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    The Astrophysical Journal, 567:L63L66, 2002 March 1 2002. The American Astronomical Society. All rights reserved. Printed in U.S.A.


    Krzysztof Belczynski,1,2,3 Tomasz Bulik,2 and Wodzimierz Kluzniak4

    Received 2002 January 24; accepted 2002 January 28; published 2002 February 15


    We compute and present the distribution in mass of single and binary neutron stars, strange stars, and blackholes. The calculations were performed using a stellar population synthesis code. We follow the evolution ofmassive single stars as well as binaries with high-mass primaries. The final product of the latter evolution canbe either a binary composed of a white dwarf and a compact object (a neutron star, black hole, or strange star),two compact objects in a binary, or two single stars if the system was disrupted. We find in binaries a populationof black holes that are more massive than single black holes that are a product of either binary or single evolution.We also find that if quark stars exist at all, their population can be as large as the population of black holes.

    Subject headings: binaries: close stars: fundamental parameters


    Binary population synthesis is a useful tool for studying thestatistical properties of stars, including the compact objects

    (e.g., Pols & Marinus 1994; Bethe & Brown 1998; PortegiesZwart & Yungelson 1998; Bloom, Sigurdsson, & Pols 1999;Belczynski & Bulik 1999). Compact objects are stellar rem-nants of a much smaller size than that of white dwarfs, soaccording to current views they could be either black holes orneutron stars or, possibly, quark stars.

    We wish to address the following questions: What is thedistribution of the masses of compact objects formed alongdifferent evolutionary paths? Given the distribution of compactobject masses, what are the relative numbers of different typesof objects (neutron stars, quark stars, black holes) both singleand in binaries? What fraction of binaries gives rise to singlecompact objects, and what fraction survives as binaries and ofwhat type?

    In 2 we briefly describe the population synthesis code usedhere, and in 3 we summarize what is known about the massesof neutron stars, quark stars, and black holes. In 4 we discussthe constraints on the masses of compact objects, in 5 wepresent the results, and, finally, in 6 we give the conclusions.


    We use STARTRACK, a stellar binary population synthesiscode consisting of two parts. The single-star evolution is basedon the formulae from Hurley, Pols, & Tout (2000), modifiedas follows. We have changed the prescription for the mass ofthe compact object formed in a supernova explosion. We usethe original Hurley et al. (2000) formulae to obtain the final

    CO core mass. We use the models of Woosley (1986) to cal-culate the final FeNi core mass (for a given CO core mass),which will collapse and form a compact object during super-nova explosion. Finally, we include calculations of Fryer &Kalogera (2001) to take into account black hole formationthrough both direct collapse and partial fallback.

    The binary evolution is described in Belczynski, Kalogera, &

    1 Northwestern University, Department of Physics and Astronomy, 2145Sheridan Road, F325, Evanston, IL 60208.

    2 Nicolaus Copernicus Astronomical Center, Bartycka 18, 00-716 Warszawa,Poland.

    3 Lindheimer Fellow.4 Institute of Astronomy, Zielona Gora University, Ul. Lubuska 2, 65-265

    Zielona Gora, Poland.

    Bulik (2002) and Belczynski (2001). We evolve only binariesin which at least one star will undergo a supernova explosionand form a compact object. The evolution starts at zero-age mainsequence (ZAMS). During the course of evolution we includethe following effects as appropriate: wind mass loss (standard,Wolf-Rayet, luminous blue variables), the tidal circularization ofa binary orbit, conservative/nonconservative mass transfer, com-mon envelope evolution, rejuvenation, hyperaccretion onto com-pact objects, and detailed supernova explosion treatment.

    Many binaries are disrupted in supernova explosions as aresult of mass loss and the natal kick. For supernova kicks weuse the distribution presented by Cordes & Chernoff (1998).We use smaller kicks when the compact object is a black holeformed via partial fallback and no kicks for the black holesformed through direct collapse; for details see Belczynski etal. (2002). We continue to evolve each star, until the formationof a stellar remnant. At the endpoint of binary evolution either

    two single remnants are left or a binaryin either case at leastone of the remnants is a compact object, while the other iseither a compact object or a white dwarf.


