4 Evolutionary Scenario for Compact Binaries with Neutron Star or Black Hole Components

4.1 Compact binaries with neutron stars

Compact binaries with NS and BH components are descendants of initially massive binaries with . The scenario of the evolution of massive binaries from a pair of main-sequence stars to a relativistic binary consisting of NSs or BHs produced by core-collapse SN was independently elaborated by Tutukov and Yungelson [782*, 472]²⁰ and van den Heuvel et al. [801, 201]. This scenario is fully confirmed by more than 30 years of astronomical observations and is now considered as “standard”.

Figure 7: Evolutionary scenario for the formation of neutron stars or black holes in close binaries. T is the typical time scale of an evolutionary stage, N is the estimated number of objects in the given evolutionary stage.

Certain modifications to the original scenario were introduced after Pfahl et al. [583] noticed that Be/X-ray binaries harboring NSs fall into two classes: objects with highly eccentric orbits () and and objects with and . Almost simultaneously, van den Heuvel [796] noted that 5 out of 7 known neutron star binaries and one massive WD + NS system in the Galactic disk have and the measured or estimated masses of second-born neutron stars in most of these systems are close to (for recent confirmation of these observations see, e.g., [685, 369, 799, 198*]). A neutron star would have mass if it is a product of an electron-capture supernova (ECSN), which was accompanied by the loss of binding energy equivalent to . These discoveries lead van den Heuvel [796] to include ECSN in the evolutionary scenario for the formation of neutron star binaries. A distinct feature of NSs formed via ECNS is their low natal kicks (typically, a Maxwellian velocity distribution with is inferred instead of ).

In Figure 7* we present the “standard” evolutionary scenario for the formation of neutron star binaries. Other versions of this scenario may differ by the types of supernovae occurring in them or in the order of their formation, which is defined by the initial masses of components and the orbital period of the binary.

It is convenient to consider subsequent stages of the evolution of a binary system according to the physical state of its components, including phases of mass exchange between them.

1. Initially, the pair of high-mass OB main-sequence stars is detached and stars are inside their Roche lobes. Tidal interaction is very effective and the possible initial eccentricity vanishes before the primary star fills its Roche lobe. The duration of this stage is determined by the hydrogen burning time of the primary, more massive component, and typically is  10 Myr (for massive main-sequence stars, the time of core hydrogen burning is ). The star burns out hydrogen in its central parts, so that a dense central helium core with mass forms by the time the star leaves the main sequence. The expected number of such binaries in the Galaxy is about 10⁴.
2. After core hydrogen exhaustion, the primary leaves the main-sequence and starts to expand rapidly. When its radius approaches the Roche lobe, mass transfer onto the secondary, less massive star, which still resides on the main-sequence, begins. Depending on the masses of the components and the evolutionary state of the donor, the mass-transfer may proceed via non-conservative but stable RLOF or via a common envelope. Even if the common envelope is avoided and the first mass exchange event proceeds on the thermal time scale of the donor , its duration for typical stars is rather short, on the order of 10⁴ yr, so only several dozens of such binaries are expected to be present in the Galaxy.
3. Mass transfer ends when most of the primary’s hydrogen envelope is lost, so a naked helium core is left. This core can be observed as a Wolf–Rayet (WR) star with an intense stellar wind if its mass exceeds  [540, 184, 185]. The duration of the WR stage is several 10⁵ yr, so the Galactic number of such binaries should be several hundreds.

   During the mass-exchange episode the secondary star acquires a large angular momentum carried by the infalling matter, so that its outer envelope can be spun up to an angular velocity close to the limiting (Keplerian) value. Such massive rapidly-rotating stars are observed as Be-stars.

4. Stars more massive than end their evolution by forming a NS. ZAMS mass range , is a “transitional” one in which NS are formed via ECSN at the lower masses and via core collapses at the higher masses, as discussed in more detail in Section 1.

   Supernovae associated with massive naked He stars (almost devoid of H-envelopes) are usually associated with SN Ib/c. The inferred Galactic-type SN Ib/c rate is  [417]. At least half of them should be in binaries. As mentioned in Section 1 ECSN may be progenitors of the faintest type II-P supernovae, because they produce only a small amount of radioactive Ni. Expected properties of core-collapse SN and ECSN are compared in Table 5 [685]. Peculiarly, the historical Crab SN in our Galaxy is suggested to be an ECSN [720].

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   Table 5: Comparison of Fe-Core Collapse and e-Capture Supernovae. Table reproduced with permission from [685], copyright by AAS.

