Strange quark stars in binaries: formation rates, mergers and explosive phenomena
Grzegorz Wiktorowicz, Alessandro Drago, Giuseppe Pagliara, Sergei B. Popov
DDraft version September 25, 2017
Preprint typeset using L A TEX style emulateapj v. 12/16/11
STRANGE QUARK STARS IN BINARIES:FORMATION RATES, MERGERS AND EXPLOSIVE PHENOMENA
G. Wiktorowicz , A. Drago , G. Pagliara , S.B. Popov Astronomical Observatory, University of Warsaw, Al. Ujazdowskie 4, 00-478 Warsaw, Poland ( [email protected] ) Dip. di Fisica e Scienze della Terra dell’Universit`a di Ferrara and INFN Sez. di Ferrara, Via Saragat 1, I-44100 Ferrara, Italy Sternberg Astronomical Institute, Lomonosov Moscow State University, Universitetsky prospekt 13, 119234, Moscow, Russia
Draft version September 25, 2017
ABSTRACTRecently, the possible co-existence of a first family composed of ”normal” neutron stars with a secondfamily of strange quark stars has been proposed as a solution of problems related to the maximummass and to the minimal radius of these compact stellar objects. In this paper we study the massdistribution of compact objects formed in binary systems and the relative fractions of quark andneutron stars in different subpopulations. We incorporate the strange quark star formation modelprovided by the two-families scenario and we perform a large-scale population synthesis study inorder to obtain the population characteristics. According to our results, the main channel for strangequark star formation in binary systems is accretion from a secondary companion on a neutron star.Therefore, a rather large number of strange quark stars form by accretion in low-mass X-ray binariesand this opens the possibility of having explosive GRB-like phenomena not related to supernovae andnot due to the merger of two neutron stars. The number of double strange quark star’s systems israther small with only a tiny fraction which merge within a Hubble time. This drastically limits theflux of strangelets produced by the merger, which turns out to be compatible with all limits stemmingfrom Earth and Lunar experiments. Moreover, this value of the flux rules out at least one relevantchannel for the transformation of all neutron stars into strange quark stars by strangelets’ absorption.
Subject headings:
Stars: neutron stars, strange quark stars, X-ray: binaries, Methods: statistical INTRODUCTION
The discovery, in 2010, of a pulsar with a mass ofabout two solar masses (Demorest et al. 2010) has stim-ulated many theoretical studies, in the nuclear astro-physics community, concerning its possible compositionand the properties of the equation of state of dense mat-ter. It is clear indeed that the center of this stellar objectcould be the site of the most dense form of nuclear matterwe are aware of: depending on the adopted model for theequation of state, the central density of this star couldbe larger than about 3 times the nuclear saturation den-sity. There are many different ideas on the compositionof matter at such a high density: for instance, hyperons(Chatterjee & Vida˜na 2016) or delta resonances (Dragoet al. 2014b) could form, or a phase transition to quarkmatter could occur (Alford et al. 2015). The need to ful-fill the two solar mass limit provides tight microphysicalconstraints on those scenarios.It is clear that precise mass measurements represent apowerful and reliable tool to investigate the propertiesof dense matter. Future observations (e.g. by the FAST(Nan et al. 2011) and SKA radio telescopes (Carilli &Rawlings 2004)) could possibly prove the existence ofeven larger masses. However, some information could beobtained also by considering the (much more uncertain)measurements of the radii. Unfortunately, up to now,only in a few cases mass and radius of a same compactstar have been estimated from x-ray analysis and more-over with large systematic uncertainties ( ¨Ozel & Freire2016; Miller & Lamb 2016). Although still under de-bate, there are a few indications of the possible existenceof stellar objects with radii smaller than about 11 km (Guillot & Rutledge 2014; ¨Ozel et al. 2016), thus verycompact. Very small radii for stars having masses ofabout 1.4 – 1.5 M (cid:12) are obtained only if the equation ofstate of dense matter is very soft at densities of about 2 –3 times nuclear matter saturation density. On the otherhand, so soft equations of state lead to maximum massessignificantly smaller than 2 M (cid:12) , because to reach verylarge masses would imply an extreme stiffening of theequation of state at larger densities, saturating the limitof causality, a situation that is not very realistic (Alfordet al. 2015). If future observations with new facilities,such as the NICER experiment on-board ISS (Gendreauet al. 2012), will confirm the existence of very compactstars, then one has to explain how the equation of stateof dense matter could be at the same time very soft (toexplain the very compact configurations) and very stiff(to explain the very massive configurations).In Drago et al. (2014a); Drago et al. (2016); Drago &Pagliara (2016), a possible solution to this puzzle hasbeen proposed. It is based on the existence of two fami-lies of compact stars: neutron Stars (NSs indicating bothstars made of nucleons and stars containing hyperons)which are compact and light, and strange quark stars(QSs) (Alcock et al. 1986; Haensel et al. 1986) whichare large and massive (a 2 M (cid:12) star would be a QS for in-stance). In this scenario, strange quark matter composedof three flavors: up, down, and strange, is the true groundstate and hadronic matter is instead metastable. A NScould therefore convert into a QS once a significant frac-tion of strangeness is formed in its interior through