Direct formation of millisecond pulsars from rotationally delayed accretion-induced collapse of massive white dwarfs
aa r X i v : . [ a s t r o - ph . S R ] M a y Mon. Not. R. Astron. Soc. , 000–000 (0000) Printed 28 July 2018 (MN L A TEX style file v2.2)
Direct formation of millisecond pulsars from rotationally delayedaccretion-induced collapse of massive white dwarfs
Paulo C. C. Freire ⋆ , Thomas M. Tauris , ⋆ Max-Planck-Institut f¨ur Radioastronomie, Auf dem H¨ugel 69, D-53121 Bonn, Germany Argelander-Institut f¨ur Astronomie, Universit¨at Bonn, Auf dem H¨ugel 71, D-53121 Bonn, Germany.
28 July 2018
ABSTRACT
Millisecond pulsars (MSPs) are believed to be old neutron stars, formed via type Ib/c core-collapse supernovae, which have subsequently been spun up to high rotation rates via ac-cretion from a companion star in a highly circularised low-mass X-ray binary. The recentdiscoveries of Galactic field binary MSPs in eccentric orbits, and mass functions compatiblewith that expected for helium white dwarf companions, PSR J2234+06 and PSR J1946+3417,therefore challenge this picture.ms.tex Here we present a hypothesis for producing this newclass of systems, where the MSPs are formed directly from a rotationally-delayed accretion-induced collapse of a super-Chandrasekhar mass white dwarf. We compute the orbital proper-ties of the MSPs formed in such events and demonstrate that our hypothesis can reproduce theobserved eccentricities, masses and orbital periods of the white dwarfs, as well as forecastingthe pulsar masses and velocities. Finally, we compare this hypothesis to a triple star scenario.
Key words: stars: neutron — white dwarfs — stars: rotation — X-rays: binaries — super-novae: general — pulsars: general
Almost since the discovery of PSR B1937+21, the first mil-lisecond pulsar (MSP, Backer et al. 1982), it has been suggestedthat these objects are old neutron stars spun up to high spinfrequencies of several hundred Hz via accretion of mass andangular momentum from a companion star (Alpar et al. 1982;Bhattacharya & van den Heuvel 1991). In this so-called recyclingphase, the system is first observable as a low-mass X-ray binary(LMXB, e.g. Bildsten et al. 1997), later as an accreting X-ray MSP(Wijnands & van der Klis 1998), or even as an MSP in the transi-tion phase between an accretion powered MSP and a rotation pow-ered radio MSP (Archibald et al. 2009; Papitto et al. 2013; Tauris2012).An inevitable consequence of a long phase ( − yr )of recycling in an LMXB, where tidal forces operate, is thatit should leave a fossil record of a highly circular system(Phinney & Kulkarni 1994). And indeed, until recently, all of themore than 100 observed, fully recycled MSPs (here defined aspulsars with spin periods less than 20 ms), in binaries with he-lium white dwarf (He WD) companions and located outside ofglobular clusters, have very small eccentricities in the range e =10 − − − ( ATNF Pulsar Catalogue , Manchester et al. 2005).Pulsar systems in globular clusters, on the other hand, often havetheir orbits perturbed after the recycling phase terminates becauseof their location in a dense environment (Rasio & Heggie 1995; ⋆ E-mails: [email protected] / [email protected]
Heggie & Rasio 1996). Until the start of 2013, the only known fullyrecycled MSP with a high eccentricity, and located in the Galacticfield, was PSR J1903+0327 ( e = 0 . , Champion et al. 2008). ThisMSP has a G-type main-sequence companion star and is thoughtto have originated from a hierarchical triple system that ejectedone of its members (Freire et al. 2011; Portegies Zwart et al. 2011;Pijloo et al. 2012). Recently, Deneva et al. (2013) presented the discovery ofPSR J2234+06 and soon afterwards Barr et al. (2013) an-nounced the discovery of PSR J1946+3417. PSR J2234+06 andPSR J1946+3417 are of special interest because they resemble eachother and share very unusual properties. Both of these Galactic fieldpulsars (see Table 1) have a spin period, P ≃ , an orbital pe-riod, P orb ≃
30 days and a companion mass, M ≃ . M ⊙ .All these values are within typical ranges expected for MSPs withHe WD companions. However, both of these MSP binaries are alsoeccentric, e ≃ . , which is unusual and unexpected from currentformation theories of MSPs, as explained above. Therefore, it isclear that these system must have a formation history which is dif-ferent from the “normal” MSP-WD systems in the Galactic field.We notice that the two median expectations for the companionmasses of PSR J2234+06 and PSR J1946+3417 are very similar toeach other and close to the values expected from the correlationbetween WD mass and orbital period for post-LMXB systems (e.g.Tauris & Savonije 1999); in particular if the slight widening of the c (cid:13) Freire & Tauris
Table 1.
