the nature of millisecond pulsars with helium white dwarf ... · the data for all bmsps with helium...

MNRAS 437, 2217–2229 (2014) doi:10.1093/mnras/stt2030Advance Access publication 2013 November 22

The nature of millisecond pulsars with helium white dwarf companions

Sarah L. Smedley,1‹ Christopher A. Tout,1,2,3 Lilia Ferrario3

and Dayal T. Wickramasinghe3

1Institute of Astronomy, The Observatories, Madingley Road, Cambridge CB3 0HA, UK2Monash Centre for Astrophysics, School of Mathematics, Building 28, Monash University, Clayton, VIC 3800, Australia3Mathematical Sciences Institute, The Australian National University, ACT 0200, Australia

Accepted 2013 October 21. Received 2013 October 16; in original form 2012 December 21

ABSTRACTWe examine the growing data set of binary millisecond pulsars that are thought to have a heliumwhite dwarf companion. These systems are believed to form when a low- to intermediate-mass companion to a neutron star fills its Roche lobe between central hydrogen exhaustionand core helium ignition. We confirm that our own stellar models reproduce a well-definedperiod–companion mass relation irrespective of the details of the mass transfer process. Withmagnetic braking, this relation extends to periods of less than 1 d for a 1 M� giant donor.With this and the measured binary mass functions, we calculate the orbital inclination of eachsystem for a given pulsar mass. We expect these inclinations to be randomly oriented in space.If the masses of the pulsars were typically 1.35 M�, then there would appear to be a distinctdearth of high-inclination systems. However, if the pulsar masses are more typically from 1.55to 1.65 M�, then the distribution of inclinations is indeed indistinguishable from random.If it were as much as 1.75 M�, then there would appear to be an excess of high-inclinationsystems. Thus, with the available data, we can argue that the neutron star masses in binarymillisecond pulsars recycled by mass transfer from a red giant typically lie around 1.6 M�and that there is no preferred inclination at which these systems are observed. Hence, there isreason to believe that pulsar beams are either sufficiently broad or show no preferred directionrelative to the pulsar’s spin axis which is aligned with the binary orbit. This is contrary to someprevious claims, based on a subset of the data available today, that there might be a tendencyfor the pulsar beams to be perpendicular to their spin.

Key words: binaries: close – stars: evolution – stars: mass-loss – stars: neutron – pulsars:general.

1 IN T RO D U C T I O N

Millisecond pulsars (MSPs) are radio pulsars with pulse periodscanonically defined to be less than 30 ms. About 90 per cent of MSPsare found in binary systems (BMSPs). From their spin-down rates,these are estimated to have magnetic fields between 107 and 109 G,significantly lower than the 1012–1013 G of ordinary radio pulsars(Lorimer 2008). In the standard rejuvenation model of BMSPs(Alpar et al. 1982; Radhakrishnan & Srinivasan 1982), the field de-cays and the pulsar spins up as matter is accreted from a companionstar during a previous phase of evolution as a low- or intermediate-mass X-ray binary (LMXB/IMXB). The discovery of several X-raypulsars with millisecond periods in LMXBs and of IGR J18245–2452, an LMXB with a rotation period of 3.9 ms that alternatesbetween a rotation-powered radio source and an accretion-poweredX-ray source (Papitto et al. 2013), has provided strong evidence insupport of a rejuvenation model. The precise nature of the evolu-

� E-mail: [email protected]

tion or decay of the magnetic field during accretion and the amountof mass accreted along the different binary pathways remains un-resolved but must depend on both the origin of the field and theevolutionary history of BMSPs.

The lowest-mass neutron stars are expected to form by electroncapture in the cores of the most massive asymptotic giant branchstars (Sugimoto & Nomoto 1980). An electron degenerate corecollapses, as electrons are captured by 24Mg and 20Ne at a massof 1.375 M� (Nomoto 1984). The cores of more massive starscollapse only after burning to iron-group elements and exceedingthe Chandrasekhar limit. Some of this baryonic mass, depending onthe equation of state of neutron stars, is lost to gravitational bindingenergy. Schwab, Podsiadlowski & Rappaport (2010) propose thatneutron stars formed by electron capture are typically born withgravitational masses of around 1.25 M�, while those formed byiron core collapse with masses around 1.35 M�. Rejuvenation andspin-up to millisecond periods requires accretion of at least 0.1 M�or so of material from the inner edge of a Keplerian accretion disc.Zhang et al. (2011) find measured masses of the neutron stars in

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2218 S. L. Smedley et al.

MSPs now to be around 1.55 M�. Here, we are able to test whetherthe data for all BMSPs with helium white dwarf companions areconsistent with such masses.

In an early study of the fastest BMSPs based on 30 stars, it wasnoted that some 30 per cent showed the presence of a strong in-terpulse separated by 180◦ from the main pulse, while this is thecase for only a few per cent of the general sample of radio pulsars(Jayawardhana & Grindlay 1996). This was interpreted as evidencefor nearly orthogonal rotators, in which the magnetic dipole axisis perpendicular to the spin axis. Subsequently, Chen, Ruderman& Zhu (1998) argued for a bimodal distribution of both nearly or-thogonal and nearly aligned rotators among the Galactic BMSPs.Ruderman & Chen (1999) presented theoretical arguments in favourof these findings based on a specific superfluid and superconduct-ing model for the evolution of field in spinning accreting neutronstars. Subsequently, Backer (1998) analysed the minimum massesof a sample of low-mass BMSPs assuming that the orbital incli-nations were randomly oriented in space but with further specificassumptions on the neutron star and white dwarf masses, and pre-sented evidence apparently lending support to the orthogonal rotatormodel.

The stellar properties of any companion along with its orbit pointto the evolutionary pathway by which the BMSP reached its currentstate. Hurley et al. (2010) made comprehensive binary populationsyntheses and described the numerous routes to MSPs and the ex-pected relation between the final companion mass and the orbitalperiod after mass transfer. Notably, they included the possibility thatthe neutron star began accreting as a massive ONe white dwarf thatcollapsed to the pulsar after accreting sufficient mass (accretion-induced collapse). Here, we focus on what appears to be a rathercommon route in which an evolved low- to intermediate-mass starfills its Roche lobe before core helium ignition but after core hydro-gen exhaustion. This sequence is apparent in the figures of Hurleyet al. (2010) whether or not the neutron star is formed by accretion-induced collapse.

In this paper, we have undertaken a detailed statistical study ofthe significantly enhanced current sample of MSPs with heliumwhite dwarf companions with their well-determined mass functionsderived from quantities given in the Australia Telescope NationalFacility (ATNF) Pulsar Catalogue and Manchester et al. (2005).1

Our study is similar to that of Stairs et al. (2005), but the in-crease in data means we can now make somewhat more significantconclusions.