theappearance of hyperons and the conversion time turnsout to be of the order of ten seconds (Drago et al. 2007;Herzog & R¨opke 2011; Niebergal et al. 2010; Pagliara a r X i v : . [ a s t r o - ph . H E ] S e p et al. 2013; Drago & Pagliara 2015). The critical den-sity for such a transition is thus close to the thresh-old of hyperons’ formation . The exact value of thedensity at which hyperons start forming depends on themicrophysics of the equation of state and it determinesthe maximum mass and the minimum radius of NSs (inthis paper we do not consider rotating configurations).The smaller the minimum radius the smaller the maxi-mum mass. Since we are interested in radii smaller thanabout 11 km, then the limiting gravitational mass of aNS is M H max ∼ . . M (cid:12) . As shown in Fig.1, the un-stable hadronic star forms a QS having the same bary-onic, but a smaller gravitational mass. The “mass de-fect” is ∆ M ∼ . . M (cid:12) . One can infer thereforethat within the two-families scenario the mass distribu-tion would be qualitatively different with respect to theone of the standard one-family scenario: we expect, inparticular, an enhancement in the number of stars withmasses in the range ( M H max − ∆ M ) (cid:54) M (cid:54) M H max (co-existence range), compensated by a depopulation of thethe region of masses larger than M H max , a feature thatcould be possibly tested by means of future observationsof the FAST and SKA radio telescopes.We will focus here on binary systems with QSs, as theyallow for the dynamical mass measurement of a compactobject. There exist three general ways a QS can formduring the evolution of a binary system :1. The binary components may not interact duringtheir evolution (single-star-like evolution). It re-quires a star having an initial mass M ZAMS ≈
18 – 22 M (cid:12) ;2. The star may become a NS with a mass M NS We performed a simulation of 2 million binaries us-ing the StarTrack population synthesis code (Belczynskiet al. 2002b, 2008) with some further amendments (seeWiktorowicz et al. 2015, and references therein). Theselarge-scale simulation were obtained with a use of theUniverse@Home project . Population synthesis methodwas previously widely used to similar tasks (e.g. Popov& Prokhorov 2007)We simulated a grid of six models with three differentmetallicities: Z (cid:12) (solar metallicity; Z (cid:12) = 0 . 02; Villanteet al. 2014), Z (cid:12) /10, and Z (cid:12) /100; and two values of M H max parameter: 1 . . M (cid:12) .For initial stellar mass distribution we used the Kroupaet al. (1993) broken power-law with α = − . M (cid:12) . For a primary we chose a mass rangeof 6 – 150 M (cid:12) to involve all possible progenitors of com-pact objects. For secondaries we studied a wider rangeof 0 . 08 – 150 M (cid:12) keeping the mass ratio distribution uni-form ( P ( q ) = const). Initial binary separations hadthe log-uniform distribution — so-called ¨Opik law, —( P ( a ) ∼ /a ; Abt 1983), whereas, the eccentricity distri-bution was assumed to be thermal ( P ( e ) = 2 e ; Duquen-noy & Mayor 1991). We assumed that the natal kickacts only during NS formation (single-mode Maxwelliandistribution with σ = 265 km s − ). Strange quark star formation In this study every NS with a mass M NS ≥ M H max trans-forms into a QS. The transition is so rapid that it occurswithin a single time step of our simulation (Drago &Pagliara 2015). In our results the maximum post-SN NSmass was 1 . M (cid:12) , which transforms into a 1 . M (cid:12) QS. (initially more massive objects form BH). However,mass accretion may make QSs even heavier.To calculate the post-transition QS mass we implementthe conservation of the baryonic mass (Bombaci & Datta2000) while the gravitational mass changes due to thedifferent binding energies of NSs and of QSs (see Fig. 1).The radius in the model we use is larger for a QS. It isirrelevant for the present study, but quite crucial in theinterpretation of the two-families model. The maximumgravitational mass of a QS is not well-determined. In ourcalculations we assume the value of 2 . M (cid:12) (thus wellabove all known massive NSs: Demorest et al. (2010);Antoniadis et al. (2013)). A possible way of determining http://universeathome.pl M Q max is through the analyses of the extended emissionof short GRBs (Lasky et al. 2014; Lu et al. 2015; Liet al. 2016). In particular in Lasky et al. (2014) theexpected mass distribution for the post-merger remnantis M = 2 . +0 . − . M (cid:12) . Although this limit includes alsosupramassive stars, it represents a hint of the existenceof stars with masses significantly larger than 2 M (cid:12) .A very crucial feature of our scheme is that the ra-dius of the compact star increases during the conversion.This is compatible with a combustion mode which is notdriven by pressure but by diffusion (Olinto 1987) andis strongly accelerated by Rayleigh-Taylor hydrodynam-ical instabilities, as discussed in Horvath & Benvenuto(1988); Drago et al. (2007); Herzog & R¨opke (2011);Pagliara et al. (2013); Drago & Pagliara (2015); Furu-sawa et al. (2016). However, these instabilities halt be-low a certain critical density and the conversion of themost external layer is much slower, see also the recentsimulations of Ouyed et al. (2017).The present analysis does not contain two potentiallyrelevant phenomena which can take place in associationwith quark deconfinement. First, the impact of quarksdeconfinement on the SN explosion is not discussed inthis paper: we only assume that if the compact star pro-duced by the SN has a mass larger than M H max then itimmediately becomes a QS. On the other hand, quarkdeconfinement could help heavy progenitors to explode(Drago et al. 2008; Drago & Pagliara 2016). This mech-anism could in principle produce compact stars withhigher masses. Second, we do not take into account therelation between mass accretion and angular momentumaccretion. More explicitly, in the present paper rotationis not considered. M / M s un M G -QS-X=3.5M B M G -SFHo -H ∆ M B M G -SFHo - ∆ M B Guillot et al. 201490% conf. level Bogdanov 2012 - 1 σ Lu et al. 2015 Hambaryan et al. 2014Bogdanovet al. 2016 Fig. 1.