Implications of the measured parameters of PSR J1903+0327 for its progenitor neutron star
^{1}
N. Copernicus Astronomical Center, Polish Academy of Sciences, Bartycka 18, 00716 Warszawa, Poland
email: bejger@camk.edu.pl; fortin@camk.edu.pl; haensel@camk.edu.pl; jlz@camk.edu.pl
^{2}
LUTh, UMR 8102 du CNRS, Observatoire de Paris, 92195 Meudon Cedex, France
Received: 13 June 2011
Accepted: 19 October 2011
Context. The millisecond pulsar PSR J1903+0327 rotating at 465 Hz has the second highest precisely measured mass (1.67 M_{⊙}) and a weak surface magnetic field (≃2 × 10^{8} G). It is located in the Galactic plane, bound in a highly eccentric (e = 0.44) orbit in a binary system with a solarmass mainsequence star. These observational findings pose a challenge for the theory of stellar evolution.
Aims. Using the intrinsic parameters of PSR J1903+0327 evaluated from radio observations (mass M, rotation period P, and magnetic field B deduced from P and Ṗ) and a model of spin evolution during the “recycling” phase (spinup by accretion from a lowmass companion lost afterwards) that takes into account the accretioninduced magnetic field decay, we aim to calculate the mass of its neutron star progenitor, M_{i}, at the onset of accretion. In addition, we derive constraints on the average accretion rate Ṁ and the preaccretion magnetic field B_{i}. We also seek for the imprint of the poorly known equation of state of dense matter at supranuclear densities on the spinup tracks and the progenitor neutron star.
Methods. Spinup is modeled by accretion from a thin magnetized disk, using the magnetictorque diskpulsar coupling model proposed by Kluźniak and Rappaport. We adopt an observationally motivated model of the surface magnetic field dissipation caused by accretion. We consider three equations of state of dense matter, which are consistent with the existence of 2.0 M_{⊙} neutron star. Orbital parameters in the accretion disk are obtained using the spacetime generated by a rotating neutron star within the framework of general relativity.
Results. Constraints on the progenitor neutron star parameters and the accretion itself are obtained. The minimum average accretion rate should be higher than 2−8 × 10^{10} M_{⊙} yr^{1}, the highest lower bound corresponding to the stiffest equation of state. Allowed B_{i}dependent values of M_{i} are within 1.0 − 1.4 M_{⊙}, much lower than the oversimplified but widely used B ≡ 0 result, where one gets M_{i} > 1.55 M_{⊙}.
Conclusions. The influence of magnetic field in the “recycling” process is crucial – it leads to a significant decrease in the spinup rate and higher accreted masses, in comparison to the B = 0 model. The estimated initial neutronstar mass depends on the assumed densematter equation of state. We also show that the otherwise necessary relativistic corrections to the Newtonian model of Kluźniak and Rappaport, related to the existence of the marginallystable circular orbit, can be neglected in the case of PSR J1903+0327.
Key words: stars: neutron / equation of state / accretion, accretion disks / dense matter
© ESO, 2011
1. Introduction
The discovery of PSR J1903+0327, which was the first millisecond radio pulsar found in a binary system with a 1.03 M_{⊙} mainsequence (MS) companion in an eccentric (e = 0.44) 95day orbit, poses a challenge to formation theories of millisecond pulsars (Champion et al. 2008). On the one hand, its timing parameters – spin frequency f = 465 Hz and spin period derivative Ṗ = 1.88 × 10^{20} s s^{1} – are nothing unusual for millisecond pulsars. On the other hand however, the high eccentricity of the orbit and the nature of the companion (mainsequence star), as well as its location in the vicinity of the Galactic plane make it unique. Pulsarclock stability and the high eccentricity of the orbit enabled the pulsar mass measurement to be determined (by means of the Shapiro delay) with formidable precision. An analysis of the timing data obtained from 2006 December through to 2010 January using the 305m Arecibo radio telescope and the 105m Green Bank Telescope, found that the pulsar mass is 1.67 ± 0.021 M_{⊙} (at the 99.7% confidence limit) and showed strong evidence that effects other than spacetime curvature, e.g., stellar winds or tidal forces acting on MS companion, are negligible (Freire et al. 2011).
According to the current theory of neutronstar (NS) evolution, millisecond radio pulsars (P < 10 ms) originate from “radiodead” pulsars via the accretioncaused spinup in lowmass Xray binaries (LMXB, see Alpar et al. 1982; Radhakrishnan & Srinivasan 1982). This socalled “recycling” process is believed to facilitate the spinup from the initial ~0.1 Hz frequency to ~500 Hz in ~10^{9} yrs and is associated with accretion of ~0.1 M_{⊙}. This idea was corroborated by the detection of millisecond Xray pulsations in LMXBs, which were interpreted as the manifestation of rotating and accreting NSs (Wijnands & van der Klis 1998). These millisecond Xray pulsars are thought to become radio millisecond pulsars after the accretion process expires. Radio millisecond pulsars are extremely stable rotators, with Ṗ ~ 10^{20} − 10^{19} s s^{1}. The surface magnetic fields estimated from the timing properties are three – four orders of magnitude weaker than those in normal radio pulsars, for which B ≃ 10^{12} G. This is explained either by the “burying” of the original magnetic field under a layer of accreted ~0.1 M_{⊙} material (BisnovatyiKogan & Komberg 1974; Taam & van den Heuvel 1986; Cumming et al. 2001) or/and by the Ohmic dissipation of electric currents in the accretionheated crust (Romani 1990; Geppert & Urpin 1994). The “recycling” in LMXBs is a particularly efficient mechanism in dense stellar systems such as globular clusters and is in accordance with specific statistics of radio millisecond pulsars (Lorimer 2008). Out of a total of 213 millisecond pulsars, some 100 (i.e., nearly half) are indeed located within binaries, while among the 1850 radio pulsars, only 141 (i.e., 8%) are in binaries. Moreover, out of all radio millisecond pulsars, more than half (130) are found in 26 Galactic globular clusters! Finally, as much as 40% of the 130 globularcluster radio millisecond pulsars are in binaries.
Efficient spinup to the kHz frequencies is possible because the lowmass companion remains for ~10^{8} − 10^{9} yrs in the postMS phase, overflowing its Roche lobe and feeding the accretion disk around a NS. Consequently, tidal friction within the extended companion has ample time to circularize the orbit. Only an additional dynamic perturbation involving a third star, an occurrence that is not so rare within a globular cluster, can make the orbit highly eccentric or even tear the system apart. This may be the origin of eccentric binaries as well as some isolated millisecond pulsars, both in globular clusters and the Galactic disk.
However, PSR J1903+0327 does not fit the above picture because neither its orbit is circular, nor its companion a postMS star. It is thus a first specimen of a new group; constructing a viable scenario of its formation turns out to be a difficult challenge. We briefly and critically review the principal scenarios that have been proposed since the discovery of the PSR J1903+0327 binary in Sect. 2. All but one are excluded although the surviving triple system scenario is not free of its own problems either (Freire et al. 2011; Portegies Zwart et al. 2011). Nevertheless, it is clear that PSR J1903+0327 was not “recycled” by its current MS companion. In the following, we focus on the LMXB stage of the pulsar evolution that certainly preceded the formation of the presently observed binary. We consider the millisecondpulsar formation model in order to (hopefully) deduce the parameters of the preaccretion progenitor NS and to place constraints on the poorly known equation of state (EOS) of dense matter.
We provide a brief introduction to accretion spinup scenario and the dissipation of pulsar magnetic field due to accretion in Sects. 3 and 4. In Sect. 5, we describe the main elements of the Kluźniak & Rappaport (2007) magnetictorque model (hereafter referred to as KR) acting on a NS accreting from a thin accretion disk. We report on efforts to extend the Newtonian KR model to include the effects of the spacetime curvature, especially to take into account the existence of the marginallystable circular orbit predicted by General Relativity. In our “recycling” simulations, we use three EOS of dense matter consistent with the existence of a 2.0 M_{⊙} NS (Demorest et al. 2010) – these equations are briefly described in Sect. 6, while the results of our simulations are presented in Sect. 7. As a final result, we place constraints on the average accretion rate during the spinup stage, the initial magnetic field, and the mass of the progenitor NS. In Sect. 8, we summarize our main results. An Appendix contains a terse overview of the influence of the spacetime curvature effects, something that was neglected in the original KR model, on the spinup tracks of PSR J1903+0327.
2. Proposed scenarios for the formation of the PSR J1903+0327 binary
Three formation scenarios, denoted below as I − III, were previously proposed in the discovery paper (Champion et al. 2008), and the first scenario was developed further by Liu & Li (2009). These three scenarios were ruled out by new observational data combined with theoretical modeling and a fourth scenario was advanced by Freire et al. (2011) and Portegies Zwart et al. (2011). These scenarios, together with their critique, are briefly discussed below:
I. Rapid rotation at birth
The pulsar was born spinning rapidly in a corecollapse supernova, in a binary system in the Galactic disk, with a MS companion, the supernova kick making the orbit strongly eccentric (Liu & Li 2009). A spin frequency of 465 Hz was reached as a consequence of accretion from the fallback disk, which requires less than 10^{3} − 10^{4} yrs before the accretion (or rather hyperaccretion – the mean rate is ~10^{4} M_{⊙} yr^{1}) stops and the pulsar becomes an active radio emitter. This scenario was considered unlikely by Champion et al. (2008), because of the very low surface magnetic field (B ~ 10^{8} G) implied by the measured value of Ṗ. Liu & Li (2009) proposed that the low magnetic field value was caused by its dissipation associated with accretion. However, they used a model of a magnetic field dissipation developed originally for slow accretion in an LMXB, which is not valid for a hyperaccreting newly born NS. Additional arguments, in particular observational ones, against this scenario can be found in Freire et al. (2011) and Portegies Zwart et al. (2011), which is therefore unacceptable.
II. Hierarchical triple system
The pulsar was recycled in a LMXB and is currently a member of a hierarchical triple system containing a white dwarf (inner binary with the pulsar) and a MS star (outer binary). The high eccentricity of the pulsar orbit is generated by the Kozai resonance (Kozai 1962) between the inner and outer binaries. The scenario is ruled out by the precise measurement of the optical spectra of the MS companion of the pulsar permitting the calculation of the companion radial velocity, which turned out to be consistent with predictions based on the orbital parameters determined from the pulsar timing. The optical data lead to an independent estimate of the mass ratio in the binary (Freire et al. 2011). Finally, there was no time dependence of the eccentricity e in the TEMPO2 timing analysis, while e growth is expected owing to the Kozai resonance in the triple hierarchical system (Gopakumar et al. 2009). The scenario should therefore be ruled out as inconsistent with observations (see also Freire et al. 2011; Portegies Zwart et al. 2011)
III. Pulsar spun up in a LMXB and then ejected into the Galactic disk together with a newly captured MS companion
The pulsar was spun up in an LMXB within a globular cluster and then exchanged the evolved companion in a process of interaction with a MS star in the cluster core, its binary orbit becoming very eccentric. The kickoff in the threebody interaction was sufficient to eject the final NSMS binary out of the globular cluster and into the Galactic disk. However, measurements of the proper threedimensional velocity of the binary allowed to track its position back in time (Freire et al. 2011). The binary was found to always be within 270 pc from the Galactic disk and farther than 3 kpc from the Galactic center. Therefore, an exchange interaction (where an evolved companion in a LMXB is exchanged for a MS star), which is probable in a dense stellar environment such as a globular cluster or the Galactic center, is so unlikely that it leads to the rejection of the scenario (Freire et al. 2011).
IV. Pulsar in a triple (tertiary) system with two MS stars of different masses, the pulsar being spunup by accretion from the evolved, more massive companion that was afterwards expelled from the tertiary to make it a binary
This scenario was proposed by Freire et al. (2011) and Portegies Zwart et al. (2011). A starting point is a tertiary composed of a massive star and two MS stars of different masses. The massive star collapses, giving birth to a pulsar, the tertiary still being bound. The initially more massive MS star (MS1) evolves forming a LMXB with the pulsar and spinning it up by accretion. After a substantial mass loss, MS1 is removed from the system owing to the ablation/accretion induced by the interaction with the millisecond pulsar or ejected from the system by means of the threebody interactions (Freire et al. 2011). This leaves a system as the observed one, with a 1.03 M_{⊙} companion still at the MS stage, in an eccentric orbit. The ablation/accretion removal of the companion was previously proposed as a mechanism producing isolated recycled pulsars in the Galactic disk. Ablation by the NS radiation indeed operates in the PSR B1957+20 binary (Black Widow), whose companion mass was reduced to 0.025 M_{⊙} (Fruchter et al. 1988; Kluźniak et al. 1988). However, details of the ablation process are rather uncertain and the timescale needed to completely vaporize the companion can be longer than the Hubble time (Levinson & Eichler 1991). Alternatively, the expulsion of the lowermass companion owing to the chaotic character of a threebody interaction in the triple system is proposed. We note, however that references quoted in Freire et al. (2011) do not seem to provide an accurate explanation of PSR J1903+0327: Hut (1984) assumes equal masses for the three bodies, and Phillips (1993) studies the reflection of the pulsar radio beam from the swarm of asteroids orbiting the pulsar and derives an upper bound of ~10^{4} M_{⊙} to the mass of the asteroidlike material orbiting the Vela pulsar within 1 AU. In our opinion, the scenario presented by Freire et al. (2011) is not yet properly supported by reliable quantitative estimates. In Portegies Zwart et al. (2011), both the evolution and the dynamics of the NS+MS1+MS2 system are studied in detail. Several competing channels of evolution for the tertiary are compared using advanced numerical simulations. The available parameter space leading to the observed PSR J1903+0327+MS2 binary is estimated and the birthrate of these binaries in the Galactic disk is found to be acceptable. Portegies Zwart et al. (2011) highlighted the need to (observationally) find a “missing link” in the scenario: a wide LMXB orbited by a tertiary lowmass MS star. It is clear that the discovery of such a system would strongly fortify the triplestar scenario.
Four scenarios, reviewed critically above, have been proposed until now. Only one of them cannot be immediately ruled out by observations, but even this one lacks a solid quantitative basis. However, it is clear that PSR J1903+0327 has been “recycled” in order to explain its present spin period, low surface magnetic field, and mass, which is much higher than the average 1.4 M_{⊙}. From now on, we focus exclusively on the spinup phase of the pulsar evolution, to deduce its preaccretion parameters.
3. Spinup by accretion in LMXBs
We consider a binary consisting of a young radio pulsar (of a “canonical” surface magnetic field B_{i} ≃ 10^{12} G, and a spin period of a fraction of a second) and a lowmass MS companion. In a few million years, it spins down by means of magnetic dipole braking down to the period of a few seconds, without suffering any significant magnetic field decay. Consequently, it crosses the radio pulsar death line and disappears as a pulsar. On a much longer timescale of a billion years, the lowmass companion enters the red giant phase and fills its Roche lobe. This entails the mass transfer onto the NS via an accretion disk. The binary system becomes a LMXB, remaining in this stage for 10^{8} − 10^{9} yrs. Accretion of the plasma onto a NS increases its mass, while accelerating its rotation, as well as inducing the decay (dissipation) of its surface magnetic field.
The accretion rate is expressed in terms of the baryon mass M_{b}, since it is the parameter that can be uniquely determined for the binary system; indeed M_{b} is related to the star’s total baryon number N_{b} by M_{b} = N_{b}m_{0}, where m_{0} is the mass per nucleon of the ^{56}Fe atom. The increase in M_{b} is straightforward and proportional to the accretion rate – at time t (measured by a distant observer) it is denoted as Ṁ_{b}(t). Assuming that accretion starts at t_{i}, the integrated (total) baryon mass increase is (1)Since the detailed history of accretion is unknown, we treat Ṁ_{b} as a constant, such that ΔM_{b}(t) ≈ Ṁ_{b}(t − t_{i}). While M_{b} is a very important global stellar parameter, the quantity that is actually measured is the gravitational mass M. We define its increase as (2)It is found that ΔM depends on ΔM_{b} according to the relation between dM, dM_{b}, and the total angular momentum J (see e.g., Friedman et al. 1988) (3)where u^{t}, the time component of fluid 4velocity as seen by a distant observer, has the meaning of chemical potential (divided by m_{0}c^{2}). For ΔM(t), one then obtains (4)using the total angular momentum J evolution equation (5)where l_{tot} includes the angular momentum transferred to the star by the infalling matter and the influence of the magnetic torque (for details, see Sect. 5). The values of the gravitational mass M and spin frequency f = 1/P for a given M_{b} and J are calculated for stationary rigidly rotating twodimensional NS models (Bonazzola et al. 1993), with the rotstar code implementation from the numerical relativity library LORENE^{1}.
4. Decay of neutronstar magnetic field in LMXBs
Millisecond pulsars have typically a low surface polar magnetic field of strength B_{p} ~ 10^{8} − 10^{9} G (see e.g., Lorimer 2008), but observations do not provide any evidence of the B_{p} decay during the radiopulsar phase. However, a substantial B_{p} decay (of some four orders of magnitude) is thought to occur during the accretion “recycling” in a LMXB, leading to the formation of a millisecond pulsar (Taam & van den Heuvel 1986; for a review see Colpi et al. 2001). We relate B_{p} to measured values of P and Ṗ by a standard formula, B_{p} = 3.2 × 10^{19}^{(}PṖ 1/s^{)}^{1/2} (Manchester & Taylor 1977). This assumes a stellar moment of inertia I = 10^{45} g cm^{2} and radius R = 10 km, and enables one to replace the measured value of Ṗ by B_{p}.