Our analysis indicates that good overall agreement with the to-tality of observations of BMSPs with He white dwarf companionscan be achieved if the masses of the neutron stars lie typicallyaround 1.6 M�, which is higher than what is expected from typicalelectron-capture or core-collapse supernovae but is consistent withexpectations of the re-cycling hypothesis. There is no significant dif-ference in this consistency between low- and high-period systemsas was suggested by Stairs et al. (2005). Nor can we agree with theconclusion of Backer (1998) in support of orthogonal rotators.

2 D E TA I L E D M O D E L S O F R L O F

We compute an extensive grid of models with the Cambridge STARS

code. In all the cases, we fully compute the evolution of the low-mass donor star, while the neutron star is treated as a point mass ac-cretor. The donors begin their evolution as zero-age main-sequencestars in thermal equilibrium. With a series of models, we investigate

1 www.atnf.csiro.au/people/pulsar/psrcat

the effect of different binary parameters on the Mc−Porb relation forthe whole range of Case B Roche lobe overflow (RLOF). In the ter-minology of Kippenhahn & Weigert (1967), Case B refers to masstransfer that begins after the exhaustion of hydrogen fuel in a star’score but before ignition of helium. We refer to late Case B when masstransfer begins after the star is established on the giant branch, earlyCase B when it begins when the star is crossing the Hertzsprung gap(HG) and very early Case B when it begins at the tip of the mainsequence once the star has a helium core but before it begins itsexcursion to the red in the Hertzsprung–Russell (H–R) diagram.

2.1 Cambridge STARS CODE

We use a version of the Cambridge STARS code (Eggleton 1971;Pols et al. 1995) updated by Stancliffe & Eldridge (2009). Thecode features a non-Lagrangian mesh. Convection is according tothe mixing-length theory of Bohm-Vitense (1958) with αMLT = 2and convective overshooting is included as described by Schroder,Pols & Eggleton (1997). The nuclear species 1H, 3He, 4He, 12C,14N, 16O and 20Ne are evolved in detail. Opacities are from theOPAL Collaboration (Iglesias & Rogers 1996) supplemented withmolecular opacities of Alexander & Ferguson (1994) and Fergusonet al. (2005) at the lowest temperatures and by Buchler & Yueh(1976) at higher temperatures. Electron conduction is that ofHubbard & Lampe (1969) and Canuto (1970). Nuclear reactionrates are those of Caughlan & Fowler (1988) and the NACRE Col-laboration (Angulo et al. 1999).

2.2 Period–white dwarf mass relation

For the donor stars, the luminosity depends only on the helium coremass and the radius depends only on the luminosity and the totalmass (Paczynski 1971), so there is a relation between the orbitalperiod Porb and the remnant mass Mc when the system detaches(Refsdal & Weigert 1971). We do not expect the Mc−Porb relationto depend on how the system gets to this point so that the details ofhow conservative is the mass transfer do not matter. Nor does therelation depend on the neutron star mass because for low enoughmass ratios the Roche lobe radius is proportional to the ratio of theseparation to the total mass of the system and by Kepler’s third lawthe cube root of the square of the orbital period is proportional tothis same ratio. Thus, Porb is a function only of Mc. We demonstratethat this is the case in the following sections.

Though the mass transfer to MSPs is likely to be non-conservative, we first consider the canonical, perhaps extreme, fullyconservative case in which all of the envelope of the donor star istransferred to and accreted by its companion and for which totalangular momentum is conserved. We model initial donor massesof 0.875, 1 and 1.2 M�. We give the companion an initial mass of1.55 M�. We demonstrate later that the relation is not affected bythe non-conservative nature of the mass transfer. For each donormass, we evolve a set of 300 models with initial periods to cover thefull range of Case B RLOF for each of the different donor masses.Magnetic braking, as discussed in more detail in the next section, isnecessary to reach the lowest periods and core masses. Fig. 1 showsa subset of the 1 M� donor models plotted with an empirical fit forthe relation (Section 2.5). In Fig. 2, we plot the Porb−Mc relationfor a selection of models with each of the three initial donor masses.There is no perceptible difference. Lower-mass donor stars exhibitvery similar behaviour to the 1 M� donor once they evolve pastthe main sequence. However, they are not expected to have evolvedwithin the lifetime of the Galaxy. Higher-mass donors, once estab-lished on the giant branch, undergo unstable RLOF which probably

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The nature of MSPs with helium WD companions 2219

Figure 1. Our canonical Porb−Mc relation. Points are a subset of our modelsmade with 1 M� donor stars in binary systems with neutron star companionsof 1.55 M� (circles). The solid line represents our empirical fit (equation 2).

Figure 2. Variation of initial donor mass. Models made with a 0.875 M�donor are open squares and models made with a 1.2 M� donor are lightopen circles. The dot–dashed line represents our canonical relation from1 M� donors.

leads to common-envelope evolution and perhaps merging of thestars. Thus, in general, we do not expect the donor star mass tolie outside the range 0.875−1.2 M� very often and so we takethe Porb−Mc relation derived from 1 M� donors to be canonical.There are some discontinuities apparent for all donor masses arounda remnant mass of 0.225 M� caused by dredge up of the previouslyburnt stellar interior by the deepening convective envelope.

2.3 Short periods and magnetic braking

The minimum white dwarf mass that can be left by a given donordepends on the Schonberg–Chandrasekhar mass (Schonberg &

Chandrasekhar 1942) of its core. Beyond this the core cannot re-main isothermal and still support the stellar envelope. After thisthe star evolves rapidly across the HG. If evolved fully conservingmass and angular momentum our 1 M� donor leaves a minimumremnant mass of 0.22 M�.

A group of very short orbital period MSPs are thought to haveevolved from LMXBs (Fabian et al. 1983) for which the masstransfer is driven by orbital angular momentum loss rather thanstellar evolution. In these systems, the typically low-initial-mass(Md < 0.9 M�) donor star does not develop a degenerate core andfinal periods are a few hours.

With their shorter hydrogen-burning lifetime, intermediate-massdonors, those around 1 M� and more, can develop helium cores butare also expected to suffer magnetic braking if their orbital periodis short enough. If such an evolved intermediate-mass star beginsRLOF before this (very early Case B), its limiting core mass isreduced as it transfers mass and so consequently behaves as if itwere a star with a somewhat lower initial total mass. There is abifurcation period below which systems evolve as LMXBs to verylow Porb and above which they evolve to larger Porb transferringmass as red giants or subgiants. We are interested in these lattersystems which end up detached with Porb � 1 d.

We include magnetic braking at the empirical rate of Verbunt &Zwaan (1981),

J

J= −0.5 × 10−28 s2 cm−2f −2

mb

IR2d

a5

GM2

MaMds−1, (1)

where Rd is the radius of the donor star, I its moment of inertia,a the semi-major axis of the orbit, M the total mass of the binarysystem, Ma the mass of the accretor and Md is the mass of the donor.The factor fmb was chosen to fit the equatorial velocities of G- andK-type stars (Smith 1979).