— The mass-radius relations for hadronic stars and strangequark stars. The dashed lines represent the baryonic masses,whereas the solid lines represent the gravitational masses. A typ-ical transition for a NS reaching M H max is shown with violet color.The solid arrow starts at M H max and ends at M H max − ∆ M . Al-though the baryonic masses before and after the transition areequal (dashed horizontal violet line), the QS gravitational mass issmaller ( M H max − ∆ M = 1 . M (cid:12) ) than the NS gravitational mass( M H max = 1 . M (cid:12) ). The maximum mass of the QS branch M Q max issignificantly larger than M H max . The brown box is a rough approx-imation of the limits indicated in Bogdanov et al. (2016). TABLE 1Number of QS/NS in binaries Metallicity a a f QSb c f crd ALL Z (cid:12) . × . × . 01 7 . × . Z (cid:12) /10 2 . × . × . 04 7 . × . Z (cid:12) /100 1 . × . × . 01 1 . × . LMXB Z (cid:12) . × . × . 26 7 . × . Z (cid:12) /10 1 . × . × . 08 1 . × . Z (cid:12) /100 7 . × . × . 25 2 . × . DQS/DNS Z (cid:12) – 6 . × – 6 . × . Z (cid:12) /10 4 . × . × . × . Z (cid:12) /100 – 7 . × – 7 . × . Note . — QS and NS quantities per MWEG at present timefor M H max = 1 . M (cid:12) . ALL – all binaries; LMXB – mass-transferringbinaries; DQS/DNS – double QS/NS. a Number of QS ( b fraction of QSs; defined as f QS := / ( c number of NSs in the model without QSs (noQS) d change in a number of compact objects (QSs and NSs) in1 . 36 – 1 . M (cid:12) mass range; f cr := ( (cid:48) + (cid:48) ) / (cid:48) ( noQS )(mass range marked with (cid:48) ) RESULTS The results are scaled to be comparable with theMilky-Way equivalent galaxy (MWEG), which we as-sumed to have a total stellar mass of M MWEG = 6 . × M (cid:12) (e.g., Licquia & Newman 2015) and continuousstar formation. We chose M H max = 1 . M (cid:12) , as our stan-dard model and quantitative results refer to this modelunless differently stated. In Sec. 3.4 the effects of chang-ing the value of the maximum mass of hadronic stars to M H max = 1 . M (cid:12) are analyzed. We show that the resultsand conclusions are qualitatively similar for both models.The ratio of the number of QSs to NSs is between0 . 01 – 0 . 04 depending on metallicity (Tab. 1), but formass-transferring binaries (in the case of LMXBs) it ishigher: 0 . 08 – 0 . 26. This corresponds to 0 . . × QSs in a MWEG. Most of them are existing in wideand therefore non-interacting binaries, or are in pairswith low-luminosity companions, which in most cases(78 – 95%) are white dwarfs (WD).Fig. 2 demonstrates the effect of deconfinement on thecompact stars’ (QS/NS) mass distribution in compari-son to the model without deconfinement (noQS). Noticespecifically that in the range 1 . . M (cid:12) , where mostof the NSs reside, the difference is minimal. In partic-ular, the deconfinement does not affect the peak in thedistribution at ∼ . M (cid:12) . Formation of QSs Mostly, a QS forms from a primary (i.e. the more mas-sive component in the binary on ZAMS). In ∼ 72 – 96%of cases a QS forms as a consequence of MT onto aNS (route R A , acc ). Only the remaining ∼ R A , dir ). Rarely( ≤ R B , dir ; ∼ 18 – 30 M (cid:12) ). Most of these QSs donot interact with their companions, but in about 3 – 18% TABLE 2Formation of strange quark stars in binaries Typical evolutionary route a Z (cid:12) Z (cid:12) /10 Z (cid:12) /100 R A , acc CE1(6-1;12-1) MT2(12-3) AICNS1 MT2(13-3) AICQS1 8 . × . × . × R A , dir CE1(4/5-1;7/8-1) SNQS1 4 . × . × . × R B , dir MT1(2-1) SN1 CE2(14-4;14-7) SNQS2 6 . × . × . × R LMXB CE1(6-1;12-1) CE2(12-3;12-7) MT2(12-7) AICNS1 MT2(13-7/8) AICQS1 MT2(13-11/17) 1 . × . × . × R DQS MT1(4-4) CE2(7-4;7-7) SNQS1 SNQS2 – 4 . × – Note . — QS formation channels: R A , acc – QS forms from a NS due to mass-accretion; R A , dir – QS form directly after SN; R B , dir – QSforms from a secondary (less-massive star on ZAMS); R LMXB – QS in LMXB (mass-transfer present); R DQS – double QS. a Only most important evolutionary phases are present: MT1/2 – mass transfer from the primary/secondary; CE1/2 – common envelope(primary/secondary is a donor); AICSN1 – accretion induced collapse of a WD into a NS; AICQS1 – accretion induced collapse of aNS into a QS; SNQS1/2 – direct formation of a QS after supernova of the primary/secondary. Stellar types: 1 – main sequence; 2 –Hertzsprung Gap; 3 – red giant; 4 – core He-burning; 5/6 – early/thermal pulsing asymptotic giant branch; 7 – He star; 8 – evolved Hestar; 11 – Carbon-Oxygen White Dwarf; 12 – Oxygen-Neon white dwarf; 13 – neutron star; 14 – black hole; 17 – Hybrid white dwarf. TABLE 3Typical parameters for formation channels Parameter a R A , acc R A , dir R B , dir R LMXB R DQSb M a [ M (cid:12) ] 1 . . . . . . . . . . M b [ M (cid:12) ] 0 . . (cid:46) . . (cid:46) . ∼ . a [ R (cid:12) ] (cid:46) (cid:46) (cid:46) (cid:46) . . t age [ Myr] (cid:38) . . M ZAMS , a [ M (cid:12) ] 6 . . . M ZAMS , b [ M (cid:12) ] 1 . . . . . . a ZAMS [ R (cid:12) ] 560 – 2200 2900 – 4500 560 – 8000 700 – 2700 570 – 1200 Note . — The table presents the typical values of strange quark star and companion masses and their separation for the present time andZAMS. In case of the present time, the age of the system is also provided. Scenarios’ designations are explained in Tab. 2 