4.1. Decay of magnetic field – theory
Theoretical modeling of accretioninduced decay of B_{p} turns out to be a challenging task. The original idea that B_{p} decays in close binaries because it is “buried” (“screened”) by the accreted matter was proposed by BisnovatyiKogan & Komberg (1974). Romani (1990) suggested that a combination of crustal heating due to accretion, accelerating Ohmic dissipation, and advection of magnetic field lines from the poles towards the equator could explain the B_{p} decay. Some other authors focused on the realistic modeling of the acceleration of the Ohmic dissipation of the crustal B caused by the heating induced by accretion (Geppert & Urpin 1994; Urpin & Geppert 1995). The decay of B_{p} due to the diamagnetic screening by the accreted layer was studied in Zhang et al. (1994) and Zhang (1998). Dependence of the screening of crustal magnetic field on the accretion rate was studied in more detail later by Cumming et al. (2001). These authors considered in detail the interplay between the advection of magnetic field and its Ohmic diffusion, and used realistic microscopic models of the outer “ocean” (molten crust) and the crust. Magnetohydrodynamical simulation of accretionburial of B_{p} have also been performed by several authors (e.g. Payne & Melatos 2004, 2007; see also the study of Wette et al. 2010, and references therein). This very brief and incomplete (especially as far as the references are concerned) review illustrates the theoretical effort to explain the decay of B_{p} during the accretiondriven recycling of the millisecond pulsars. Unfortunately, no reliable and robust scenario based on realistic microphysics combined with complete magnetohydrodynamical treatment, and consistent with astronomical observations, is available today. Therefore, in what follows we limit our description of the B_{p} decay to simple phenomenological models, based to some extent on observations of three populations: old radio pulsars, accreting binary neutron stars, and millisecond pulsars. Our basic assumption (which is widely accepted) is that these populations are linked by the recycling process, which in turn is driven by accretion onto a magnetized neutron star in a close binary.
4.2. Decay of magnetic field – phenomenology (with some observational basis)
Taam & van den Heuvel (1986) analyzed a set of LMXBs of different ages and therefore different amounts of accreted mass. They suggested that there is a possible inverse correlation between B_{p} and the (estimated) total amount of accreted material. Their conclusion was confirmed in a later study by van den Heuvel & Bitzaraki (1995). Shibazaki et al. (1989) presented more detailed arguments based on a subset of LMXBs that enabled this inverse correlation to be quantified as (6)where B_{i} is the initial (preaccretion) magnetic field, B_{p}(ΔM_{b}) is the magnetic field after is accreted by the neutron star, and m_{B} is a constant setting the scale of dissipation of B_{p} with increasing ΔM_{b}. We note that Shibazaki et al. (1989) do not advocate any physical model leading to Eq. (6). They report astrophysically interesting bounds on m_{B}, resulting from the application of Eq. (6) to some LMXBs, Xray pulsars, and millisecond pulsars. In particular, for m_{B} ≳ 10^{3} M_{⊙}, the evolution at an Xray pulsar stage proceeds along the equilibrium spinup line, with accretion (advection) torque balanced by the magnetic torque (Ghosh & Lamb 1979; Fig. 1 of Shibazaki et al. 1989) – rapid millisecond rotation cannot be obtained in this way. If, however, m_{B} ≲ 10^{4} M_{⊙}, then the evolutionary tracks in the log B_{p} − log P plane are consistent with measured pairs of B_{p} and P for binary and isolated millisecond radio pulsars. Finally, taking m_{B} ~ 10^{4} M_{⊙} allows a reasonable representation of the inverse correlation between B_{p} and ΔM_{b} noted by Taam & van den Heuvel (1986).
One has to be aware that Eq. (6) is based on a limited and uncertain set of data referring to LMXBs. It is clear that Eq. (6) is too simplistic to describe magnetic field decay for all kinds of accreting neutron stars. It is natural to expect that the decay of B_{p} depends not only on ΔM_{b}, but also on Ṁ, which is decisive in the heating of the neutronstar interior (Wijers 1997; Urpin & Geppert 1996; Urpin et al. 1998; Cumming et al. 2001). Specifically, Wijers (1997) demonstrated that the decay law B_{p} ∝ 1/ΔM_{b} is inconsistent with a broader set of available data for accreting neutron stars in both Xray binaries and recycled millisecond pulsars.
In spite of the limitations and uncertainties discussed above, we adopt Eq. (6) as our baseline description of the B_{p} decay in LMXBs and compare it with other proposed phenomenological formulae for the accretioninduced B_{p} decay to assess the modelindependent features of our basic results.
For example, Wijers (1997) discussed the square dependence on ΔM_{b}/m_{B} in the denominator of Eq. (6) as well: (7)In this case, one has to use a lower value of m_{B} ≃ 10^{3} ÷ 10^{2} M_{⊙}, which allows to form a millisecond pulsar before the field decays to very low values.
An exponential decay of B_{p} was employed by Kiel et al. (2008) and Osłowski et al. (2011) to be (8)where B_{min} = 10^{8} G is the assumed minimal residual magnetic field, to reproduce the observed P − Ṗ distribution by means of the population synthesis studies (Osłowski et al. 2011). We discuss briefly how the final results depend on the assumed magnetic field decay model in Sect. 7.3.
5. Spinup by disk accretion onto a magnetized neutron star
We assume that the evolution of an accreting NS can be represented as a sequence of stationary rotating configurations of increasing baryon mass. We use the KR formalism to determine the circular orbit r_{0}, from where the accretion effectively takes place – the radial inner boundary of the Keplerian accretion disk in which the viscous torque is nonvanishing. Following KR, we define the corotation radius (i.e., the radial distance at which the Keplerian orbital angular frequency Ω_{K} is equal to the rotation frequency of a central star, ω_{s} = 2πf), the magnetospheric radius (where μ = B_{p}R^{3} approximates the stellar magnetic dipole moment), the magnetictocorotating radius ratio ξ ≡ r_{m}/r_{c}, and the socalled “fastness” parameter . In addition, we include relativistic effects – inevitable in the spacetime generated by a rotating NS – especially the existence of the marginally stable orbit r_{ms} and denote the Schwarzschild gravitational radius as r_{s} ≡ 2GM/c^{2}, and the corotation radius in r_{s} units, β ≡ r_{c}/r_{s}.
We then solve the equation that corresponds to the vanishing of the viscous torque to determine the disk inner boundary, namely (9)where l is the specific angular momentum of a particle calculated in a relativistic way. To avoid the arduous calculations of the NS spacetime that would be needed to obtain the specific angular momentum l at a given disk radius r in every timestep of the evolution, we exploit the result of Bejger et al. (2010), who provide an approximation of the value of l(r), defined as (10)with an approximate value of the particle orbital velocity v(11)This approximate value of l(r) deviates by less than one per cent from the true value for spin frequencies and masses similar to PSR J1903+0327 measurements (for details see Bejger et al. 2010). Equations (9) − (11) yield an algebraic equation for r_{0} of (12)with a dimensionless function f_{ms} defined as (13)which for l(r) given by Eqs. (10) and (11) equals (14)where and .
In general, Eqs. (12) and (14) allow us to calculate r_{0} in the “recycling” process of even very rapidly rotating and massive millisecond pulsars, when the existence of a relativistic marginallystable orbit cannot be neglected i.e., when the KR condition of r_{0} ≪ r_{ms} is no longer valid. The value of r_{ms} corresponds to the solution of (15)which is determined by assuming that f_{ms}(r_{ms}) = 0. The validity of employing this general, refined approach in the case of PSR J1903+0327 is studied in the Appendix.
To calculate the increase in the total stellar angular momentum J, we take into account the transfer of specific angular momentum l_{0} ≡ l(r_{0}) and the KR prescription for the magnetic torque (16)
6. Equations of state
The EOS of dense cores of NSs remains poorly constrained. This is due to, on the one hand, a lack of knowledge of strong interactions in dense matter, and on the other hand, deficiencies in the available manybody theories of dense matter. This uncertainty has been reflected as a rather broad scatter in the theoretically derived and EOS dependent maximum allowable masses for NSs, M_{max}(EOS) (see, e.g. Haensel et al. 2007). Fortunately, the measurement of the mass of PSR J16142230, of 1.97 ± 0.04 M_{⊙} (Demorest et al. 2010), introduces a rather strong constraint of M_{max} ≥ 2.0 M_{⊙}. This means that the true EOS is rather stiff. To illustrate a remaining uncertainty in the stiffness, we considered three different models of the EOS. In all cases, the simplest composition of matter was assumed – neutrons, protons, electrons, and muons in βequilibrium (npeμ):