Fig. 3 shows details of the low-period end of the Porb−Mc relationfor systems evolved with and without magnetic braking. The formof the magnetic-braking law is such that the initial orbital periods forwhich it can produce lower period systems with He white dwarfs

Figure 3. Details of the period–core mass relation for critically brakedsystems (open circles) compared with non-braked systems (solid circles).All models have a 1.55 M� neutron star that accreted from an initially1 M� donor.

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Figure 4. A H–R diagram, in the bolometric luminosity (L)–effective tem-perature (Teff) plane, for the evolution of an isolated 1 M� star (labelledA), along with the evolution of similar stars in binary systems with neutronstar companions of 1.55 M�. All have active magnetic braking. Model Bhas an initial orbital period of 2 d, model C – 2.14 d and D – 20 d.

is finely balanced between 2 and 2.2 d to produce final periodsbetween 0.5 and 6 d.

2.4 Examples of evolution

Fig. 4 shows a H–R diagram for three binary star models (B, C andD) that begin RLOF at different stages along with a track (A) forsingle star evolution. Model B begins RLOF only once establishedas a red giant, late Case B. Model C is crossing the HG and beginsRLOF as a subgiant, early Case B, while model D begins masstransfer with an isothermal gas-pressure-supported core, very earlyCase B. By the time the stars in models C and D detach, they haveestablished degenerate He cores and are ascending the red giantbranch. The larger the initial period, the further up the giant branchthe RLOF begins. Mass transfer strips the envelope to reveal thestars’ cores which then cool to become white dwarfs.

2.5 Empirical fit

A good empirical fit to Porb−Mc relation for all the models with1 M� donors, including those with magnetic braking, is given by

Porb/d = exp(a + (b/(Mc/M�)) + c(ln(Mc/M�))), (2)

with

(a, b, c)

=

⎧⎪⎪⎨⎪⎪⎩

(22.902, 3.097, 23.483) for 0.17 ≤ Mc < 0.225

(14.443, 0.335, 9.481) for 0.225 ≤ Mc < 0.28

(11.842, −3.831, −4.146) for 0.28 ≤ Mc < 0.48.

(3)

2.6 Measured pulsar masses

Table 1 lists observed MSP – white dwarf binary systems that havemeasured pulsar masses and companion masses. These masses aregenerally higher than those found for isolated pulsars (Zhang et al.2011). This may be due to either a range in initial neutron star massor accretion during the mass transfer. Tauris & Savonije (1999)noticed a negative correlation between the orbital period and themass of the pulsar. In Fig. 5, we plot the data for the systems inTable 1. Any correlation is actually rather weak. Though we listall the measured masses here, we note that all of those with orbitalperiods of less than 1 d are excluded from our analysis becausethey may have undergone Case A mass transfer, while the star isstill burning hydrogen in its core, rather than Case B mass transfer.Indeed B1957+20 is a black widow pulsar (Fruchter et al. 1988)with a low-mass main-sequence star rather than a white dwarf.J1910–5959A is also excluded because it is a member of a globularcluster (Corongiu et al. 2012). J2016+1948 is excluded because ithas Pspin > 30 ms and J1614−2230 is excluded because it has acompanion mass larger than 0.472 M�.

De Vito & Benvenuto (2010) looked at the effect of the neutronstar mass on these systems and found that, for systems of giveninitial orbital period and initial donor star mass, the mass of theneutron star heavily affects the evolution. However, they concludethat the Mc−Porb relation is insensitive to the initial neutron starmass, confirming the prediction made by Rappaport et al. (1995).We tested the effect of the neutron star mass on the evolution ofour binary systems by computing models that initially had a 1 M�donor star and neutron star masses of 1.35, 1.55 and 1.75 M�.

Table 1. BMSPs with measured pulsar and companion masses. Two systems with Pspin > 30 ms are includedfor completeness but are excluded from our analysis.

Name Pspin/ms Porb/d Mp/M� Mcomp/M� Reference

J0348+0432 39.12 0.103 2.01 ± 0.04 0.172 ± 0.003 Antoniadis et al. (2013)J0751+1807 3.48 0.26 1.26 ± 0.14 0.191 ± 0.015 Nice, Stairs, & Kasian (2008)J1738+0333 5.85 0.35 1.47 ± 0.06 0.181 ± 0.006 Antoniadis et al. (2012)B1957+20 1.61 0.38 2.40 ± 0.12 0.035 ± 0.002 Gonzalez et al. (2011)J1012+5307 5.26 0.60 1.6 ± 0.2 0.16 ± 0.02 van Kerkwijk et al. (2005)J1910−5959A 3.27 0.84 1.33 ± 0.11 0.180 ± 0.018 Corongiu et al. (2012)J1909−3744 2.95 1.53 1.438 ± 0.024 0.2038 ± 0.0022 Jacoby et al. (2005)J0437−4715 5.76 5.74 1.76 ± 0.02 0.254 ± 0.014 Verbiest et al. (2008)J1614−2230 3.15 8.69 1.97 ± 0.04 0.500 ± 0.006 Demorest et al. (2010)B1855+09 5.36 12.33 1.6 ± 0.2 0.270 ± 0.025 Splaver (2004)J1910+1256 4.99 58.47 1.6 ± 0.6 0.30 − 0.33 Gonzalez et al. (2011)J1713+0747 7.99 67.83 1.3 ± 0.2 0.28 ± 0.03 Splaver et al. (2005)J1853+1303 4.09 115.65 1.4 ± 0.7 0.33 − 0.37 Gonzalez et al. (2011)J2016+1948 64.95 635.04 1.0 ± 0.5 0.43 − 0.47 Gonzalez et al. (2011)

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Figure 5. Neutron star masses and orbital periods of the observed BMSPslisted in Table 1. Solid circles represent the systems that fit our selectioncriteria, while open circles represent those that are excluded for variousreasons. Vertical lines indicate masses of 1.35, 1.55 and 1.75 M�.

Figure 6. The effect of neutron star mass on the Mc−Porb relation. Threecases are shown here, all models have a 1 M� donor star and were computedwith initial periods from 2 to 2.4 d. Open squares have an initial neutronstar mass of 1.35 M�, open circles 1.55 M� and open diamonds 1.75 M�.The dot–dashed line represents our empirical fit. As expected, the neutronstar mass has no effect on the Porb−Mc relation.

We made 100 models for each case with initial orbital periodsspanning the entire Mc−Porb relation. Fig. 6 shows the Mc−Porb

relations for a subset of these models with initial periods between 2and 2.4 d. The Mc−Porb relation itself is not affected by the neutronstar mass but, for a given initial orbital period, the higher the neutronstar mass, the more massive the predicted white dwarf remnant. Thehigher mass of neutron star lies further up the relation but not off it.