and in Sec. 3.1. a M a – QS mass; M b – companion mass; a – separation; t age – age of the system (time since ZAMS) b Both components are QSs of cases a LMXB can form ( R LMXB ; Sec. 3.2). If themetallicity is moderate ( Z = Z (cid:12) / 10) double QSs formas a result of binary evolution ( R DQS ; Sec. 3.3). Tables2 and 3 summarize the most typical evolutionary routesfor all scenarios. Although models for different metallic-ities share the same trends, there are differences in thetotal number and relative abundances of QSs formed viadifferent channels. R A , acc ; QS forms through accretion onto a NS — This is themost typical formation scenario of QSs in binaries. In atypical case, a primary is about 7 . M (cid:12) and a secondaryis 1 . M (cid:12) . The primary evolves faster and fills the Rochelobe (RL) while being on the asymptotic giant branch af-ter 53 Myr. The MT is usually unstable due to large massratio and a common envelope (CE) occurs. If the binarysurvives this phase, the primary is ripped off its hydrogenenvelope and becomes an Oxygen-Neon WD with a massof about 1 . M (cid:12) . Afterwards, the secondary evolves, be-comes a red giant (RG) and fills its RL. A MT from thesecondary increases the mass of the WD up to 1 . M (cid:12) .Then the primary collapses and becomes a 1 . M (cid:12) NS.Afterwards, the secondary re-fills the RL and commencesa MT again. The system becomes a LMXB. The NS massmay rise up to 1 . M (cid:12) due to accretion and the decon-finement transforms it into a QS with a mass of 1 . M (cid:12) (Fig. 4, upper plot). The MT may proceed further,what will allow the QS to reach a higher mass (typically up to 1 . M (cid:12) ). The evolution leads to the formation ofa QS-WD system, which is usually too wide to interactanymore.As already remarked before, here we do not considerthe effect of rotation on the structure of compact stars.As discussed in Bejger et al. (2011), the central den-sity during mass accretion could increase marginally dueto the simultaneous increase of the angular momentum.Therefore the conversion of the NS could occur eitherduring the mass accretion stage or after the end of massaccretion during the spin down. R A , dir ; Direct collapse to a QS after a SN explosion — Theinitial binary is more massive than in R A , acc scenario.The primary’s initial mass is about 16 – 28 M (cid:12) and thesecondary’s mass is ∼ . . M (cid:12) . When the primaryfills its RL, the MT is unstable, so the CE phase com-mences. The secondary is massive enough to eject the en-velope and the system survives with a much shorter orbit(due to orbital angular momentum loss). Additionally,the outer envelope of the primary is ripped off. The SNexplosion, which occurs shortly after, may significantlychange the separation. Usually no further interaction isobserved and the secondary evolves unaffected and formsa WD after ∼ . . . . . . . . M NS / QS d N / d M NS QS Z = Z (cid:12) QSNSNS (noQS) . . . . . . . . M NS / QS d N / d M NS QS Z = Z (cid:12) / QSNSNS (noQS) . . . . . . . . M NS / QS d N / d M NS QS Z = Z (cid:12) / QSNSNS (noQS) Fig. 2.— The distribution of masses for NS (blue) and QS (red)for metallicity Z = Z (cid:12) (upper plot), Z = Z (cid:12) / 10 (middle plot), and Z = Z (cid:12) / 100 (lower plot). The black line marks the distribution ofNSs in model without QSs (noQS). Features seen in the QS massdistribution are related to post-QS-formation evolution and are nota subject of this study. ZAMS. Progenitors of QSs in R A , acc are lighter, there-fore, are more abundant (approximately twice) in theinitial populations, then in the case of R A , dir . Addition-ally, in the case of a heavy primary ( R A , dir ) it is hard forthe secondary to eject the massive envelope of a primary during CE phase, which frequently leads to a merger.On the other hand, without a CE a system is still wideduring a SN. Consequently, it is frequently disrupted (in ∼ 98% of cases).The companions in R A , acc are usually low-mass WDs,in agreement with the results by Popov & Prokhorov(2005), who found that many of the most massive uncol-lapsed compact objects might be observed at the stageof accretion from WD companions. The QS is formed atthe stage of accretion which lasts for a long time. Afteraccretion is over the QS is spun-up and can be observedas a millisecond radio pulsar. R B , dir ; QSs from secondaries — QSs formed from secon-daries, i.e. less massive binary components on ZAMS,constitute only a small fraction ( ≤ ∼ 41 – 77 M (cid:12) primaryand ∼ 18 – 30 M (cid:12) secondary. The separation is shortenough that when the primary enters the Hertzsprung-Gap (HG) phase its RL is filled. The MT is non-conservative. A BH forms directly after a SN explosionwith small mass loss (typically, BHs obtain low natalkicks). Its mass is between 7 . M (cid:12) . A few Myrlater, the secondary fills its RL while being on the corehelium burning. The CE occurs in which the separationis shortened and the star losses its outer hydrogen en-velope. The second SN results in a direct formation ofa QS and the binary becomes a double compact object(BH-QS). Low-mass X-ray binaries We performed a separate analysis for accreting QS. Weconsidered all mass-transferring binaries with NS/QS ac-cretors. Initial donor masses are in general below 3 M (cid:12) ,as heavier companions usually provide dynamically un-stable MT. Properties of X-ray emission from accretingNSs or QSs are similar (Alcock et al. 1986), therefore,accreting QSs constitute a subgroup of LMXBs.Although most of the QSs form through accretion froma RG companion ( R A , acc ), the MT phase is relativelyshort and the resulting QS-WD binary is too wide tocommence Roche lobe overflow. Consequently, the