Physical parameters of two newly discovered binary MSPs (datataken from Deneva et al. 2013; Barr et al. 2013).Pulsar
P P orb M , med e J2234+06 3.6 ms 32 days . M ⊙ . M ⊙ orbit from the event that imparted the eccentricity is accounted for,see Section 3. This provides confidence that the current companionsare indeed He WDs which have lost their hydrogen envelopes viastable Roche-lobe overflow. Optical detections would confirm this. By analogy with PSR J1903+0327, one could advance the hy-pothesis that both PSRs J2234+06 and J1946+3417 originated ashierarchical triple systems, which evolved to produce a neutronstar orbited by two F/G-type dwarfs. Because of the widening ofthe inner orbit during the subsequent neutron star accretion in theLMXB phase, the systems later became dynamically unstable (e.g.Mikkola 2008) and one of the components was eventually ejected.The only difference being that it was the donor star (the WDprogenitor) in the inner binary which was ejected in the case ofPSR J1903+0327 (Freire et al. 2011; Portegies Zwart et al. 2011;Pijloo et al. 2012), whereas it would have been the outer tertiarystar in the cases of PSR J2234+06 and PSR J1946+3417. In a triplesystem, the Kozai process (Kozai 1962) may lead to large cyclicvariations in the inner orbital eccentricity prior to ejection of thetertiary star (e.g., Mardling & Aarseth 2001). Hence, one may ex-pect a wide range of eccentricities of the surviving MSP-WD bina-ries. Whether or not triple star evolution, or formation and ejectionof a binary system from a dense cluster, is plausible for the rela-tively small eccentricities ( e ≃ . ) observed in PSRs J2234+06and J1946+3417 (compared to e = 0 . for PSR J1903+0327) re-quires detailed modelling beyond the scope of this Letter.Here we advocate for another solution. In Section 2 we presenta hypothesis of a new direct formation channel of MSPs which canexactly explain both the unusual properties and the similarities ofthe recently discovered MSPs. In Section 3 we present simulationsand make further falsifiable predictions about these systems whichcan be tested in the near future. In Section 4 we discuss future per-spectives and we summarise our conclusions in Section 5. Besides from formation via core-collapse supernovae, it has beensuggested for many years that neutron stars may also be producedfrom accretion-induced collapse (AIC) of a massive ONeMg WD ina close binary (Nomoto et al. 1979; Taam & van den Heuvel 1986).The properties of such neutron stars are unknown. It has beensuggested that AIC events cannot produce MSPs directly since r-mode instabilities would spin-down any young, hot MSP on a veryshort timescale (Andersson et al. 1999). However, if the scenariodescribed here is confirmed by further observations, then the roleof r-mode instabilities has to be revised.In the following, we rely on the results of the recent modellingby Tauris et al. (2013). They investigated a scenario where MSPs are produced indirectly via AIC, i.e. the AIC leaves behind a nor-mal neutron star which is subsequently recycled to become an MSP,once the mass-transfer resumes after the donor star refills its Rochelobe and continues LMXB evolution until the end. Their main resultis that as a consequence of the finetuned mass-transfer rate neces-sary to make the WD grow in mass, the resultant MSPs created viathe AIC channel preferentially form with < P orb <
60 days ,clustering more at P orb ≃ −
40 days . Furthermore, the mod-elling of these systems produced He WD companions with masses, M WD ≃ . − . M ⊙ . These values are interesting since theymatch exactly the observed values of P orb and M WD for the newlydiscovered MSPs in eccentric orbits (Table 1). However, in theTauris et al. (2013) scenario of indirect formation of MSPs, con-tinued post-AIC mass transfer leads to highly circularised systems.Therefore, that scenario cannot explain the newly discovered MSPswith e ∼ . . In case a mass-gaining WD is spun up to rapid rotation vianear-Keplerian disk accretion (Langer et al. 2000), it can avoidAIC (Yoon & Langer 2004, 2005) and evolve further to super-Chandrasekhar mass values via continuous