ters are shown in Fig. 10. For a given duration of the ac-cretion phase, one can then derive the amount of matteraccreted by the NS to reach its current configuration. Thenthe birth mass can be simply derived and the pre-accretionmagnetic field is given by Eq. (6).For high ˙ M , the value of A depends sensitively on thesolution of Eq. (2), and the function A ( ˙ M ) changes its signat the point ˙ M min corresponding to (cid:18) r r c (cid:19) / = 23 , i . e ., r c = 1 . r . (10)For ˙ M > ˙ M min , the accreted mass ∆ M is a decreas-ing function of τ acc (see inserts in Figs. 4 and 7). In ourcase, ˙ M min ≃ × − M ⊙ / yr and thus is more thanone order of magnitude lower than the Eddington limit ≃ × − M ⊙ / yr . The minimum mass needed to spin apulsar up to its observed frequency depends on the valueof the specific angular momentum of a particle at the coro-tation radius and is given by ∆ M min = 0 . J f /l (0 . r c ) .This quantity does not depend on the accretion rate. References
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