evo-lutionary route leading to the formation of a LMXB witha QS is different. We found that a typical companion isa WD with a very small mass ( (cid:46) . M (cid:12) ). It is a di-rect consequence of a prolonged mass transfer onto theprimary. The separation which allows for a Roche lobeoverflow is small ( (cid:46) . R (cid:12) ) and the period is very short( P (cid:46) R LMXB ; Accretion onto a QS — The initial evolution to-wards the formation of a LMXB with a QS is in generalsimilar to R A , acc , however, secondaries are on the av-erage more massive on ZAMS. As a result, the heavierRG companion after the first CE provides unstable MT,which results in a second CE phase (this time the sec-ondary is a donor). The orbit shrinks and the secondaryloses much of its mass ( ∼ M b ≈ . M (cid:12) ) re-fills the RL and transfers massonto a heavy WD primary ( M a ≈ . M (cid:12) ). Afterwards,the latter becomes a NS and after another 20 Myr a QS.As a result of the mass loss, the secondary becomes aWD. The separation is very small ( ∼ . R (cid:12) ) due to twoearlier CE phases, so the secondary is able to fill its RLagain due to gravitational radiation (GR). A long andstable MT phase proceeds during which the mass of thedonor drops below 0 . M (cid:12) (Fig. 4, middle plot).The mass distributions of compact objects in LMXBs(Fig. 3) differ from those including all binary systems(Fig. 2). In models involving formation of QSs, we ob-tained a higher number of systems (22 – 67%) in the coex-istence range. In general, the increase is more significantfor higher values of M H max . In spite of this excess, themain peak of the mass distribution is still at ∼ . M (cid:12) (so outside of the coexistence range) and its position andmagnitude are unaffected by the deconfinement. Indeed,most of the mass measurements are outside of the coex-istence range, thus we cannot investigate the presence ofthis difference. Double QSs The main hindrance to the formation of double com-pact objects is the natal kick that may disrupt the bi-nary during either of SN explosions. Nevertheless, wefound an evolutionary route leading to the formation ofdouble QS (DQS). Noteworthy, such scenario may be re-alized only in stellar populations with moderate metal-licity ( Z = Z (cid:12) / f cr ; see Tab. 1). For Z (cid:12) , or Z (cid:12) /100 metallicity,the f cr fraction is < 1, which marks the fact that the de-confinement in general hinders the formation of doublecompact objects by ∼ . . R DQS ; Double QS binary — Typically, a DQS originatesfrom a binary which on ZAMS consists of two stars withmasses ∼ 20 – 24 M (cid:12) , where the primary is on the aver-age only slightly ( ∼ M (cid:12) ) heavier than the secondary.The orbit must be wide enough to accommodate thesestars (570 – 1000 R (cid:12) ). In a typical system, at the age of ∼ ∼ . M (cid:12) helium star with ∼ M (cid:12) core helium burning compan-ion. After 200 kyr, the secondary expands, fills the RLand commences the CE phase. The separation shrinksto a few R (cid:12) and the double helium star forms. The pri-mary and the secondary sequentially (after 300 and 500kyr, correspondingly) explode and form two QSs directly(like in R A , dir scenario). The system have a high chanceof surviving and forming a DQS on an orbit of 7 . R (cid:12) (Fig. 4, lower plot).The presented scenario does not work for solar metal-licity ( Z (cid:12) ). The reason for that is a strong expansion ofhigh- Z stars. The RL is filled earlier for the same ini-tial separation (when the primary is on the HG), thenfor Z = Z (cid:12) / 10 (primary is on a core helium burning).Consequently, the mass-loss is faster and longer, so thehelium star forms earlier. This results in the SN explo-sion happening before the CE phase may commence andthis shortens significantly the separation. Therefore, theorbit is larger ( ∼ R (cid:12) ) in comparison to post CEsystems ( ∼ R (cid:12) ), what leads to a system disruptionduring SNe. . . . . . . . . M NS / QS d N / d M NS QS LMXB Z = Z (cid:12) QSNSNS (noQS) . . . . . . . . M NS / QS d N / d M NS QS LMXB Z = Z (cid:12) / QSNSNS (noQS) . . . . . . . . M NS / QS d N / d M NS QS LMXB Z = Z (cid:12) / QSNSNS (noQS) Fig. 3.— The distribution of masses for NS (blue) and QS (red)in LMXBs for metallicity Z = Z (cid:12) (upper plot), Z = Z (cid:12) / 10 (mid-dle plot), and Z = Z (cid:12) / 100 (lower plot). The LMXB is definedas a mass-transferring binary with NS/QS accretor. The blackline marks the distribution of NSs’ masses in model without QSs(noQS). Features seen in the QS mass distribution are related topost-QS-formation evolution and are not a subject of this study. There are no DQS in the lowest- Z environments( Z (cid:12) / Z stars. This results in a lower chance of MT occurrence age [Myr] phase M a [ M (cid:12) ] M b [ M (cid:12) ]0535700570057005700 ZAMSCEMTNSMTQS 7.26.7(1.3)1.3(1.4)1.31.3(1.5)1.4 1.21.21.2(1.0)1.01.0(0.5)0.5 a ≈ R (cid:12) MS MSAGB MS a ≈ R (cid:12) WD RGNS RG a ≈ R (cid:12) NS RGQS RG a ≈ R (cid:12) age [Myr] phase M a [ M (cid:12) ] M b [ M (cid:12) ]047180190190190210210 ZAMSCECEMTNSMTQSLMXB 7.67.1(1.3)1.31.3(1.4)1.31.3(1.5)1.41.4(1.5) 4.04.04.0(0.7)0.7(0.6)0.60.6(0.3)0.50.3(0.1) a ≈ R (cid:12) MS MSAGB MS a ≈ R (cid:12) WD RG a ≈ . R (cid:12) WD HeSNS HeS a ≈ . R (cid:12) NS HeSQS WD a ≈ . R (cid:12) QS WD∆ t ≈ M a [ M (cid:12) ] M b [ M (cid:12) ]08.99.29.39.4 ZAMSMTCESNQSSNQS 2423(8.4)8.28.1(1.7)1.7 2323(30)29(8.1)8.07.9(1.7) a ≈ R (cid:12) MS MSCHeB CHeB a ≈ R (cid:12) HeS CHeB a ≈ R (cid:12) QS HeS a ≈ . R (cid:12) , e ≈ . QS QS a ≈ . R (cid:12) , e ≈ . t merge ≈ . Fig. 4.