accretion (cf. fig. 7 inTauris et al. 2013, for the possible growth up to & M ⊙ ).Here we propose a scenario, where accretion leads to the for-mation of a super-Chandrasekhar mass ONeMg WD which initiallyavoids AIC as a result of rapid rotation. Only after the accretion hasterminated, and the WD loses sufficient spin angular momentum(see below), does it undergo AIC to directly produce an MSP. Weshall refer to this event as rotationally-delayed accretion-inducedcollapse (RD-AIC), see Fig. 1.It is important to notice that under the new hypothesis pre-sented here, accretion ceases completely before AIC occurs. Atthat stage the detached system consists of two WDs: a low-massHe WD (the remnant of the former donor star) and an ONeMg WDwith a mass above the Chandrasekhar limit, and which later under-goes RD-AIC. Hence, in this case there will be no re-circularisationafter the AIC event.
Observations of binaries confirm that accreting WDs rotatemuch faster than isolated ones (Sion 1999); in one case,HD 49798/RX J0648, there is even evidence for a WD rotatingwith a spin period of only P WD = 13 . (Mereghetti et al. 2011),corresponding to ∼ per cent of its critical (break-up) rotation fre-quency. The observational evidence for such fast rotation supportsthe increase of the mass stability limit above the standard valuefor non-rotating WDs ( . M ⊙ ), as required by our scenario. Ananalogous idea of rotationally-delayed SNe Ia explosions has beenproposed by Justham (2011) and Di Stefano et al. (2011) for mas-sive CO WDs.For WDs with rigid body rotation, the resulting super-Chandrasekhar masses are in the range . − . M ⊙ (Yoon & Langer 2004, and references therein). For differentiallyrotating WDs, the stability limit may in principle reach ∼ . M ⊙ ,although it is quite possible that efficient transport of angular mo-mentum by magnetic torques and/or baroclinic instabilities acts toensure rigid rotation (Piro 2008). On the other hand, recent obser-vations of exceptionally luminous SNe Ia (e.g. Howell et al. 2006; c (cid:13) , 000–000 irect formation of MSPs from delayed AIC Figure 1.
Illustration of the binary stellar evolution from the zero-age mainsequence (ZAMS) to the final millisecond pulsar (MSP) stage. A primary − M ⊙ star evolves to initiate Roche-lobe overflow (RLO) towards the ∼ M ⊙ secondary star, leading to dynamically unstable mass transferand the formation of a common envelope (CE; Ivanova et al. 2013). Theenvelope ejection leads to formation of an oxygen-neon-magnesium whitedwarf (ONeMg WD) from the naked core of the primary star (possibly aftera stage of Case BB RLO from the naked core – not shown). When the sec-ondary star evolves, it initiates RLO leading to a cataclysmic variable (CV)X-ray binary system. As a result of accretion the WD becomes a rapidlyspinning super-Chandrasekhar mass WD. After accretion has terminatedit loses spin angular momentum and eventually undergoes a rotationally-delayed accretion-induced collapse (RD-AIC) to directly form an MSP witha helium white dwarf (He WD) companion in an eccentric orbit. Stellarmasses given in units of M ⊙ . Scalzo et al. 2010) suggest that their WD progenitors had a mass of ∼ . − . M ⊙ . If the critical rotation frequency is obtained dur-ing accretion then further mass accumulation is prohibited, unlessangular momentum is transported from the WD back to the disk byviscous effects (Popham & Narayan 1991; Saio & Nomoto 2004).The final fate of super-Chandrasekhar ONeMg WDs dependson whether or not the effects of electron captures dominate over nu-clear burning (Nomoto et al. 1979; Nomoto & Kondo 1991). Theonset of electron captures on Mg and Ne occurs at a density of ρ ∼ × g cm − , whereas the density for the ignition of explo-sive nuclear burning (oxygen deflagration) depends on the centraltemperature. Therefore, after accretion has terminated, the final fateof a super-Chandrasekhar WD depends on the competition betweenits cooling rate and its loss of angular momentum, as demonstratedin detail by Yoon & Langer (2004, 2005). If the WD interior hascrystallised by the time its spin angular momentum decreases be-low the critical level (corresponding to J AICcrit ≃ . × erg s ,for a . M ⊙ WD) it undergoes RD-AIC.Yoon & Langer (2004, 2005) discussed the loss of WD spinangular momentum due to gravitational wave emission caused byso-called CFS instabilities to non-axisymmetric perturbations. Intheir second paper, these authors investigated 2-dimensional mod-els and found that only r-mode instabilities (Andersson 1998)are relevant for accreting WDs, whereas bar-mode instabilities(Chandrasekhar 1970; Friedman & Schutz 1978) are irrelevant be-cause the ratio of rotational to potential energy cannot reach thecritical limit of