However, the neutron star mass does matter when we computeorbital inclinations (equation 11). For now, we conclude that masses

of MSPs in systems that could have helium white dwarf companionslie between 1.35 and 1.75 M� and cluster around 1.55 M�.

2.7 RLOF efficiency

Relaxing conservative evolution further, we allow mass to be lostfrom either star during mass transfer. The total angular momentumJ is the sum of the orbital angular momentum and the rotational (orspin) angular momentum of each star. The latter is typically only1−2 per cent of the orbital angular momentum and so can usually beignored though we do include it in our detailed stellar models. Tauris& Savonije (1999) argued that the accretion process is inefficient,with only a small portion of transferred matter accreted by theneutron star. The rest is ejected. Burderi et al. (2005) confirmed thatthe accretion process in these systems is non-conservative. Only0.1 M� of material from the inner edge of the disc is needed to spinup the neutron star to periods below 1 ms. De Vito & Benvenuto(2012) computed the evolution of a set of models with initial donorstar between 0.5 and 3.5 M�, initial orbital periods between 0.175and 12 d and initial neutron stars masses between 0.8 and 1.4 M�,with varying degrees of mass transfer efficiency. Following theirapproach, we define efficiency parameters α, the fraction of masslost by the donor that is transferred to the accretor and β, theamount of matter actually captured by the accretor. A fraction 1 − α

of the mass lost by the donor then carries off the specific angularmomentum of the donor while a fraction α(1 − β) of the masslost by the donor carries off the specific angular momentum of theaccretor. So

Ma = −αβMd, (4)

and

J = (1 − α)Mda22ω + α(1 − β)Mda

21ω. (5)

Thence,

J

J= (1 − α)Mda

22ω + α(1 − β)Mda

21ω

(MaMd/(Ma + Md))a2ω, (6)

where J = MaMda2ω/(Ma + Md) is the orbital angular momentumfor semimajor axis a and orbital angular velocity ω = 2π/Porb. Thissimplifies to

J

J= Md

Md

[1 − α + α(1 − β)q2

1 − q

], (7)

where the mass ratio q = Md/Ma.We evolved sets of 100 models for combinations of α and β,

with initial orbital periods in the range 2–3 d. Fig. 7 illustrates theeffects on the Mc−Porb relation. As expected, the relation betweenMc and Porb is not much affected by the non-conservative natureof the RLOF but, for a given initial orbital period, increasing α ordecreasing β leaves a more massive core and consequently a longerfinal period because, even though the escaping material carries awayangular momentum, the total system mass falls.

2.8 Comparison of detailed models and data

The systems with measured neutron star masses listed in Table 1also have measured companion masses. We again ignore systemswith Porb < 1 d. Note that J1614−2230 is also excluded becauseto fit the data and the Porb−Mc relation would require too small aneutron star mass (see Section 4). This is encouraging because itis indeed found to have a companion of 0.5 M�, too massive for aHe white dwarf. We plot the remaining data in Fig. 8 and find that

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Figure 7. The effect of varying the RLOF efficiency on the Mc−Porb rela-tion. Five cases are shown (1) α = β = 1 (solid circles), (2) α = β = 0 (opencircles), (3) α = 1 and β = 0.6 (diamonds), (4) α = 0.6 and β = 0 (crosses)and (5) α = 1 and β = 0 (triangles). Each set initially has Md = 1 M�,Ma = 1.55 M� and 2 < Porb/d < 2.19 . The dot–dashed line represents ourcanonical relation.

Figure 8. MSPs with measured companion masses in the Mc−Porb plane.The solid line represents our canonical relation. We do not include systemsexcluded from our analysis except for the 635 d system J2016−2230 (shownby +). It appears to lie precisely on the relation but has a spin period of65 ms.

the fit with our canonical relation is good given the measurementerrors. Note that J2016+1948 is also formally excluded becauseits spin period exceeds 30 ms but we plot it here because it lieson the canonical Porb−Mc relation. This suggests that it has passedthrough a similar evolution but has just not been spun up so much.

Figure 9. The effect of donor star metallicity on the Mc−Porb relation.Models are all for accretion on to a 1.55 M� neutron star from a 1 M�donor. The solid line has Z = 0.02, the short dashed line Z = 0.01 and thelong dashed line Z = 0.03. The effect of changing metallicity over the rangethat might be expected in the Galactic field is still rather small but largerthan any effect owing to the nature of the mass transfer.

2.9 Donor star metallicity

We are concentrating on BMSPs in the Galactic field and so do notexpect the metallicity to vary much from solar. However, we inves-tigated how the metallicity of the donor star affects our Mc−Porb

relation by evolving two further sets of models with Z = 0.01, halfsolar and Z = 0.03, one and a half solar. As metallicity falls, opacitydrops and stars are smaller for the same luminosity and hence coremass, so the final orbital period for a given white dwarf mass issmaller. The results are shown in Fig. 9. The change in the rela-tion with metallicity is larger than for any of the other effects wehave tested, but all three models remain in good agreement with theobserved systems.

2.10 Comparison with previous work

The relation between the mass of the white dwarf remnant and theorbital period after mass transfer (Mc−Porb) has been computed anumber of times in the past. Rappaport et al. (1995) used 11 stellarmodels with various masses between 0.8 and 2 M�, computed byHan, Podsiadlowski & Eggleton (1994) with Eggleton’s STARS code(Eggleton 1971) to obtain the relation between the radius of a star onthe red giant branch and its core mass in the range 0.15–1.15 M�.They then used this to derive an Mc−Porb relation. Tauris & Savonije(1999) conducted a detailed study of the thermal response of thedonor star to mass loss in order to examine the evolution of the masstransfer (non-conservative in this case) and to monitor its stability.They used Eggleton’s STARS code to compute 121 models of theevolution of different binary systems, with a low-mass donor and aneutron star companion, through stable mass transfer. Their donormasses were between 1 and 2 M�, a neutron star of 1.3 M� andinitial orbital periods between 2 and 800 d. They obtained Mc−Porb

relations and devised fitting formulae for various chemical com-positions. Podsiadlowski, Rappaport & Pfahl (2002) made a sys-tematic study of low- and intermediate-mass binary systems withdonor masses from 0.6 to 7 M� (a much wider range than Tauris &

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Savonije 1999), a neutron star of 1.4 M� and initial orbital periodsbetween 0.17 and 100 d. They used an updated version of the stel-lar evolution code described by Kippenhahn, Weigert & Hofmeister(1967) to make 100 binary star models, in which half of the mass lostby the donor was accreted by the companion. Their models showedan enormous variety of evolutionary channels which they suggestmay be the reason behind the diversity in the observed populationof these systems. De Vito & Benvenuto (2010) made many detailedevolutionary models with a wide range of donor (0.5–3.5 M�) andaccretor (0.8–1.4 M�) masses and orbital periods at the onset ofRLOF from 0.5 to 12 d with a code described by Benvenuto &De Vito (2003). Their models show signs of departure from theRappaport Mc−Porb relation at core masses below 0.25 M� but arein better agreement with the Mc−Porb relation found by Tauris &Savonije (1999). They studied the dependence of the neutron starmass on the Mc−Porb relation and later the dependence of the masstransfer efficiency on the Mc−Porb relation (De Vito & Benvenuto2012). Shao & Li (2012) explored low- and intermediate-mass sys-tems, similarly to Podsiadlowski et al. (2002), with the CambridgeSTARS code, with donor masses of 1–6 M� and accretor masses of1.0–1.8 M�. They modelled the Mc−Porb relation for donors from1 to 5 M�, compared their relations to observed possible MSP –helium white binaries – and explored magnetic braking in thesesystems.