DH
This model of Douchin & Haensel (2001) isnonrelativistic and its energy density functional is based on theSLy4 effective nuclear interaction. The model describes in aunified way both the dense liquid core of NS and its crust, yieldingM_{max} = 2.05 M_{⊙} and a circumferential radius at the maximum mass of R_{Mmax} = 10.0 km.

APR
= A18 + δv + UIX^{ ∗ } from Akmal et al. (1998) is a nonrelativistic model with some relativistic corrections. It consists of a twonucleon Argonne potential A18 with relativistic boost corrections δv and an adjusted threenucleon Urbana UIX* potential. A variational solution of the manybody problem yields M_{max} = 2.21 M_{⊙} and a circumferential radius at the maximum mass of R_{Mmax} = 10.0 km.

BM
= TM16S0 (with some minor changes) was drawn from a set of relativistic models of Bednarek & Mańka (2009). It consists of a Lorentzcovariant effective nonlinear Lagrangian including up to quartic terms in meson fields, based on chiralsymmetry breaking expansions. The EOS was calculated in the mean field approximation, yielding M_{max} = 2.11 M_{⊙} and a circumferential radius at the maximum mass of R_{Mmax} = 11.95 km.
7. Results
To illustrate the laws and relations governing the “recycling” process, we begin by constraining the final spin frequency, f = 465 Hz. We then calculate sets of the evolutionary tracks labeled with the NS initial parameters – M_{i}, P_{i}, and B_{i} – covering a broad range of possible values. If not stated otherwise, the figures relate to results obtained using the DH EOS. In what follows, we denote B_{p} by B and Ṁ_{b} by Ṁ.
7.1. Constraining the parameter space: final frequency f = 465 Hz
Fig. 1 Massradius relation for accreting stars with different initial magnetic field B_{i} and accretion rate Ṁ. Solid black curve denotes static configurations. Evolutionary tracks (arrows mark the direction of evolution) correspond to the following initial parameters (B_{i} [G], Ṁ [M_{⊙}/yr] ) – a): (10^{12}, 3 × 10^{11}), b): (10^{12}, 10^{10}), c): (10^{12}, 10^{9}), d): (10^{11}, 10^{11}), and e): (10^{11}, 10^{9}). Tracks c) and d) coincide, as explained in Sect. 7.1 (color online). 