    Bodmer (1971) suggested that stars composed of up, down,and strange quarks (in roughly equal numbers) may exist ifquark plasma is the ground state of matter. Relativistic modelsof strange stars composed of such self-bound quark matterwere first computed by Brecher & Caporaso (1976), Witten(1984), Alcock, Farhi, & Olinto (1986), and Haensel, Zdunik,& Schaeffer (1986). Alcock et al. (1986) give a detailed dis-cussion of the possible avenues of the formation of quark stars.If they are formed through a phase transition after a certain

    critical density is exceeded in the core of a neutron star, or ifthey are formed in a supernova of a star that has captured aseed of strange matter, quark stars could be more massivethan neutron stars. In other scenarios, no neutron stars at allwould exist, or the abundance of quark stars need not be afunction of their mass. The astrophysics of quark stars hasrecently been reviewed by Cheng, Dai, & Lu (1998) and Mad-sen (1999).

    It has been argued that young, glitching, pulsars cannot bestrange stars (Alpar 1987). Madsen (1988) and Caldwell &Friedman (1991) argue that strange stars in Hulse-Taylortypebinaries would eventually contaminate the entire Galaxy withstrange matter as a result of their binary coalescence, and thuspreclude the formation of young neutron stars. But Kluzniak






    Fig. 2.Cumulative fraction of compact objects corresponding to the dif-ferential distributions of Fig. 1. The top curve is for compact objects arisingfrom single stars. The remaining curves describe the outcome of binary evo-lutionnote that the most likely fate of a compact object born in a binary isto be single and that to end up as a companion to a white dwarf is more likelythan to be in a binary with another compact object.

    the neutron star (Friedman, Parker, & Ipser 1986; Cook, Sha-piro, & Teukolsky 1992). The corresponding increase of max-imum mass in rapidly rotating quark stars is even larger (Ster-gioulas, Kluzniak, & Bulik 1999). However, in this discussionwe neglect these effects of stellar rotation; i.e., we assume thatnone of the neutron stars (or quark stars) formed has a periodof less than 10 ms.

    Of course, there is no theoretical maximum for the mass ofa black hole. The maximum masses we find in our calculationssimply reflect the formation route of the black hole. We findthat the most massive black holes survive in binaries (Fig. 1;compare panels labeled group 0 or group 1 with the oneslabeled group 2 or group 3).

    Observations of binary stars yield direct information on themasses of some compact objects. Neutron stars in the Hulse-Taylortype binaries have accurately measured masses of 1.44and 1.39M,. Millisecond pulsars have been analyzed by Thor-sett & Chakrabarty (1999), who found that they are consistentwith all being in the narrow mass range of .1.34 0.04 M,Among the neutron stars that exhibit X-ray bursts, the mass ofCyg X-2 has been quoted as (Orosz &1.78 0.23 M,

    Kuulkers 1999). However, for most LMXBs the masses remainunknown.

    There is a class of LMXBs for which bright X-ray emissionis transient and the masses of the compact objects cluster inthe range of 5.5 to 7.5 M,. These are thought to be blackholes. At present our code does not yield an excess of blackholes in this mass range with a white dwarf companion in thebinary; instead, a peak at about 10 M, results (see Fig. 1).However, we note that according to our results, single blackholes are particularly abundant at , and there is aM5 M,deficit of single compact objects in the mass range of about2.55 M,. If the black hole LMXBs were formed throughbinary capture in globular clusters, our results would be con-sistent with the measured masses of the transient sources.

    5. RESULTS

    Compact objects may be formed through both single andbinary stellar evolution. Single compact objects may be de-scendants of massive single stars but also of components of abinary system disrupted in a supernova explosion. We willdenote the single compact objects formed from primordial sin-gle stars as group 0, and those formed as a result of the binaryevolution as group 1. Under favorable conditions some binariessurvive supernova explosions, and they finally form tight sys-tems with compact object/objects. Most of these binaries willconsist of a white dwarf and a compact object, and the restwill form binaries with two compact objects (we will denotethe compact object in binaries with white dwarfs as group 2and the double compact objects as group 3).