T /W = 0 . (corresponding to J = 4 × erg s ). The estimated timescale of removing (or redistribut-ing) angular momentum has been estimated to be in the range − yr , depending on T /W and the degree of differen-
Figure 2.
Schematic evolutionary track (blue solid line) in the ( M WD , J )–plane calculated for an accreting ONeMg WD in a close binary system. Af-ter termination of the mass-transfer process, the super-Chandrasekhar massWD is rapidly spinning, which prevents its collapse. From this point, thespin evolution is solely determined by loss of angular momentum (e.g. ascaused by r-mode instabilities and magnetodipole radiation). The light-bluehatched area marks the critical region, J J AICcrit (Yoon & Langer 2005) atwhich boundary the WD undergoes a RD-AIC event and produces a MSP.The red dashed line indicates critical break-up rotation. tial rotation of the WD (Lindblom 1999; Yoon & Langer 2004,2005). However, recent work by Ilkov & Soker (2012) questionsthe efficiency of r-mode instabilities and hence they advocate fora very long delay timescale > . This would give the super-Chandrasekhar mass WD plenty of time to cool down, crystalliseand undergo RD-AIC, thus favouring our scenario.To summarize, we postulate that MSPs can be formed directly(without any need for further spin up from a companion star) in anRD-AIC event that happens up to ∼ after termination of themass-transfer phase.In Fig. 2 we show an evolutionary track of a rapidly spinningWD undergoing RD-AIC (see fig. 11 in Yoon & Langer 2005, formore detailed tracks). The WD is assumed to be non-spinning ini-tially and have a mass of . M ⊙ prior to accretion from its com-panion star. We assumed rigid rotation and efficient angular mo-mentum accretion at the Keplerian disk value. The r-mode instabil-ities (giving rise to loss of rotational energy via gravitational waves)were calculated during accretion following Lindblom (1999). If thetimescale of loss of spin angular momentum, from the terminationof the accretion phase until the WD has a spin angular momentum, J < J
AICcrit , is sufficiently long ( ∼ yr ) then the result is an AICevent (Yoon & Langer 2005). The implosion of a WD with a radius of about 3000 km and an as-sumed surface magnetic flux density, B ∼ G (e.g. Jordan et al.2007) into a neutron star with a radius of ∼
10 km should pro-duce, by conservation of magnetic flux, an MSP surface B-fieldof G × (3000 / ∼ G . The resultant neutron star musthave a spin rate below the break-up limit and for a typical MSP spinperiod of a few ms, it is expected that it must lose spin angular mo-mentum during the AIC, possibly by ejection of a few . M ⊙ ofbaryonic matter in a circumstellar disk. According to modelling byDessart et al. (2006); Kitaura et al. (2006); Metzger et al. (2009); c (cid:13) , 000–000 Freire & Tauris
Darbha et al. (2010), up to a few . M ⊙ of material is ejectedin the AIC event, possibly leading to synthesis of Ni in the diskwhich may result in a radioactively powered, short-lived SN-liketransient (that peaks within and with a bolometric lumi-nosity ≃ erg s − ).The RD-AIC hypothesis makes several very precise, easilyfalsifiable predictions: • As already mentioned, the He WD companions in our RD-AIC scenario are expected to have masses in the range M ≃ . − . M ⊙ (up to . M ⊙ for low-metallicity WD progen-itors) and orbital periods of −
60 days . In rare cases, we expectWD masses up to ∼ . M ⊙ , if the donor star had a ZAMS mass > . M ⊙ (Tauris et al. 2013). • The binding energy of a neutron star can be expressed as: E b ≃ .