   Properties	Iron Core Collapse	ECSN
   Supernova Properties
   Explosion energy	ergs
   Ejecta	rich in heavy elements (Fe, Si, O)	few heavy elements
   Neutron Star Properties
   Masses	range of masses	characteristic mass
   Neutron star kick	large standard kick ()	low kick
   Binary Properties
   Occurrence	single or binaries	preferentially in binaries
   Eccentricity	high	low
   Recycled pulsar spin	misaligned-aligned with orbit	aligned with orbit
   	(e.g., geodetic precession)

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   Disruption of the binary due to the second SN in the system is very likely (e.g., if the mass lost during the symmetric SN explosion exceeds 50% of the total mass of the pre-SN binary, or it is even smaller in the presence of the kick; see Section 3.4 above). Population synthesis estimates show that (4 – 10)% of initial binaries survive the first core-collapse SN explosion in the system, depending on the assumed kick distribution [135, 607*, 878, 396]. Some runaway Galactic OB-stars must have been formed in this way. Currently, only one candidate O-star with a non-interacting NS companion is known, thanks to multi-wavelength observations – HD 164816, a late O-type spectroscopic binary [772]. Null-results of earlier searches for similar objects [585, 675], though being dependent on assumptions on the beaming factor of pulsars and their magnetic field evolution, are consistent with a very low fraction of surviving systems, but may be also due to obscuration of radio emission by the winds of massive stars.

5. If the system survives the first SN explosion, a rapidly rotating Be star in orbit with a young NS appears. Orbital evolution following the SN explosion is described by Eqs. (46* – 51*). The orbital eccentricity after the SN explosion is high, so enhanced accretion onto the NS occurs at the periastron passages. Most of about 100 Galactic Be/X-ray binaries [622] may be formed in this way. Post-ECSN binaries have a larger chance for survival thanks to low kicks. It is possible that a significant fraction of Be/X-ray binaries belong to this group of objects. The duration of Be/X-ray stage depends on the binary parameters, but in all cases it is limited by the time left for the (now more massive) secondary to burn hydrogen in its core.

   An important parameter of NS evolution is the surface magnetic field strength. In binary systems, magnetic field, in combination with NS spin period and accretion rate onto the NS surface, determines the observational manifestation of the neutron star (see [423] for more details). Accretion of matter onto the NS can reduce the surface magnetic field and spin-up the NS rotation (pulsar recycling) [54, 654, 655, 52*].

6. Evolving secondary expands to engulf the NS in its own turn. Formation of a common envelope is, apparently, inevitable due to the large mass ratio of the components. The common envelope stage after  10³ yr ends up with the formation of a WR star with a compact companion surrounded by an expanding envelope (Cyg X-3 may serve as an example), or the NS merges with the helium core during the common envelope to form a still hypothetical Thorne–Żytkow (TZ) object [759].

   Cygnus X-3 – a WR star with a black hole or neutron star companion is unique in the Galaxy, because of the high probability of merger of the components in CE, the short lifetime of surviving massive WR-stars and high velocity ( 1000 km s^–1) of their stellar winds, which prevents the formation of accretion disks [177, 442*, 422]. On the other hand, it is suggested that there may exist in the Galaxy a population of  100 He-stars of between 1 and with relativistic companions, which do not reveal themselves, because these He-stars do not have strong-enough winds [442].

   The possibility of the existence of TZ-stars remains unclear (see [25]). It was suggested that the merger products first become supergiants, but rapidly lose their envelopes due to heavy winds and become WR stars. Peculiar Wolf–Rayet stars of WN8 subtype were suggested as their observed counterparts [202]. These stars tend to have large spatial velocities, the overwhelming majority of them are single and they are the most variable among all single WR stars. The estimated observed number of them in the Galaxy is  10. A single (possibly, massive) NS or BH should descend from them.

   A note should be made concerning the phase when a common envelope engulfs the first-formed NS and the core of the secondary. Colgate [116] and Zel’dovich et al. [884] have shown that hyper-Eddington accretion onto a neutron star is possible if the gravitational energy released in accretion is lost by neutrinos. Chevalier [104] suggested that this may be the case for the accretion in common envelopes. Since the accretion rates in this case may be as high as , the NS may collapse into a BH inside the common envelope. An essential caveat is that the accretion in the hyper-Eddington regime may be prevented by the angular momentum of the captured matter. The magnetic field of the NS may also be a complication. The possibility of hyper-critical accretion still has to be studied. Nevertheless, implications of this hypothesis for different types of relativistic binaries were explored in great detail by Bethe and Brown and their coauthors (see, e.g., [71] and references therein). Also, the possibility of hyper-Eddington accretion was included in several population synthesis studies with the evident result of diminishing the population of NS + NS binaries in favor of neutron stars in pairs with low-mass black holes (see, e.g., [607*, 34*]).