The fits of Tauris & Savonije (1999) have been most extensivelyused in the past so we compare our models with theirs in Fig. 10.Their fits are of the form

Mwd

M�=

(Porb

b

)1/a

+ c, (8)

with

(a, b, c) ={

(4.50, 1.2 × 105, 0.120) Pop I,

(5.00, 1.0 × 105, 0.100) Pop II,(9)

for Population I stars and Population II stars. Our models comparewell with their Population I stars. Despite the fact that lower metal-licities are a better fit to two of the measured companion masses,

Figure 10. Comparison with the work of Tauris & Savonije (1999). Thedot–dashed line represents our canonical relation. The two solid curvesrepresent fits of Tauris & Savonije (1999) for Population I stars (leftmost)and Population II stars (rightmost).

we use solar metallicity because we exclude BMSPs in globularclusters. The remaining systems fit better or equally well at solarmetallicity.

3 SE L E C T I O N O F B M S P s

The ATNF catalogue (Manchester et al. 2005) currently lists 293BMSPs with Pspin ≤ 30 ms. Of these, 101 are in the Galactic fieldand have measured binary mass functions. It is important to excludeall BMSPs that have not been recycled by accretion from a red gi-ant with a helium core but otherwise we wish to test the hypothesisthat all remaining systems did indeed form by this route. We ex-clude any BMSPs in globular clusters because their formation canbe more convoluted (Lorimer 2008). B1257+12 can be excludeddirectly because the orbit is for its planet of mass only 0.02 M⊕(Konacki & Wolszczan 2003) and J1502−6752 is excluded becauseit has an ultralight companion (Manchester et al. 2005). Nor did weinclude B1620−26 because it is identified as being a multiple sys-tem in the ATNF Pulsar Catalogue so the other components in thesystem could have altered its evolution and J1903+0327 becauseit has a main-sequence star companion and is in a multiple system.Stars below 2.3 M� suffer a helium flash when their degeneratecore reaches 0.472 M�. This is the maximum mass that can bereached by a helium white dwarf companion and corresponds toa final orbital period of 940 d. Stars shrink during core heliumburning and by the time they have grown sufficiently to fill theirRoche lobes again their cores are generally above 0.5 M�. There istherefore a distinction between MSPs recycled by red giants withhelium cores (Porb < 940 d) and those recycled by asymptotic giants(Porb > 940 d). This distinction is apparent in the figures of Hurleyet al. (2010). We therefore impose an upper period limit of 940 d.At low orbital periods, we must exclude systems in which masstransfer began while the donor star was still on the main sequence(Case A). The observed orbital period distribution for BMSPs ap-pears to have a gap between 0.73 and 1.19 d (Fig. 11). Above this

Figure 11. The distribution of all BMSP orbital periods less than 1000 d.Those systems with Porb > 940 d are not expected to have helium whitedwarf companions. Below the gap apparent around Porb = 1 d, we can expectcontamination from systems that have undergone Case A mass transfer andnow have low-mass hydrogen-rich companions.

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we can easily form systems with helium white dwarf companionswhen magnetic braking is included in our models. Below it there islikely to be contamination by systems that have undergone Case Amass transfer and still have a hydrogen-rich companion. We assumethat the gap represents a good discriminator between the two popu-lations and so exclude all systems with Porb < 1 d. All the remainingsystems are listed in Table A1 which can be found in the Appendix.In practice, we might expect most of the MSPs to have formed fromrelatively low-mass progenitors in electron-capture supernovae. Insuch stars, degenerate cores collapse at 1.375 M� (Nomoto 1984).They may lose about 0.1 M� of their gravitational mass dependingon the equation of state of the neutron star but then probably needto accrete a further 0.1 M� to be spun up to millisecond periods.The measured masses listed in Table 1 suggest that the MSPs usu-ally accrete more than this. Nevertheless, in the next section, wedescribe how we further exclude systems that cannot fit the modelwith a neutron star less massive than 1.2 M� and we use this ratherconservative minimum to avoid excluding systems with the lowestmeasured masses of pulsars (Zhang et al. 2011). Only three systemsneed to have a pulsar mass 1.2 < Ma/M� < 1.35.

4 O R B I TA L I N C L I NAT I O N S

In the recycling model, the MSP has been spun up by accretionthrough a disc of material transferred from a binary companiontidally locked to the binary orbit. The spin of the pulsar should thenbe aligned with the orbit. We expect orbital inclinations i to theline of sight to be uniformly distributed over a sphere so that thedistribution of cos i should be flat.

Backer (1998) explored the spin periods, ages and magnetic fieldsof Galactic helium white dwarf MSPs. Modelling fixed masses forboth the neutron star and the white dwarf, he concluded that ob-served inclinations are not randomly distributed but rather tend to behigh. To explain this, he suggested that pulsar beams preferentiallylie parallel to the orbital plane and so perpendicular to the pulsarspin axis. The following year, Thorsett & Chakrabarty (1999) usedthe Mc−Porb relation derived by Rappaport et al. (1995) to concludethat the observed inclinations of binary MSPs are consistent witha random distribution. Stairs et al. (2005) similarly investigated alarger data set with the Mc−Porb relations of both Tauris & Savonije(1999) and Rappaport et al. (1995). They concluded that the exist-ing forms of the Mc−Porb relations overestimate the white dwarfmasses at large periods and are also in conflict for the shorter periodsystem. With a Kolmogorov–Smirnov (KS) test on the cumulativedistribution of orbital inclinations, they found that the Mc−Porb re-lation of Tauris & Savonije (1999) is incompatible at the 99.5 percent level when they drew the pulsar masses from a Gaussian cen-tred on 1.35 M� with width 0.04 M� (mirroring the neutron starmass distribution found by Thorsett & Chakrabarty 1999). Repeat-ing the same investigation for neutron stars picked from a Gaussiancentred on 1.75 M� with width 0.04 M�, they reached agreementat the 50 per cent level. Our analysis here is equivalent to that ofStairs et al. (2005) applied to a larger data set.