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Figure 1 presents the massradius relation for accreting NSs for different values of accretion rate Ṁ and initial magnetic field B_{i}. The initial mass is fixed at 1.4 M_{⊙}. The upper ends of each curve correspond to the final frequency, 465 Hz. For B_{i} = 10^{12} G, the lowest accretion rate considered is 3 × 10^{11} M_{⊙}/yr because for lower rates the configurations enter the axisymmetricperturbation instability region already for frequencies lower than 465 Hz. The same configurations are shown in Fig. 2, where the rotational frequency as a function of mass is presented for different accretion rates. For some evolutionary tracks in Fig. 2, the initial frequency is set to f_{i} = 50 Hz, the value corresponding to a typical frequency expected for newlyborn pulsars (Table 7 of FaucherGiguere & Kaspi 2006). These curves are indistinguishable from those corresponding to initially nonrotating configurations, as the braking timescale is by many orders of magnitude shorter than the spinup time. The difference is visible on a logarithmic scale in Fig. 3, where we present the frequency evolution of an accreting star for two initial configurations: nonrotating NS and a configuration rotating initially with frequency 50 Hz. The accretion rate here is fixed to Ṁ = 10^{9} M_{⊙}/yr. For B_{i} ~ 10^{12} G, the spindown predominates for ~10^{3} yrs, and afterwards the accretion spinsup the star. The amount of accreted material depends very sensitively on the strength of the initial magnetic field; correspondingly, for B_{i} = 10^{11} G the spindown timescale is two orders of magnitude longer (10^{5} yrs) than for B_{i} = 10^{12} G. This is a direct consequence of the quadratic dependence of the magnetic torque on the magnetic field B, Eq. (16). The minimum value of rotation frequency corresponds to the exact balancing of angular momentum l_{0}(r_{0}) at the accretion disk edge, r_{0}, by the magnetic torque. In Figs. 4 and 5, we show the magnetic field vs. mass dependence in such a case (the error bar corresponds to the 3σ measurement of the mass, M = 1.667 ± 0.021 M_{⊙}).
Fig. 2 Spin frequency evolution during accretion, for initial frequencies zero and 50 Hz, as a function of stellar mass. Three different cases of accretion rates (in M_{⊙}/yr) for initial magnetic field B_{i} = 10^{12} G (solid line), 10^{11} G (dashed line) are shown (color online). 