084 ( M NS /M ⊙ ) M ⊙ c (Lattimer & Yahil 1989), where M NS is its gravitational mass. The collapse of super-Chandrasekharmass WDs of . − . M ⊙ (for rigid rotation) therefore leads toMSPs with gravitational masses of . − . M ⊙ , if we assumethat . M ⊙ of baryonic material is lost during the AIC. • The sudden release of gravitational binding energy (and massejection into a disk) increases the orbital period and imposes an ec-centricity to the system given by (Bhattacharya & van den Heuvel1991): e = ∆ M/ ( M NS + M ) , if the AIC is symmetric and no kickis imparted to the newborn MSP (see below). Here we assume thatthe pre-AIC binary orbit is circular, which is a good assumption forX-ray binaries were tidal torques circularise the system on a shorttimescale. For the ranges of M NS and M given above, this leads toa remarkable narrow range of post-AIC eccentricities: . − . .(The exact values depend on the still unknown equation-of-state ofneutron stars.) This result is in excellent agreement with the sys-tems presented in Table 1, cf. Section 3.1 for a discussion. • The momentum kick imparted to a newborn neutron star viaan AIC event is expected to be small. This follows from detailedsimulations of AIC events which imply explosion energies signif-icantly smaller than those inferred for standard iron-core collapsesupernovae (Kitaura et al. 2006; Dessart et al. 2006), and also be-cause of the small ejecta mass and the short timescale of the event(compared to the timescales of the non-radial hydrodynamic in-stabilities producing large kicks), e.g. Podsiadlowski et al. (2004);Janka (2012). Our hypothesis therefore predicts that eccentric bi-nary MSPs with He WDs will have small peculiar space velocities. ( P orb , e ) –plane The spread of eccentricities and orbital periods of the resultant sys-tems formed via RD-AIC is extremely sensitive to any kick givento the MSP during the AIC event. In Fig. 3 we demonstrate this byshowing a Monte Carlo simulation of the expected eccentricitiesand orbital periods using the range of pre-AIC parameters givenabove and adding small kick velocities of w
10 km s − . Thedynamical effects were calculated following the formulae of Hills(1983). The properties of systems undergoing RD-AIC events areseen to be surprisingly similar to the characteristics of the recentlydiscovered MSPs in eccentric orbits (Table 1). If the WD companions happen to be bright, then a study of theirspectral lines will yield the mass ratio of the binary components, q . Furthermore, given the eccentric orbits of these MSPs, we will Figure 3.