— Schematic representation of a typical binary evolutionleading to the formation of a QS (upper plot), LMXB with a QS(middle plot), or a double QS (lower plot). a is a separation and e is an eccentricity. Abbreviation are explained in Tab. 2. For detailssee Sec. 3.1. during the core helium burning phase. As a consequence,after the MT phase, the primary is still massive andforms a BH. For lower-mass primaries the problem isstill present as the expansion is small and it is hard tocommence a CE phase and to shrink the orbit before thesecond SN. As a result, the system becomes disruptedwhen the second SN occurs.The estimated merger rate of DQSs due to GR (Peters1964) is ∼ 71 events Gyr − M − . An average timeto coalescence after the formation of the second QS isabout 10 , 000 Gyr and only a few systems are compactenough to merge within the Hubble time. In the case ofthe MW galaxy, low- Z stars are present mainly in thebulge, which constitutes only ∼ / ∼ 12 events Gyr − assuminga constant star formation rate. Importance of the value of the limitinggravitational mass of NSs ( M H max ) We found that our results change only quantita-tively with different values of M H max . The formationof QSs occurs through the same evolutionary routesand the differences in the coexistence range, which for M H max = 1 . M (cid:12) is between 1 . 46 – 1 . M (cid:12) , are also small(30 – 72%). Tab. 4 provides results for the two values of M H max = 1 . . M (cid:12) . The ∼ 2. The R A , acc is the main evolutionary route for bothmodels with more than 72% of QSs forming through thisscenario. The fraction of LMXBs is similar for both val-ues of M H max (difference of (cid:46) M H max requires higher initial stellarmasses, as both QSs form directly after the SN (not as aresult of mass accretion). Massive stars are less numer-ous on ZAMS, therefore DQSs have less progenitors forhigher M H max . DISCUSSION A comparison with a previous study Belczynski et al. (2002a) performed a population syn-thesis study of a QS population with the use of the ear-lier version of the StarTrack code. They found that QSsmay constitute ∼ 10% of all compact objects and notedthat most of them in the Galaxy will be single ratherthan bound with companions. The current version of the StarTrack code has been significantly updated since thatpaper (see Sec. 2). Moreover, we incorporated a muchmore realistic model of QSs based on the two-familiesscenario. Belczynski et al. (2002a) just assumed that afraction of stars in a particular mass range represent QSs.Nevertheless, our results mostly agree with those ofBelczynski et al. (2002a). We also found that the ma-jority of QSs in the MW galaxy exist as single stars andthat their number, although being significantly smallerthan the number of NSs, is comparable with the numberof BHs. The present study is also broader as it involvesadditionally an analysis of the formation scenarios, DQSmergers, and LMXBs. Comparison with observations TABLE 4Comparison of models with M H max = 1 . M (cid:12) and . M (cid:12) Z = Z (cid:12) Z = Z (cid:12) / Z = Z (cid:12) / M H max [ M (cid:12) ] 1.5 1.6 1.5 1.6 1.5 1.6 . × (1%) 4 . × (1%) 2 . × (4%) 1 . × (2%) 1 . × (1%) 2 . × (2%) R A , acc . × (96%) 4 . × (94%) 2 . × (79%) 9 . × (72%) 1 . × (80%) 1 . × (93%) R A , dir . × (4%) 1 . × (2%) 2 . × (10%) 8 . × (8%) 3 . × (2%) 9 . × (1%) R B , dir . × ( (cid:46) . × (4%) 2 . × (11%) 2 . × (20%) 2 . × (18%) 1 . × (6%) R LMXB . × (18%) 6 . × (14%) 1 . × (4%) 6 . × (5%) 7 . × (5%) 5 . × (3%) R DQS – – 4 . × (8%) 3 . × (1%) – – Note . — The table shows present numbers of QSs per MWEG for models with different limiting mass ( M H max ) and different metallicities.Results are shown both for the entire population and specific evolutionary routes. Numbers in parenthesis represent: for f QS ); evolutionary routes: fraction of R DQS : fraction of all double compact objects (NS or QS). SeeSec. 3.4 for discussion. The deconfinement process modifies the mass distribu-tion of compact stars in the coexistence range. (Figs. 2and 3). Our calculations predict 10 – 57% more bina-ries in models involving deconfinement than in modelswithout it (noQS). For LMXBs this excess is even morepronounced (22 – 61%). For M H max = 1 . M H max . Even if aQS formed with a RG companion ( R A , acc ), the counter-part will mostly become a WD at some age. Therefore,QSs will spend typically most of theirs life with a WDcompanion. As far as observations are concerned, Lat-timer (2012) provided a list of NS mass measurements inbinaries and the most typical companions appeared to beWDs. What makes WD the most typical companion isthe long duration of this evolutionary stage. Therefore,we have higher chance of observing the system in thattime. Although WDs are significantly lighter than QSsduring MT, the resulting orbit expansion is counteractedby WD expansion and GR, what allows for a prolongedMT. It is easier to fill the RL by a companion whichexpands significantly due to nuclear evolution (e.g. MS,RG), however, a WD, if it manages to fill its RL, willprovide a much longer MT phase.In the near future more accreting compact objects canbe identified in an X-ray survey made by eROSITA (Pre-dehl et al. 2010) on-board Spectrum-RG satellite (to belaunched in 2018). Systems with WD donors are of spe-cial interest, as in such cases accretors are expected to bemassive. Accordingly, our simulations predict that WDsshould be the most typical donors to LMXBs with QSs. After accretion is over a compact object can be observedas a millisecond radio pulsar. It is expected that the newradio telescope FAST (Nan et al. 2011) can provide moresources of this kind. Phenomenology of QSs in LMXBs As one can notice from Tab. 4, the fraction of LMXBscontaining a QS is not negligible, ranging from a few per-cent for low metallicities to