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Fig. 3 Spin frequency evolution for initial spin frequencies of zero and 50 Hz and for an initial magnetic field of B_{i} = 10^{12} G and 10^{11} G as a function of the accreted mass (upper axis) or time (lower axis) calculated for a constant accretion rate Ṁ = 10^{9} M_{⊙}/yr, on a logarithmic scale (color online). 

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Fig. 4 Final magnetic field B_{f} versus gravitational mass M for configurations rotating at f = 465 Hz. Colors correspond to different accretion rates. For a given accretion rate (in M_{⊙}/yr), different curves are defined by a given initial configuration (i.e., central density or initial mass). Along each curve, the initial magnetic field increases upwards. The error bar corresponds to the uncertainty (3σ) in the mass measurement (color online). 

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We note that the problem under consideration is degenerate with respect to the constant (17)(see Fig. 4), as a consequence of Eqs. (12) and (16) depending on the quantity q, and not B and Ṁ separately.
To obtain the results for some Ṁ_{2}, the magnetic field B corresponding to Ṁ_{1} should be multiplied by the factor . This relation allows us to determine the lower bound to the accretion rate for given observed values of B and M or, assuming an accretion rate, to determine the maximum value of the final magnetic field e.g., if Ṁ = Ṁ_{10} = 10^{10} M_{⊙}/yr (red curve in Fig. 4), B_{max} = 1.16 × 10^{8} G for M = 1.67 M_{⊙}. From the observations, we infer that B_{f} ≃ 2 × 10^{8} G; since the condition B_{max} > B_{f} should be fulfilled, we use the scaling law (18)to obtain a lower limit to the accretion rate (19)
Fig. 5 Final magnetic field B_{f} vs. the gravitational mass for a star rotating at f = 465 Hz. Different colors correspond to different EOSs (solid red – DH, dashed blue – APR, dotted green – BM). The accretion rate is Ṁ = 3 × 10^{10} M_{⊙}/yr. The error bar reflects the uncertainty (3σ) in the mass measurement (color online). 

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In Fig. 5, we present the relation between the magnetic field and the gravitational stellar mass for the configuration spun up to 465 Hz, for three different EOSs described in Sect. 6. The lower limit to Ṁ assessed above is marginally consistent with the DH EOS for the measured values of M = 1.67 M_{⊙} and B = 2 × 10^{8} G.
7.2. Constraining the parameter space even further: f = 465 Hz and M = 1.67 M_{⊙}
Lines presented in Fig. 6 correspond to different initial parameters (magnetic field B_{i} and mass M_{i}) that lead to configurations of M = 1.67 M_{⊙} at the spin frequency f = 465 Hz. We employ two fixed accretion rates of Ṁ = 10^{10} M_{⊙}/yr and Ṁ = 10^{9} M_{⊙}/yr. The scaling relation holds and allows us to set limits on Ṁ for a given value of the magnetic field.
Fig. 6 Final magnetic field, B_{f}, versus initial gravitational mass, for a star with final parameters M = 1.67 M_{⊙} and the frequency f = 465 Hz, for APR, DH, and BM EOSs. The accretion rate is Ṁ = 10^{10} M_{⊙}/yr (solid curves) and Ṁ = 10^{9} M_{⊙}/yr (dashed curves). Scaling relation is fulfilled, i.e., dashed curves match the solid ones multiplied by (color online). 

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The differences between the DH and APR EOSs are small in comparison to the difference of them both with a much stiffer BM EOS. The value of the stellar moment of inertia is responsible for such a discrepancy: I(M) for the BM EOS is about 25% higher than the corresponding values for the DH and APR EOS (for the same M). Consequently, to obtain the same final frequency one needs a larger J and a larger amount of accreted mass, which leads to the lower value of B_{f}.
7.3. From a progenitor NS to PSR J1903+0327: M = 1.67 M_{⊙}, f = 465 Hz, and B = 2 × 10^{8} G
One can obtain a stringent limit on the parameters of the progenitor NS by fixing the final value of the magnetic field. Some tracks considered in this subsection are shown in Fig. 7, labeled by the initial value of the magnetic field B_{i} and the accretion rate Ṁ. The final magnetic field is fixed to B = 2 × 10^{8} G. For comparison, we present the case in which we neglect the effect of the magnetic field entirely: the accretion takes place from the marginallystable orbit calculated exactly in accordance with General Relativity (see, e.g., Zdunik et al. 2002).
Fig. 7 Spinup tracks of the accreting NSs leading to the final configuration rotating at f = 465 Hz and with gravitational mass M = 1.67 M_{⊙} and magnetic field B = 2 × 10^{8} G. Curves are labeled by the average accretion rate (in M_{⊙}/yr and initial value of the magnetic field (in G). For comparison the spinup for B = 0 (dashed line), via accretion from marginally stable orbit, is shown (color online). 