If the standard recycling model is correct, we expect the systemsto be oriented randomly in space so that the probability P(i) di offinding an inclination i between i and i + di should be such thatP(i) ∝ sin i. We are interested exclusively in the systems that haveperiods within range of the Mc−Porb relations consistent with CaseB RLOF. In the ATNF catalogue, the minimum possible mass Mm,given the observed mass function and a pulsar mass of 1.35 M�

is listed. This can be transformed back to the binary mass functionf by

f = (Mc sin i)3

(Ma + Mc)2= M3

m

(1.35 M� + Mm)2, (10)

where Ma is the actual mass of the pulsar. We list the mass functionsf in Table A1, along with the companion masses Mc obtained fromPorb with our canonical relation by a spline interpolation. For thecombination of f and Mc, there is a maximum mass Mamax of pulsarthat can be accommodated in the system. We list this in Table A1 too.When Mamax < 1.2 M�, we exclude the system from the analysis.Such cases are marked with bullets. Note that all systems foundto have an ATNF minimum companion mass Mc > 0.5 M�, toolarge to be a He white dwarf and marked with stars in Table A1, areautomatically excluded by this cut. Inclinations are then given by

i = sin−1

((f (Ma + Mc)2)1/3

Mc

), (11)

and we expect cos i to be uniformly distributed between 0 and 1if the inclinations are consistent with being picked from a uniformdistribution.

For each neutron star mass Ma ∈ {1.35, 1.55, 1.75}M�, we cal-culate the inclinations for all remaining systems in Table A1. Whenthe maximum permitted pulsar mass is less than the chosen Ma,we assign i = 90◦ and thereby use Ma = Mamax for such systems.Figs 12–14 are histograms of the numbers of BMSPs over cos isplit into two orbital period ranges for the three different choicesof neutron star mass. There are 28 systems with Porb < 12 d and29 systems with longer periods. The distributions are reasonablyuniform but show a distinct dearth of high-inclination systems forMa = 1.35 M�. This is less extreme as Ma is raised to 1.55 M�and becomes an excess when Ma = 1.75 M�. To examine the sig-nificance of these trends, we use KS tests on the cumulative distri-butions of cos i.

Figure 12. Histogram of the number density of the MSP binary systemswith respect to cos i, assuming a current neutron star of mass 1.35 M�. Thesolid histogram represents just the low-period systems (1–12 d), the lightlyshaded histogram adds all the remaining high-period systems.

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Figure 13. Histogram of the number density of the MSP binary systemswith respect to cos i, assuming a current neutron star of mass 1.55 M�.The solid histogram represents just the low-period systems (1–12 d) and thelightly shaded histogram adds all the remaining high-period systems.

Figure 14. Histogram of the number density of the MSP binary systemswith respect to cos i, assuming a current neutron star of mass 1.75 M�.The solid histogram represents just the low-period systems (1–12 d) and thelightly shaded histogram adds all the remaining high-period systems.

4.1 KS tests

The KS test compares an observed cumulative distribution with amodel. In this case, we compare the observed distribution of cos iwith a uniform distribution. The KS statistic D is the maximumdifference between the observed cumulative distribution and theuniform cumulative distribution for any cos i and with this is asso-ciated the probability that D of this size or larger would be found fora sample of observations chosen at random from the uniform distri-bution (Press et al. 1992). Fig. 15 shows the cumulative distributionsof cos i for the three pulsar masses compared with the uniform dis-tribution. Recall that all the systems for which Ma > Mamax have

Figure 15. Cumulative distributions of cos i. The expected uniform distri-bution is the dashed line. The blue solid line represents the data when weassume the pulsar masses are 1.55 M� whenever possible and as large aspossible otherwise. The dotted line represents for 1.35 M� pulsars and thedot–dashed line for 1.75 M� pulsars. The upturn at cos i = 0 is artificialbecause any system that cannot accommodate a neutron star of the appro-priate mass is placed there. This artificial difference is not used for our KStests.

Table 2. KS test on cos i for Ma = 1.35 M�.

Ma/M� Porb/d range D Probability Nsys

1.35 Full 0.242 0.000 201 571.35 <12.3 0.257 0.0406 281.35 >12.4 0.266 0.0307 28



1.55 Full 0.120 0.359 571.55 <12.3 0.181 0.286 281.55 >12.4 0.142 0.591 28



1.75 Full 0.266 0.000 470 571.75 <12.3 0.122 0.772 281.75 >12.4 0.236 0.0747 28

been assigned cos i = 0 and this raises the cos i = 0 end of each dis-tribution in a perhaps biased manner. The results for Ma = 1.35 M�are however unaffected because the largest difference D is at largercos i. For the other two cases, we take D to be the largest differencebeyond the first data point that has cos i > 0. This eliminates thebias by treating the highest inclination systems as if they were asuniformly distributed as possible.

Tables 2–4 list the D and its significance for the three cases for allsystems combined and split into orbital period ranges. We discardthe system in the middle with Porb = 12.33 d so that the two ranges

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have precisely the same number of systems and it is meaningful tocompare the KS probabilities. For Ma = 1.35 M�, there is a signif-icant lack of high-inclination systems and for Ma = 1.75 M�, thereis a significant excess and both of these can be ruled out given theirvery low KS probabilities. For Ma = 1.55 M�, a distribution as dif-ferent as observed would be found 35 times in 100 random samplesso that the inclinations are consistent with a uniform distribution oforientations in space. Equivalent analysis for Ma = 1.45 M� givesa KS probability of 0.040, while for Ma = 1.65 M� it is 0.173.Neither of these are small enough to really rule out such neutronstar masses, but we conclude that anything in the range around1.55 ≤ Ma/M� ≤ 1.65 is quite acceptable. The KS test is weak forsmall data sets, but this conclusion remains apparent for the low andhigh systems taken independently. When Ma = 1.55 M�, the highprobabilities of around one- and two-thirds mean there is no reasonto suggest that the distribution or orbital inclinations change withperiod in this case. However, when Ma = 1.75 M�, the fit for thelow-period group is somewhat better than for the high-period group.This is consistent with what Stairs et al. (2005) found. However, thenumbers are still too small for this to be a significant conclusion,while the full distribution, likely to occur at random only once inabout 2000 samples, does significantly rule out such a large neutronstar mass.

For comparison, we apply the same analysis to the sample ofMSPs used by Backer (1998) and with Ma = 1.4 M�, as he assumed.Fig. 16 shows the cumulative distributions of cos i modelled withour canonical Porb−Mc and a neutron star mass of 1.4 M�, aschosen by Backer (1998). We exclude systems according to the samecriteria we have used for the full sample and assign inclinations ofi = 90◦ to systems with Mamax < 1.4 M�. The KS probability ofobtaining D = 0.334 or larger is 3.44 × 10−2 so that the data usedby Backer (1998) actually show dearth of high inclinations just aswe find with Ma = 1.35 M�. Backer (1998) based his conclusionon a comparison between the distribution of minimum masses and auniform distribution of sin i. However, it is the ratio of that minimummass to the actual mass of the companion, that is, sin i. In Fig. 17,

Figure 16. Cumulative distribution of cos i for the sample of 15 MSPs usedby Backer (1998) that fit our selection criteria. The dot–dashed line is theCDF of the expected uniform distribution. The solid line is for Backer’ssample of systems with Porb > 1 d obtained with our canonical Porb−Mc

relation and a neutron star mass of 1.4 M�.