    Figure 1 shows the number of compact objects per massinterval formed along each route. We start forming compactobjects at mass 1.2 M,, and their number falls off with themass of the final compact object, as expected for our assumedinitial mass function M2.7. The peak in the distribution inFigure 1 around 10M, reflects the relation we obtain betweenthe ZAMS mass of a progenitor and the final mass of a compactobject. This relation for a wide range of progenitor ZAMSmasses results in a final compact object mass of10M, (Bel-czynski et al. 2002). This is an effect of stellar wind, whichincreases with the mass of the star, and thus decreases the finalmass of a compact object for large initial stellar masses. As aresult, the mass of the FeNi core is a weak function of theinitial mass of the star for a wide range of the ZAMS masses.

    In Figure 2 we present, separately for each formation route,the cumulative fraction of compact objects as a function oftheir final mass. The normalization is such that a fraction ofunity corresponds to the total number of stars used in the sim-ulation (we used a total of binaries and single6 67# 10 7# 10stars), and we assumed a binary fraction of 50%; i.e., we as-sumed that (initially) out of every three stars, one is single and

    the other two are in a binary system. The single star and theprimary mass in binaries were in the range 5100 M,, and themass of the secondary was found assuming a flat mass ratio qdistribution and a 2.7 slope of the initial mass function. Eachdistribution rises quickly in the small mass range, which is alsoseen in Figure 1. Thus, within each group even a small masswindowM2M1may yield a significant number of quark stars,if they constitute a sizable fraction of the objects in that masswindow. For example, if quark stars are formed in the narrowmass range , with and(M , M ) M p 1.7 M M p1 2 1 , 2

    , and no neutron stars of that mass exist, the fraction1.8 M,of quark stars in each group will be from a few percent to10%. This fraction is comparable to that of black holes in anygiven group, which is about 15%20% in groups 0, 1, and 2


    50% in group 3. We also note that most of the strangestars in the Galaxy should exist as single objects and that onlya small fraction of them, 105 to 104, are going to be indouble compact object binaries.

    We have listed in Table 1 the numbers of binaries with com-pact object components obtained in our simulation. The binariesare classified according to their component masses, and forillustrative purpose we have labeled the objects in the massrange as strange stars. The table allows1.7 M !M!2.5 M, ,one to read the relative numbers of objects of different types.


    We have shown the effects of the binary evolution on thedistribution of masses of compact objects. As expected, the




    TABLE 1

    Number of Coalescing Double Compact Objects Obtained from 67# 10Initial Binaries

    Primary Mass


    Secondary Type

    White Dwarfs Neutron Starsa Strange Starsb Black Holesc

    . . . . . .1.3 ! M! 1.7 32956 5533

    . . . . . .1.7 ! M! 2.5 11305 4738 166 . . . . . . . . . . . .2.5 ! M 9650 2216 1186 6291

    a Compact object with mass .1.3 ! M! 1.7b Compact object with mass .1.7 ! M! 2.5c Compact object wi th mass .2.5 ! M

    bulk of the population of compact objects have masses below. While for single stellar evolution there exists a unique2 M,

    relation between the stellar mass and the compact object mass,there is no such relation when the binary evolution is takeninto account. Binary evolution works both ways: the mass ofa compact object formed from a particular star in a binary canbe smaller or larger than that formed from an identical starundergoing single stellar evolution.

    In the low-mass range, the cumulative fraction of compactobjects rises steeply with increasing mass (see Fig. 2). Thus,

    even in a small mass interval (M1, M2), the fraction of stars inthe compact object population can be large. On the other hand,the fraction of black holes hardly depends on their minimummass (the cumulative curves flatten above ). We conclude3 M,that the population of quark stars can easily be as large as thepopulation of black holes, even if there is only a small masswindow for their formation.

    The low-mass peak in the differential distribution of Fig-ure 1 is less pronounced for double compact object binaries(group 3). Thus, the chance of finding quark stars in Hulse-Taylortype objects is slim, primarily because of the smallnumber of such objects known so far. The prospects look better

    for compact objects in binaries with dwarfs. Although Thorsett& Chakrabaty (1999) show that the masses of these objects areconsistent with being constrained to a narrow range, the massfunction for individual objects allows for different (higher orlower) masses. However, the observed number of these sourcesis not large, and the search here may suffer from small numberstatistics. Our results show that most quark stars should existas single objects, yet it is most difficult to measure the massesand radii for them. Therefore, in the search for quark stars wemay have to concentrate on single compact objects, such as

    pulsars.It is interesting to note that the most massive blac...


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