   Recently, Chevalier [105] pointed to the possible connection of CE with a neutron star with some luminous and peculiar type IIn supernovae. The neutron star engulfed by the massive companion may serve as a trigger of the SN explosion in dense environments due to a violent mass-loss in the preceding CE phase. However, presently our understanding of the evolution of neutron stars inside CE is insufficient to test this hypothesis.

7. The secondary He-star ultimately explodes as a supernova leaving behind a NS binary, or the system disrupts to form two single high-velocity NSs or BHs. Even for a symmetric SN explosion the disruption of binaries after the second SN explosion could result in the observed high average velocities of radio pulsars (see Section 3.5 above). In the surviving close NS binary system, the older NS is expected to have a faster rotation velocity (and possibly higher mass) than the younger one because of the recycling at the preceding accretion stage. The subsequent orbital evolution of such NS binary systems is entirely due to GW emission (see Section 3.1.4) and ultimately leads to the coalescence of the components.

Detailed studies of possible evolutionary channels that produce merging NS binaries can be found in the literature, e.g., [787*, 788*, 430*, 607*, 19, 135, 834, 34, 294*, 314, 815*, 149, 839, 61, 332, 357, 356, 548, 383, 548, 157*, 198]). We emphasize that the above-described scenario applies only to close binaries, that have components massive enough to produce ECSN or core-collapse SN, but not so massive that the loss of the hydrogen envelope by stellar wind and an associated widening of the orbit via the Jeans mode of mass ejection may prevent RLOF by the primary. This limits the relevant mass range by [470, 807].

There also exists a population of NSs accompanied by low-mass companions. A scenario similar to the one presented in Figure 7* may be sketched for them too, with the difference that the secondary component stably transfers mass onto the companion (see, e.g., [305, 334, 335, 777]). This scenario is similar to the one for low- and intermediate-mass binaries considered in Section 7, with the WD replaced by a NS or a BH. Compact low-mass binaries with NSs may be dynamically formed in dense stellar environments, for example in globular clusters. The dynamical evolution of binaries in globular clusters is beyond the scope of this review; see, e.g., [39] and [52] for more detail and further references.

At the end of numerical modeling of the evolution of massive binaries outlined above, one arrives at the population of NS binaries, the main parameter of which is distribution over orbital periods and eccentricities . In the upper panel of Figure 8* we show, as an example, a model of the probability distribution of Galactic NS + NS binaries with in plane at birth; the lower panel of the same figure shows a model probability distribution of the present day numbers of NS + NS systems younger than 10 Gyr [607*]. It is clear that a significant fraction of systems born with merge. A detailed discussion of general-relativity simulations of NS + NS mergers including the effects of magnetic fields and micro-physics is presented in [181].

Figure 8: Upper panel: the probability distribution for the orbital parameters of the NS + NS binaries with day at the moment of birth. The darkest shade corresponds to a birthrate of . Lower panel: the probability distribution for the present-day orbital parameters of the Galactic disc NS + NS binaries younger than 10 Gyr. The grey scaling represents numbers in the Galaxy. The darkest shade corresponds to 1100 binaries with given combination of and .

As noted, important phases in the above described scenario are the stages in which one of the components is a WR-star. Evolution in close binaries is a channel for formation of a significant fraction of them, since RLOF is able to remove hydrogen envelopes from much lower-mass stars than stellar wind. As WR-stars are important contributors to the UV-light of the galaxies (as well as their lower-mass counterparts – hot subdwarfs), this points to the necessity of including binary evolution effects in the spectrophotometric–population-synthesis models, see, e.g., [419].

4.2 Black-hole–formation parameters

So far, we have considered the formation of NSs and binaries with NSs. It is believed that very massive stars end their evolution by forming stellar-mass black holes. We will now discuss their formation.

In the analysis of BH formation, new important parameters appear. The first is the threshold mass beginning from which a main-sequence star, after the completion of its nuclear evolution, can collapse into a BH. This mass is not well known; different authors suggest different values: van den Heuvel and Habets [802] – ; Woosley et al. [852] – ; Portegies Zwart, Verbunt, and Ergma [606], Ergma & van den Heuvel [176], Brown et al. [70] – . A simple physical argument usually put forward in the literature is that the mantle of the main-sequence star with before the collapse has a binding energy well above 10⁵¹ erg (the typical supernova energy observed), so that the supernova shock is not strong enough to expel the mantle [210*, 211].