Figure 17. Histogram of the ratio of the minimum companion mass toa pulsar of 1.4 M� to its actual mass if it lies on our Porb−Mc relationcompared to the distribution of Mcmin /Mc = sin i for randomly orientedorbits (solid line).

we show the distribution of this ratio when we assume that the actualcompanion mass is known from Porb. This distribution then showsa dearth of systems at large sin i quite contrary to his original claim.The conclusion that pulsar beams tend to be perpendicular to theirspin axis is therefore no longer tenable.

5 C O N C L U S I O N

We have demonstrated that the relation between orbital period andcompanion white dwarf mass of recycled MSPs that have evolvedvia Case B RLOF is largely independent of initial donor mass,neutron star mass and how conservative is the mass transfer. Itdoes however depend on the metallicity of the donor star becausethis changes the radius to core mass relation for red giants. For acompanion of 1 M�, about the smallest that can be expected toevolve within the Galactic field, the lowest orbital period BMSPsformed without additional angular momentum loss have Porb ≈6.2 d. However, with magnetic braking, or some alternative angularmomentum loss mechanism, the relation can be extended down toPorb < 1 d.

We selected all Galactic BMSPs, with Pspin < 30 ms, that couldhave been recycled by accretion from a red giant with a heliumcore that is now a He white dwarf companion from those listedin the ATNF catalogue. With our Porb−Mc relation, we calculatedthe orbital inclinations of these systems, assuming a set of pulsarmasses. We find that when the pulsar mass is taken to be as largeas possible, up to a maximum of 1.55 M�, the distribution of incli-nations is consistent with random orientation of the orbits in space.If the maximum pulsar mass is reduced to 1.35 M�, there appearsa dearth of high-inclination systems, and if it is raised to 1.75 M�,there appears an excess. Hence, we deduce that all systems, selectedand listed in Table A1 without bullets, are consistent with havingpulsars of mass around 1.6 M�, having been recycled by accretionfrom a red giant with a helium core and having no observationalbias towards particular orbital orientation. Given that the selec-tion of particular orbital inclinations would be encouraged if MSP

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magnetic axes and hence their beams were preferentially alignedwith or orthogonal to their spin axes, our result is consistent withrandom orientation of magnetic axes too. Numbers of systems re-main too small to make conclusions about systems in differentorbital period ranges.

AC K N OW L E D G E M E N T S

SLS thanks STFC for her studentship. CAT thanks Churchill Col-lege for his fellowship. LF and DTW thank the Institute of Astron-omy for supporting visits. SLS thanks Philip Hall for help in makingdetailed stellar models with mass transfer. We also thank the refereefor many suggestions that have led to significant improvements tothis work.

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APPENDI X A : LI ST O F SELECTED BMSPs

Table A1 lists all ATNF MSPs, with Pspin < 30 ms and Porb > 1 din the Galactic field. The fourth column lists the companion massderived from our Porb−Mc relation. The fifth column is the measuredmass function, reconstructed from the minimum mass given in theATNF catalogue. The sixth column is the maximum neutron starmass that could be accommodated with the given mass functionand derived companion mass. Systems in which Mamax < 1.2 M�are marked and excluded from further analysis. The final columnindicates whether there is any determination of the companion’stype.

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Table A1. Binary MSPs in the ATNF Pulsar Catalogue in the Galactic field with Porb > 1 d. Systems with superscript B were in thesample of Backer (1998) and those with a subscript C have measured masses (see Table 1). The � symbol denotes systems with minimummasses in the ATNF catalogue larger than 0.5 M�. The pulsar spin period is Pspin and Porb is the orbital period. The companionmass Mc is computed from Porb with our canonical relation. The measured mass function f is calculated from Mm given in the ATNFcatalogue and Mamax is then the maximum pulsar mass that can be accommodated given f and Mc. The • symbol denotes systems forwhich Mamax < 1.20 M� and therefore do not fit the model. The final column lists any information on or suggestion of the type of thecompanion to be found in references given in the ATNF catalogue. MS = Main-sequence star, CO = CO or ONeMg white dwarf, He =He white dwarf, UL = ultralight companion or planet (Mc < 0.08 M�) and U = undetermined companion type. A letter T at the enddenotes that there are more than two components in the system. We excluded the systems with UL companions, MS companions andmultiple systems. Many of the MSPs were thought to have CO companions have minimum masses in the ATNF catalogue larger than0.5 M�. All are excluded because they require a minimum neutron star mass of less than 1.20 M�.

Name Pspin/ms Porb/d Mc/M� f /M� Mamax /M� Comp. type

J0613−0200B 3.061 844 086 531 89 1.198 512 5753 0.187 0.000 971 85502 2.408 HeJ1435−6100� 9.347 972 210 248 1.354 885 2170 0.191 0.138 321 89 0.033• COJ2043+1711 2.379 878 960 26 1.482 290 809 0.194 0.002 092 8803 1.675 HeJ1909−3744C 2.947 108 068 107 6401 1.533 449 474 590 0.195 0.003 121 9739 1.350 HeJ0034−0534B 1.877 181 884 5850 1.589 281 801 0.197 0.001 263 4271 2.257 HeJ1622−6617 23.623 444 739 27 1.640 635 183 0.198 0.000 374 72931 4.346 UJ0101−6422 2.573 151 972 1683 1.787 596 706 0.201 0.001 653 8449 2.008 HeJ1231−1411 3.683 878 711 077 1.860 143 882 0.202 0.002 644 5979 1.559 HeJ1949+3106� 13.138 183 343 7040 1.949 535 0.203 0.109 386 10 0.073• COJ0218+4232B 2.323 090 467 8309 2.028 846 084 0.204 0.002 038 4352 1.832 HeJ1439−5501� 28.634 888 190 455 2.117 942 520 0.205 0.227 594 46 –0.011• COJ2017+0603 2.896 215 815 562 2.198 481 129 0.205 0.002 342 6589 1.716 HeJ2317+1439B 3.445 251 071 0225 2.459 331 464 0.208 0.002 199 4253 1.808 HeJ1502−6752 26.744 423 7673 2.484 4570 0.208 5.570 8571e−06 39.908 ULB1802−07 23.100 855 284 30 2.616 763 35 0.209 0.009 449 1902 0.774• UJ1911−1114B 3.625 745 571 3977 2.716 557 61 0.210 0.000 797 098 48 3.196 HeJ1431−5740 4.110 543 956 7658 2.726 855 823 0.210 0.001 688 7826 2.131 UJ1748−3009 9.684 273 2.933 8198 0.212 0.000 286 955 98 5.545 HeJ1216−6410 3.539 375 658 423 4.036 727 18 0.220 0.001 669 4467 2.300 HeJ1045−4509B 7.474 224 226 211 33 4.083 529 2547 0.220 0.001 764 9364 2.235 HeJ1337−6423� 9.423 406 713 4.785 334 07 0.224 0.105 082 01 0.103• COJ1745−0952 19.376 303 441 1294 4.943 453 386 0.225 0.000 591 263 44 4.170 HeJ1732−5049 5.312 550 289 078 45 5.262 997 206 0.226 0.002 449 0954 1.951 HeJ0721−2038 15.542 394 974 00 5.460 832 80 0.227 0.029 704 982 0.400• UJ0437−4715B