almost 20% for solar metal-licity. We estimated the rate of formation of QSs inLMXBs to be 19.5 (12.7) / 23.9 (16.7) / 15.8 (11.6)events Myr − MWEG − for Z (cid:12) / 10% of Z (cid:12) / 1% of Z (cid:12) , respectively (numbers in parenthesis refer to modelwith M H max = 1 . M (cid:12) ). Assuming that Milky-Way (MW)galaxy consists of 1/6 Population II stars and 5/6 Popu-lation I, we get an estimated number of ∼ 13 – 20 eventsMyr − in MW connected with NS to QS transition.There are at least two possible observational implica-tions of this result: the emission of a powerful electro-magnetic signal in correspondence with the formation ofa QS and the spin distribution of the pulsars in LMXBs.The formation of a QS in a LMXB is a stronglyexothermic process (releasing order of 10 erg) and itcan take place in a millisecond radio pulsar. These twoproperties strongly suggest a connection between the for-mation of a QS in a LMXB and at least a sub-class ofGRBs within the protomagnetar model Metzger et al.(2011). It is remarkable that such a GRB would notbe connected with the death of a massive star and thuswith a SN. It is tempting to associate this possibilitywith the famous case of GRB060614 (Fynbo et al. 2006;Della Valle et al. 2006; Gal-Yam et al. 2006).Although it is difficult to derive a frequency from justa single event we can try to compare the observed ”rate”of GRBs lacking a SN with the rate of GRBs associatedwith the formation of a QS in a LMXB (mainly route R A , acc ). • The rate of NS to QS transitions in LMXBs is ofthe order of ∼ 13 – 20 events Myr − . • A significant fraction, order of few tens percent, ofcompact stars in LMXBs rotates very rapidly andcould possibly generate a GRB (always through theprotomagnetar mechanism). This translates into arate of GRBs associated with R LMXB of the orderof one every 10 years. • The fraction of long GRBs lacking a SN in respectto the GRBs for which an association with a SNhas been clearly established to be of the order of10% (Hjorth & Bloom 2012). • The rate of long GRBs has been estimated tobe of the order of one every 10 − years pergalaxy(Podsiadlowski et al. 2004), therefore therate of long GRBs non associated with a SN couldbe of the order of one every 10 − years. Onecan notice that the rate estimated in our model isfairly close to the observed one.One should note that within the protomagnetar modela very strong magnetic field is needed. If this magneticfield is present before the formation of the QS, it mayhinder the mass accretion. The magnetic field could in-stead be generated during the combustion from hadronsto quarks, which lasts a few seconds Drago & Pagliara(2015). During the combustion the moment of inertiaincreases significantly (Pili et al. 2016) and it leads tothe development of a strong differential rotation whichin turn could generate the needed high magnetic field(Bucciantini et al. 2017).The second possible phenomenological implication con-cerns the spin distribution of fast rotating pulsars inLMXBs. The increase of the moment of inertia resultingfrom the conversion of a NS into a QS implies a signif-icant spin-down of the pulsar. In Pili et al. (2016) achange of the moment of inertia was large, up to a fac-tor of two, implying a reduction of the spin frequencyagain by a factor of two. It is tempting to connect thiseffect with the bimodal distribution of the spin frequencyfound recently by Patruno et al. (2017) where the slowestcomponent would contain a significant fraction of QSs inour scheme. Strangelets pollution The rate of mergers of DQS is crucial in order to esti-mate the production of strangelets, i.e. of lumps of stablestrange quark matter, significantly smaller than a star.There are two known mechanisms by which strangeletscould be produced: they could be produced at the timeof primordial baryogenesis (when the temperature didfall below about 150 MeV) or they could be produced bypartial fragmentation of at least one of the QSs at thebeginning of the merging process of a DQS system . Thefirst process is uncertain (it has been criticized e.g. in Al-cock & Farhi (1985)), but the second is very relevant asa potential source of strangelets. The existence of a sig-nificant flux of strangelets could trigger deconfinementin all compact stars at the moment of their formation(Madsen 1988), implying that only QSs can exist andtherefore invalidating the two-families scenario. In orderto clarify this issue two crucial information have to beprovided: the rate of DQS mergers not directly collaps-ing into a BH and the probability of forming fragments(strangelets) in the mass range indicated above. An es-timate of the first number has been obtained in this sim-ulation. First, the number of DQS mergers is about 12 A further possible mechanism for strangelets production wouldbe connected with an explosive conversion of hadronic stars intoQSs Jaikumar et al. (2007) but we follow the scheme presented inSec. 