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The spinup in the B = 0 case is very efficient, one needs ≃ 0.1 M_{⊙} only to reach the observed frequency of 465 Hz. Neglecting a magnetic field therefore leads to a rather high progenitorNS mass, 1.58 M_{⊙}, whereas reasonable values of the progenitor B and Ṁ enable M_{f} to be reached starting from a moderate value of M_{i} ~ 1.3 M_{⊙}. The amount of accreted matter in the B = 0 case can be larger if we allow for the reduction in the angular momentum transport efficiency (parameter x_{l} ≪ 1 in Zdunik et al. 2002), here however we assume a 100% efficiency (x_{l} ≡ 1), in accordance with recent numerical calculations (Beckwith et al. 2008; Shafee et al. 2008).
The accretion rate as a function of M_{i} (assuming final B_{f} = 2 × 10^{8} G) is presented in Fig. 8. The lower limit to the accretion rate depends on the EOS and ranges from 2.5 × 10^{10} M_{⊙}/yr (DH, APR) to 8.5 × 10^{10} M_{⊙}/yr (BM). For a given EOS, the required Ṁ depends weakly on the accreted mass needed to spin up the star to 465 Hz, provided that ΔM > 0.2 M_{⊙}. For ΔM < 0.1 M_{⊙}, the required accretion rate increases dramatically with decreasing ΔM. Using , we obtain the time needed to spin up the star to its presently observed frequency. The result is presented in Fig. 9.
Fig. 8 Average accretion rate vs. initial mass needed to evolve a NS by disk accretion to the frequency f = 465 Hz, mass M = 1.67 M_{⊙}, and magnetic field B = 2 × 10^{8} G for the three EOSs under consideration (color online). 

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Fig. 9 Accretion rate versus time of accretion needed to reach observable configuration (color online). 

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We also performed calculations for other B(ΔM) dependences (Eqs. (7) and (8) in Sect. 4). The results are qualitatively similar in the case of an inverse quadratic dependence (Eq. (7)); the value of Ṁ at the “plateau” region in Figs. 8 and 9 is slightly higher, 3.6 × 10^{10} M_{⊙}/yr, than 3 × 10^{10} M_{⊙}/yr, while the Eq. (6) is adopted. For the exponential field decay (Eq. (8)) however, the resulting Ṁ is 8 × 10^{10} M_{⊙}/yr (for the mass decay scale m_{B} = 0.05 M_{⊙}, which according to Osłowski et al., gives the closest agreement between their population synthesis model predictions and the observed P − Ṗ data for Galactic pulsars). This discrepancy is due to the different behavior of B_{p}(ΔM) close to the final B_{p} value, where Eq. (8) gives significantly larger values of B_{p} than Eq. (6). For an efficient spinup, the “magnetic term” proportional to in l_{tot} should be small (Eq. (16)), hence the relatively large value of B_{p} has to be compensated for by a higher value of Ṁ – for example, assuming B_{i} = 10^{12} G, Eq. (6) yields ΔM = 0.5 M_{⊙} in 1.7 × 10^{9} yrs, while the exponential decay of Osłowski et al. (2011) gives 0.46 M_{⊙} in 6 × 10^{8} yrs, that is, the accretion of almost the same amount of matter but somewhat more rapidly. One should however treat this kind of empirical formulae with caution, since in the case of parameters obtained by population synthesis methods they may be uncontrollably affected by other assumptions.
8. Conclusions
Our simulations have allowed to estimate the intrinsic parameters of the progenitor NS required to reach, in the process of recycling, the measured parameters of PSR J1903+0327. To some extent, as we have shown, the progenitor NS parameters depend on the EOS of dense matter. This was studied using three EOSs consistent with the measurement of 2 M_{⊙} NS (Demorest et al. 2010).
We have found that the mean accretion rate Ṁ during recycling should be larger than (2.5 ÷ 8.5) × 10^{10} M_{⊙}/yr, the highest lower bound being obtained for the stiffest EOS.
For each EOS, the required mean accretion rate is approximately constant for a broad range of initial masses, 1 M_{⊙} ÷ 1.4 M_{⊙}. Therefore, depending on the initial magnetic field B_{i} we can reproduce parameters of PSR J1903+0327, or more generally, observed millisecond pulsars, for a specific range of initial masses. In other words, the present parameters of a recycled millisecond pulsar do not allow us to determine its initial mass, in contrast to the case of recycling by accretion with B = 0. Simulations that neglect the magnetic field (following the seminal paper of Cook et al. 1994) give a rather high value of the lower bound to the progenitor NS mass of M_{i} > 1.55 M_{⊙} (1.58 M_{⊙} for DH and APR EOS, and 1.55 M_{⊙} for the BM EOS). This is a direct consequence of the finding that the magnetic field significantly decreases the spinup rate (effectively decreasing the efficiency of angular momentum transfer onto the star, as in to the B = 0 and x_{l} ≪ 1 case). Accounting for the magnetic field effect (magnetic torque) is therefore crucial for the understanding of the observable pulsar population properties, especially in view of the proposal that gravitational wave emission is a major dissipative agent that prevents efficient spinup (Arras et al. 2003; Watts et al. 2008; Watts & Krishnan 2009). In other words, for evolutionary reasons there may not be sufficient time and/or a sufficient amount of matter to accrete, to form a rapidlyspinning pulsar with a magnetic field sufficiently strong to produce a detectable radio beam.
The framework presented here is suitable for testing the global properties of the Galactic pulsar population, as well as for studying other millisecond pulsars with precisely measured masses, e.g., massive millisecond pulsars recycled in intermediatemass Xray binaries, where the recycling process is much shorter than the one studied here (Tauris et al. 2011). Our analysis can also be easily extended to take into account timevarying accretion rates and more detailed multiparameter magnetic field decay prescriptions – this paper is the first of a series devoted to these studies.
Fig. 10 Upper panel: spinup tracks of accreting NSs leading to presently observed PSR J1903+0327 parameters on radiusspin frequency plane. The relevant characteristic radii are: stellar radius (solid curves), the radius of a marginally stable orbit (dotted curves), and the radius of the inner boundary of accretion disk (dashed curves). Colors correspond to the specific values of accretion rate and the initial value of the magnetic field. Lower panel: effective radial potential V(r,l(r)) in the energy (mc^{2}) units for an exact, approximate, and Schwarzschild solution; the dot marks the final r_{0}. See the text for more details (color online). 