The upper mass limit for BH formation (with the caveat that the role of magnetic-field effects is not considered) is, predominantly, a function of stellar-wind mass loss in the core-hydrogen, hydrogen-shell, and core-helium burning stages. For a specific combination of winds in different evolutionary stages and assumptions on metallicity it is possible to find the types of stellar remnants as a function of initial mass (see, for instance [274]). Since stellar winds are mass (or luminosity) and metallicity-dependent, a peculiar consequence of mass-loss implementation in the latter study is that for the mass-range of precursors of black holes is constrained to , while more massive stars form NSs because of heavy mass loss. The recent discovery of the possible magnetar in the young stellar cluster Westerlund 1 [499] hints at the reality of such a scenario. Note, however, that the estimates of the stellar wind mass-loss rate are rather uncertain, especially for the most massive stars, mainly because of clumping in the winds (see, e.g., [390, 123, 263]). Current reassessment of the role of clumping generally results in the reduction of previous mass-loss estimates. Other factors that have to be taken into account in the estimates of the masses of progenitors of BHs are rotation and magnetic fields.

The second parameter is the mass of the nascent BH. There are various studies as for what the mass of the BH should be (see, e.g., [762*, 44*, 210, 215*]). In some papers a typical BH mass was found to be not much higher than the upper limit for the NS mass (Oppenheimer–Volkoff limit , depending on the unknown equation of state for NS matter) even if the fall-back accretion onto the supernova remnant is allowed [762]. Modern measurements of black hole masses in binaries suggest a broad range of them –  [542, 475, 633]. A continuous range of BH masses up to was derived in calculations [215]. However, as stressed in Section 2.3, in view of the absence of robust “first-principle” calculations of stellar core collapses, both the BH progenitor’s mass and that of the formed BH itself remain major parameters. Here many possibilities remain, including the interesting suggestion by Kochanek [371] that the absence of high mass giants () as the progenitors of Type IIP supernovae may indicate BH formation starting from progenitor masses as low as .

There is observational evidence that the dynamically-determined BH masses in extragalactic HMXBs (M 33 X-7, NGC 300 X-1, and IC10 X-1) residing in low-metallicity galaxies are higher () than in the Milky Way surroundings [124]. Unless it is a selection effect (the brightest X-ray sources are observed first), this may indicate the dependence of BH formation on the progenitor’s metallicity. This dependence is actually expected in the current models of evolution of single stars [235] and HMXB evolution [421].

It is still a challenge to reproduce successful supernova explosions in numerical calculations, especially in 3D (see, for example, [497], and the discussion in [568]). In the current numerical calculations, spectra of BH masses and kicks received by nascent BHs are still model dependent, see, e.g., [213, 321]. Therefore, in the further discussion we will parameterize the BH mass by the fraction of the pre-supernova mass that collapses into the BH: . In fact, the pre-supernova mass is directly related to , but the form of this relationship is somewhat different in different scenarios for massive star evolution, mainly because of different mass-loss prescriptions. According to our parameterization, the minimal BH mass can be , where itself depends on . The parameter can vary in a wide range.

The third parameter, similar to the case of NS formation, is the possible kick velocity imparted to the newly formed BH (see the end of Section 3.5). In general, one expects that the BH should acquire a smaller kick velocity than a NS, as black holes are more massive than neutron stars. A possible relation (as adopted, e.g., in calculations [430*]) reads

where is the maximum NS mass. When is close to , the ratio approaches 1, and the low-mass black holes acquire kick velocities similar to those of neutron stars. When is significantly larger than , the parameter , and the BH kick velocity becomes vanishingly small. The allowance for a quite moderate can increase the coalescence rate of BH binaries [430].

The possible kick velocity imparted to newly-born black holes makes the orbits of survived systems highly eccentric. It is important to stress that some fraction of such BH binaries can retain their large eccentricities up to the late stages of their coalescence. This signature should be reflected in their emitted waveforms and should be modeled in templates.

Asymmetric explosions accompanied by a kick change the space orientation of the orbital angular momentum. On the other hand, the star’s spin axis remains fixed (unless the kick was off-center). As a result, some distribution of the angles between the BH spins and the orbital angular momentum (denoted by ) will be established [609, 331]. It is interesting that even for small kicks of a few tens of km/s an appreciable fraction (30 – 50%) of the merging BH binary can have . This means that in these binaries the orbital angular momentum vector is oriented almost oppositely to the black hole spin. This is one more signature of imparted kicks that can be tested observationally. The BH spin misalignment can have important consequences for BH-NS mergings [207]. The link between forthcoming observations by second and third-generation GW detectors with astrophysical scenarios of BH spin formation and evolution in compact binaries is discussed in more detail in [237].