Distribution in the ( P orb , e ) –plane of systems formed via theRD-AIC scenario. The two red stars are the recently discovered eccentricMSPs (Table 1). The black solid circle indicates a symmetric ( w = 0 ) AICfrom a . M ⊙ WD with a pre-AIC orbital period, P orb , = 24 days and a He WD companion star of mass, M = 0 . M ⊙ . The V-shapedlight-blue distribution is for the same system but applying a small kickof w = 10 km s − in a random (isotropic) direction. The indigo-violetand the yellow distributions superimposed are also for the same systembut with w = 5 and − , respectively. The wide light grey dis-tribution is for w = 5 km s − and a random selection of pre-AIC sys-tems (assuming equal probabilities), with . − . M ⊙ WDs and P orb , = 15 −
30 days (corresponding to M = 0 . − . M ⊙ ).The dark grey distribution is similar but restricted to w = 2 km s − . certainly be able to measure the rate of advance of periastron ( ˙ ω )for these systems. If the radius of the companion is small com-pared to the size of the orbit (which is the case for a WD), then ˙ ω is solely due to the effects of general relativity and can be usedto estimate the total mass of the system (Weisberg & Taylor 1981).The combination of ˙ ω and q would be enough to determine themasses of the components. Another possible solution is the mea-surement of the Shapiro delay for these systems. Even a relativelylow-precision measurement of h (Freire & Wex 2010) can, whencombined with the measurement of ˙ ω , yield very precise compo-nent masses, as in the cases of PSR J1903+0327 (Freire et al. 2011)and PSR J1807 − . − . M ⊙ . Measuring a higherMSP mass would, if not falsifying our hypothesis, require differen-tial rotation of the progenitor WD, which may be a problem withrespect to the need of a long delay timescale (Ilkov & Soker 2012).The unusual MSPs discussed in this Letter were discovered inrecent pulsar surveys (e.g. Cordes et al. 2006; Deneva et al. 2013;Barr et al. 2013) with high time and frequency resolution that havegreatly increased the number of known MSPs, revealing new rarepulsar populations. If on-going and future surveys detect many ec-centric MSPs with WD companions with e ∼ . (and P orb =10 −
60 days ), this would not only support our RD-AIC hypoth-esis; it would also imply that AIC events do not produce kicks(or at least w − , cf. Fig. 3) and that WDs rotaterigidly. Furthermore, it would imply that r-mode instabilities do notnecessarily slow down young, hot MSPs, as previously suggested(Andersson et al. 1999).Note, there may also be eccentric MSPs with WDs formedvia the triple scenario outlined in Section 1.2, which will have amuch wider distribution in the ( P orb , e ) –plane and possibly more c (cid:13) , 000–000 irect formation of MSPs from delayed AIC massive companions. Detection of an MSP with a main-sequencecompanion and e ∼ . would support a triple star scenario for theformation of MSPs with WDs and e ∼ . , and thus significantlyweaken the need for our RD-AIC hypothesis.Population synthesis investigations of MSP formation viaAIC have been performed by Hurley et al. (2010) and Chen et al.(2011). The former study concluded that, in general, the AIC chan-nel to MSP formation is important. The latter study investigateddirect MSP formation via AIC and concluded that the probabilityof forming eccentric MSPs can be ruled out (Even using high kicksthey could not produce eccentric MSPs with P orb >
20 days ), incontradiction with the new discoveries, cf. Table 1. We recommendnew population synthesis modelling using our RD-AIC scenarioin order to probe more carefully the expected number of such ec-centric MSP systems to be detected, and for the statistics of theirresulting parameter space. Ideally, the triple system scenario shouldbe modelled for comparison as well.Finally, it should be investigated under which circumstances abinary evolves via RD-AIC or follows the Tauris et al. (2013) path.The latter was calculated using a point mass accreting WD whichdid not allow for detailed spin angular momentum modelling. Forthe resulting MSPs with He WD companions, the values of P orb and M are expected to be roughly similar. The RD-AIC scenario,however, produces eccentric systems. The common scenario for the formation of MSPs via recycling inLMXBs is well established with plenty of observational evidence,as discussed in Section 1. The RD-AIC hypothesis presented inthis Letter provides an additional formation channel of MSPs thatmakes very specific predictions about future discoveries and the ex-istence of a separate population of eccentric MSPs. If this hypothe-sis is confirmed by future observations, it would also have interest-ing consequences for better understanding the direct AIC channelto produce MSPs, i.e. with respect to WD progenitor masses, (ab-sence of) momentum kicks in AIC, and possibly even constrainingneutron star equations-of-state given that the post-AIC eccentrici-ties depend on the released gravitational binding energy.
ACKNOWLEDGEMENTS
We thank the anonymous referee for very constructive commentsthat improved this manuscript and Sung-Chul Yoon for discussions.P.F. gratefully acknowledges the financial support by the EuropeanResearch Council for the ERC Starting Grant BEACON under con-tract no. 279702. T.M.T. gratefully acknowledges financial supportand hospitality at both the Argelander-Insitut f¨ur Astronomie, Uni-versit¨at Bonn and the Max-Planck-Institut f¨ur Radioastronomie.
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