C 5.757 451 924 362 137 5.741 046 46 0.227 0.001 243 1183 2.835 He

J1545−4550 3.575 288 618 847 12 6.203 064 928 0.227 0.001 588 5663 2.480 UJ1811−2405 2.660 593 316 90 6.272 302 04 0.227 0.005 069 2659 1.289 HeJ1603−7202B 14.841 952 249 089 35 6.308 629 6703 0.227 0.008 788 2681 0.925• UJ1017−7156 2.338 514 440 1138 6.511 898 8121 0.227 0.002 853 1163 1.800 HeJ2129−5721B 3.726 348 482 966 41 6.625 493 093 0.228 0.001 049 2057 3.123 HeJ1835−0114 5.116 387 644 239 6.692 5427 0.228 0.002 417 1121 1.983 HeJ2145−0750 16.052 423 914 336 60 6.838 93 0.228 0.024 105 367 0.474• COJ1022+1001� 16.452 929 950 784 05 7.805 130 2826 0.233 0.083 054 673 0.157• COJ1543−5149 2.056 960 392 42 8.060 773 04 0.234 0.004 496 8769 1.453 HeJ0621+1002 28.853 860 730 049 8.318 6813 0.235 0.027 026 827 0.457• COJ1327−0755 2.677 923 197 1205 8.439 086 019 0.235 0.004 425 1757 1.479 HeJ1614−2230C 3.150 807 653 4271 8.686 619 4196 0.236 0.020 483 367 0.565• COJ1125−6014 2.630 380 739 7848 8.752 603 53 0.236 0.008 127 9214 1.036• HeJ1400−1438 3.084 233 2007 9.547 43 0.238 0.007 034 2534 1.149• HeJ1841+0130 29.772 775 3332 10.471 626 0.241 0.000 421 290 66 5.523 HeJ1918−0642 7.645 872 883 908 84 10.913 177 7486 0.242 0.005 249 4287 1.403 HeJ1804−2717B 9.343 030 685 703 11.128 7115 0.243 0.003 346 9189 1.825 HeB1855+09B

C 5.362 12.327 19 0.246 0.005 557 3963 1.389 HeJ1900+0308 4.909 239 016 418 12.476 021 44 0.246 0.002 089 9584 2.425 HeJ2236−5527 6.907 549 392 921 12.689 187 15 0.247 0.004 507 0041 1.578 UJ1933−6211 3.543 431 438 847 12.819 406 50 0.247 0.012 103 419 0.869• UJ1600−3053 3.597 928 508 655 47 14.348 457 7709 0.250 0.003 556 0324 1.851 HeJ1741+1351 3.747 1544 16.335 0.254 0.005 399 7280 1.491 HeJ0900−3144 11.109 649 157 3889 18.737 635 76 0.259 0.015 693 887 0.795• UJ1801−3210 7.453 584 373 41 20.771 6995 0.263 0.001 185 1566 3.646 HeJ1709+2313 4.631 196 277 8409 22.711 892 38 0.265 0.007 438 2973 1.319 HeJ1618−39 11.987 313 22.8 0.265 0.002 217 7665 2.638 HeB1257+12 6.218 531 948 400 48 25.262 0.268 567 00 0.000 000 0 ∞ UL

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Table A1 – continued

Name Pspin/ms Porb/d Mc/M� f /M� Mamax /M� Comp. type

J1844+0115 4.185 543 936 664 50.645 8881 0.295 0.001 191 8677 4.341 HeJ1825−0319 4.553 527 919 736 52.630 4992 0.296 0.002 362 4627 3.020 UJ0614−3329 3.148 669 579 439 53.584 6127 0.297 0.007 895 1142 1.523 HeJ2033+1734 5.948 957 534 8502 56.307 795 27 0.299 0.002 775 9932 2.799 HeJ1910+1256C 4.983 583 940 565 11 58.466 742 029 0.300 0.002 962 8574 2.720 HeJ1713+0747B

C 4.570 136 525 082 782 67.825 129 8718 0.306 0.007 896 1829 1.595 HeJ1455−3330B 7.987 204 796 261 76.174 5676 0.310 0.006 271 5859 1.869 HeJ1125−5825 3.102 213 918 9504 76.403 2169 0.310 0.007 001 0555 1.754 HeJ2019+2425B 3.934 524 080 331 24 76.511 634 79 0.310 0.010 686 460 1.361 HeJ1737−0811 4.175 017 312 8551 79.517 379 0.312 0.000 138 039 25 14.497 UJ1850+0124 3.559 763 768 043 84.949 858 0.314 0.005 848 3603 1.989 HeJ1935+1726 4.200 101 791 882 90.763 89 0.317 0.004 260 4451 2.416 HeJ2229+2643B 2.977 819 294 7556 93.015 8934 0.318 0.000 839 499 25 5.866 HeJ1903+0327� 2.149 912 364 349 21 95.174 118 753 0.319 0.139 558 57 0.163• MSTJ1751−2857 3.914 873 196 3690 110.746 4576 0.325 0.003 013 0429 3.051 HeJ1853+1303C 4.091 797 381 456 16 115.653 786 43 0.327 0.005 439 6584 2.207 HeB1953+29B 6.133 166 510 2401 117.349 097 28 0.327 0.002 416 7741 3.485 HeJ2302+4442 5.192 324 646 411 125.935292 0.331 0.009 209 5292 1.650 UJ1643−1224B 4.621 641 516 998 18 147.017 397 76 0.338 0.000 782 972 26 6.671 HeJ1708−3506 4.505 158 948 26 149.133 18 0.338 0.001 828 7070 4.261 HeJ1640+2224B 3.163 315 817 91380 175.460 661 94 0.346 0.005 907 4572 2.302 HeB1620−26 11.075 750 914 2025 191.442 81 0.350 0.007 974 8112 1.972 HeTJ0407+1607 25.701 739 194 63 669.0704 0.437 0.002 893 1881 4.939 He

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the nature of millisecond pulsars with helium white dwarf ... · the data for all bmsps with helium...

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