2 and supported by the papers there quoted in which detona-tion is never obtained. Gyr − in our Galaxy, as stated above. Second, the totalmass of the binary system exceeds 3 M (cid:12) in most of thecases. Due to that many of these DQS systems collapsedirectly to a BH, as indicated by the analysis of Bausweinet al. (2009). The exact fraction of events in which theBH is not promptly formed is linked to M Q max , but in anycase it cannot exceed 12 events Gyr − .Assuming, as an upper limit, that each event releasesa mass of about 10 − M (cid:12) (similar to the mass ejected indouble NS mergers; the real number could be smaller bythree/four orders of magnitude) one obtains an averagestrange quark matter density ρ s in the Galaxy of about(10 − − − ) g cm − . The flux of strangelets per unitof solid angle d j s / dΩ can be estimated as follows by as-suming that they all have the same baryon number A:d j s dΩ = ρ s v πAm p , (1)where v is the average velocity of the strangelets and m p is the proton mass. By assuming that low massstrangelets have a velocity comparable to the veloc-ity of the galactic halo i.e. v = 250 km s − onegets: d j s / dΩ ∼ − ρ /A cm − s − sr − where ρ = ρ s / (10 − g / cm ) . Having estimated an upper limit to the flux ofstrangelets it is possible to compare this limit with lim-its coming from Earth and Lunar experiments and withlimits coming from astrophysics. Concerning the firsttype of limits, summarized in Price et al. (1984); De Ru-jula & Glashow (1984); Perillo Isaac et al. (1998); Weber(2005); Han et al. (2009), they are almost completely re-spected by our estimate of the flux. Only taking our veryconservative upper limit on ρ s , a small overlap with theconstraints from the Lunar Soil experiment is found.A more stringent constraint has been obtained recentlyby the PAMELA experiment (Adriani et al. 2015). Ourupper limit on the flux would violate the observationallimits for A (cid:46) . On the other hand, the more realisticestimate quoted above fully satisfies the PAMELA limits.Concerning the limits coming from astrophysics, themost relevant analysis has been done by Madsen (1988).First, even using our highest value of ρ s the probabilityof capture of strangelets by a cold NS is negligible. Thisimplies that pulsars displaying glitches (such as the Velaand the Crab pulsars) had a marginal chance to trans-form into QSs (which could not be able to glitch). Inthis way one of main mechanisms for the conversion ofall NSs into QSs is ruled out.According to Madsen, another possibility to trigger theformation of a QS is based on the capture of strangeletsby main sequence stars: the strangelets would accumu-late close to the core of the star and they would trans-form a NS into a QS soon after the SN explosion. Inorder to be captured by a main sequence star (and notto pass through it) strangelets need to have a baryonnumber smaller than ∼ . This mechanism has twoweak points. First, it is easy to demonstrate by usingdimensional arguments that the strangelets dynamicallyejected at the moment of the merger have a baryon num-ber larger than about 10 (Madsen 2002). Strangeletfragmentation through collisions, while quite efficient,could not be able to reduce the baryon number of thestrangelets initially produced by ten orders of magnitude0(Bucciantini et al. 2017). Second, the strangelet locatedin the core of the collapsing star could evaporate dueto the high temperatures reached at the moment of thebounce. Similar arguments can be applied to the case ofmolten NSs.In conclusion, the limits stemming from Earth and Lu-nar experiments can be rather easily satisfied directly byour estimate of the upper limit on the flux without mak-ing any assumption on the fragmentation mechanism ofthe strangelets. Astrophysical limits are more subtle: inparticular they depend on the ability of the strangelets tofragment into small nuggets and to survive temperaturesof the order of few MeV. Single QS population Results of presented simulations show that formationof a QS in a binary system usually results in a disrup-tion. Only in ∼ . . × single QSs (depend-ing on the model) originating from disrupted binariesshould be present currently in a MWEG. Potentiallyit is possible to form a QS through single-star evolu-tion providing the ZAMS mass of a star is in the range M ZAMS ≈ 17 – 22 M (cid:12) . However, most often so massivestar are found in binaries (e.g. binary fraction > SUMMARY AND CONCLUSIONS We performed a population synthesis study of strangequark stars (QS). The two families scenario predicts thata neutron star (NS) becomes a QS after reaching themass limit M H max (Drago et al. 2016), which we adoptedto be 1 . 5, or 1 . M (cid:12) in our modeling. Our results turnout to be rather robust respect to the variation of M H max .Notice anyway that in our analysis we have not includedthe effect of rapid rotation on the structure of the star.This will constitute the next extension of the presentwork.Our analysis of QS population may be summarized as follows: • We found that QS may constitute ∼ ∼ . − . M (cid:12) are QSs). Typically, a QS forms asa result of mass-accretion from a red giant compan-ion onto a NS, however, a direct formation (imme-diately after the supernova explosion) is also pos-sible in (cid:46) 30% of cases. • A relatively larger number of QS is predicted inlow-mass X-ray binaries (3 – 18%) and especially inthe coexistence range (22 – 72%). The effect on themass distribution of compact stars is, however, toosmall to be detected using current observations. Iffuture missions will provide better mass and radiusmeasurements, it will be possible to test our pre-dictions. • Double QSs may constitute up to 8% of doublecompact objects with components masses below2 . M (cid:12) . In most of the cases the two QSs donot merge within a Hubble time. We estimateda merger rate of ∼ 12 events Gyr − for the Galac-tic bulge. Such a low rate implies a rather small“strangelets pollution” and in turn rules out atleast one of the possible mechanisms suggested inthe literature to convert all NSs into QSs. More-over, all limits stemming from Earth and Lunarexperiments are rather easily satisfied. • The rate of conversion of a NS into a QS due tomass accretion in low-mass X-ray binaries is ratherlarge, order of one event every 10 years. This pro-cess is strongly exothermic (it releases about 10 erg) and it can take place in a rapid rotating com-pact star. 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