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Appendix: The importance of relativistic effects in the “recycling” process of PSR J1903+0327
The original framework of Kluźniak & Rappaport (2007) is restricted to the nonrelativistic limit, i.e., it does not include the effects of the existence of a marginally stable orbit of radius r_{ms}. As mentioned by these authors, their model should be used only where r_{0} ≫ r_{ms}. To establish a more general model that can be applied to compact and massive configurations near the massshedding limit, we have amended this shortcoming here by introducing Eq. (12). We demonstrate in this Appendix however, that in the case of moderately fastspinning PSR J1903+0327 this refined approach is not essential.
The upper panel of Fig. 10 shows three characteristic radii of the problem, r_{0}, stellar radius R, and r_{ms}, calculated along a spinup evolutionary track. For the final configuration, f = 465 Hz, M = 1.67 M_{⊙}, and B = 2 × 10^{8} G, where r_{0} reaches its minimum value, r_{0}/r_{ms} ≃ 2. We compare two results, one where the effect of General Relativity is mitigated by Eq. (12), and the original model of KR, f_{ms} = 1 in Eq. (12). The difference in r_{0} values is ~200 m, i.e., 0.7%. Lower panel shows the effective radial potential V(r,l(r)) (l(r) is the particle specific angular momentum on the circular orbit of the radius r; see, e.g., Eq. (2) of Bejger et al. 2010); the final r_{0} is sufficiently far from the r_{ms}, hence we conclude that in the particular case of PSR J1903+0327 the relativistic correction necessary to account for r_{ms} can be omitted.
Acknowledgments
We are grateful to W. Kluźniak for his helpful comments referring to the KluźniakRappaport model of magneticallytorqued accretion disks. We also acknowledge the helpful remarks of participants of the CompStar 2011 Workshop (Catania, Italy, 9 − 12 May, 2011), after the talk by one of the authors (JLZ). This work was partially supported by the Polish MNiSW grant No. N N203 512 838, by the LEA Astrophysics PolandFrance (AstroPF) program, and the ESF Research Networking Programme CompStar. M.B. acknowledges Marie Curie Fellowship within the 7th European Community Framework Programme (ERG2007224793).
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All Figures
Fig. 1 Massradius relation for accreting stars with different initial magnetic field B_{i} and accretion rate Ṁ. Solid black curve denotes static configurations. Evolutionary tracks (arrows mark the direction of evolution) correspond to the following initial parameters (B_{i} [G], Ṁ [M_{⊙}/yr] ) – a): (10^{12}, 3 × 10^{11}), b): (10^{12}, 10^{10}), c): (10^{12}, 10^{9}), d): (10^{11}, 10^{11}), and e): (10^{11}, 10^{9}). Tracks c) and d) coincide, as explained in Sect. 7.1 (color online). 

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In the text 
Fig. 2 Spin frequency evolution during accretion, for initial frequencies zero and 50 Hz, as a function of stellar mass. Three different cases of accretion rates (in M_{⊙}/yr) for initial magnetic field B_{i} = 10^{12} G (solid line), 10^{11} G (dashed line) are shown (color online). 

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In the text 
Fig. 3 Spin frequency evolution for initial spin frequencies of zero and 50 Hz and for an initial magnetic field of B_{i} = 10^{12} G and 10^{11} G as a function of the accreted mass (upper axis) or time (lower axis) calculated for a constant accretion rate Ṁ = 10^{9} M_{⊙}/yr, on a logarithmic scale (color online). 

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In the text 
Fig. 4 Final magnetic field B_{f} versus gravitational mass M for configurations rotating at f = 465 Hz. Colors correspond to different accretion rates. For a given accretion rate (in M_{⊙}/yr), different curves are defined by a given initial configuration (i.e., central density or initial mass). Along each curve, the initial magnetic field increases upwards. The error bar corresponds to the uncertainty (3σ) in the mass measurement (color online). 

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In the text 
Fig. 5 Final magnetic field B_{f} vs. the gravitational mass for a star rotating at f = 465 Hz. Different colors correspond to different EOSs (solid red – DH, dashed blue – APR, dotted green – BM). The accretion rate is Ṁ = 3 × 10^{10} M_{⊙}/yr. The error bar reflects the uncertainty (3σ) in the mass measurement (color online). 

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In the text 
Fig. 6 Final magnetic field, B_{f}, versus initial gravitational mass, for a star with final parameters M = 1.67 M_{⊙} and the frequency f = 465 Hz, for APR, DH, and BM EOSs. The accretion rate is Ṁ = 10^{10} M_{⊙}/yr (solid curves) and Ṁ = 10^{9} M_{⊙}/yr (dashed curves). Scaling relation is fulfilled, i.e., dashed curves match the solid ones multiplied by (color online). 

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In the text 
Fig. 7 Spinup tracks of the accreting NSs leading to the final configuration rotating at f = 465 Hz and with gravitational mass M = 1.67 M_{⊙} and magnetic field B = 2 × 10^{8} G. Curves are labeled by the average accretion rate (in M_{⊙}/yr and initial value of the magnetic field (in G). For comparison the spinup for B = 0 (dashed line), via accretion from marginally stable orbit, is shown (color online). 

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In the text 
Fig. 8 Average accretion rate vs. initial mass needed to evolve a NS by disk accretion to the frequency f = 465 Hz, mass M = 1.67 M_{⊙}, and magnetic field B = 2 × 10^{8} G for the three EOSs under consideration (color online). 

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In the text 
Fig. 9 Accretion rate versus time of accretion needed to reach observable configuration (color online). 

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In the text 
Fig. 10 Upper panel: spinup tracks of accreting NSs leading to presently observed PSR J1903+0327 parameters on radiusspin frequency plane. The relevant characteristic radii are: stellar radius (solid curves), the radius of a marginally stable orbit (dotted curves), and the radius of the inner boundary of accretion disk (dashed curves). Colors correspond to the specific values of accretion rate and the initial value of the magnetic field. Lower panel: effective radial potential V(r,l(r)) in the energy (mc^{2}) units for an exact, approximate, and Schwarzschild solution; the dot marks the final r_{0}. See the text for more details (color online). 

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In the text 