1 Introduction

Close binaries consisting of two compact stellar remnants — white dwarfs (WDs), neutron stars (NSs) or black holes (BHs) are considered primary targets of the forthcoming field of gravitational wave (GW) astronomy (see, for a review, [758, 252, 673, 672]), since their orbital evolution is entirely controlled by the emission of gravitational waves and leads to ultimate coalescence (merger) and possible explosive disruption of the components. Emission of gravitational waves accompanies the latest stages of evolution of stars and manifests instabilities in relativistic objects [10, 13]. Close compact binaries can thus serve as testbeds for theories of gravity [221]. The NS(BH) binary mergers that release ∼ 1052 erg as GWs [113, 114] should be the brightest GW events in the 10 − 1000 Hz frequency band of the existing or future ground-based GW detectors like LIGO [23], VIRGO [6], GEO600 [635], KAGRA(LCGT) [728] (see also [642] for a review of the current state of existing and 2nd- and 3rd-generation ground-based detectors). Mergers of NS(BH) binaries can be accompanied by the release of a huge amount of electromagnetic energy in a burst and manifest themselves as short gamma-ray bursts (GRBs). A lot of observational support for NS-NS/NS-BH mergers as sources of short GRBs have been obtained (see, e.g., studies of short GRB locations in the host galaxies [204, 751] and references therein). As well, relativistic jets, associated with GRB of any nature may be sources of GW in the ground-based detectors range [50].

WD binaries, especially those observed as AM CVn-stars and ultracompact X-ray binaries (UCXB), are potential GW sources within the frequency band (10−4 − 1) Hz of the space GW interferometers like (the currently cancelled) LISA [187],Footnote 1 NGO (eLISA) [10, 11], DEGIGO [857] and other proposed or planned low-frequency GW detectors [122, 40]. At the moment, eLISA is selected for the third large-class mission in ESA’s Cosmic Vision science program (L3). Its first step should be the launch of ESA’s LISA Pathfinder (LPF) mission in 2015; the launch of eLISA itself is currently planned for 2034.Footnote 2

WD binary mergers are among the primary candidate mechanisms for type Ia supernovae (SNe Ia) explosions. The NIR magnitudes of the latter are considered as the best “standard candles” [26] and, in this guise, are crucial in modern cosmological studies [638, 578]. Further improvements in the precision of standardization of SNe Ia fluxes is possible, e.g., by account of their environmental dependence [640, 641, 820]. On the other hand, usually, as an event beneath the “standard candle”, an explosion of a non-rotating WD with the Chandrasekhar mass (≃ 1.38 M) is considered. Rotation of progenitors may increase the critical mass and make SNe Ia less reliable for cosmological use [154, 155].

SNe Ia are suggested to be responsible for the production of about 50% of all heavy elements in the Universe [761]. Mergers of WD binaries are, probably, one of the main mechanisms of the formation of massive (> 0.8 M) WDs and WDs with strong magnetic fields (see, e.g., [322, 392] for recent studies and review of earlier work).

A comparison of SN Ia rates (for the different models of their progenitors) with observations may, in principle, shed light on both the star formation history and on the nature of the progenitors (see, e.g., [663, 873, 453, 874, 871, 250, 457, 249, 458]).

Compact binaries are the end products of the evolution of stellar binaries, and the main purpose of the present review is to describe the astrophysical knowledge of their formation and evolution. We shall discuss the present situation with the main parameters determining their evolution and the rates of coalescence of NS/BH binaries and WDs.

The problem is to evaluate as accurately as possible (i) the physical parameters of the coalescing binaries (masses of the components and, if possible, their spins, magnetic fields, etc.) and (ii) the occurrence rate of mergers in the Galaxy and in the local Universe. Masses of NSs in binaries are known with a rather good accuracy of 10% or better from, e.g., pulsar studies [760, 367]; see also [410, 550] for recent summaries of NS mass measurements.

The case is not so good with the rate of coalescence of relativistic stellar binaries. Unfortunately, there is no way to derive it from first principles — neither the formation rate of the progenitors for compact binaries nor stellar evolution are known well enough. However, the situation is not completely hopeless, especially in the case of NS binary systems. The natural appearance of rotating NSs with magnetic fields as radio pulsars allows one to search for binary pulsars with a secondary compact companion using powerful methods of modern radio astronomy (for example, in dedicated pulsar surveys, such as the Parkes multi-beam pulsar survey [455, 192]).

Based on observational statistics of Galactic pulsar binaries with NS companions, one can evaluate the Galactic rate of NS binary formation and merging [586, 505]. On the other hand, a direct simulation of the evolution of population of binaries in the Galaxy (the population synthesis method) can also predict the formation and merger rates of close compact binaries as a function of (numerous) parameters of stellar formation and evolution. Both kinds of estimates are plagued by badly constrained parameters or selection effects, but it is, nevertheless, encouraging that most likely Galactic rates of events obtained in two ways currently differ by a factor of ≈ 3 only: 80/Myr from observations and 30/Myr from population synthesis; see [456] for a recent review of observational and theoretical estimates, also Section 6.

No BH or NS + BH binary systems have been found so far, so merger rates of compact binaries with BHs have been evaluated as yet only from population synthesis studies.

1.1 Formation of stars and end products of their evolution

Let us briefly remind the key facts about star formation and evolution. Approximately 6% of the baryonic matter in the Universe is confined to stars [217]. Recent observational data suggests that, first, long thin filaments form inside molecular clouds and, next, these filaments fragment into protostellar cores due to gravitational instability, if their mass-per-unit length exceeds a certain threshold [14]. An object may be called a “star” if it is able to generate energy by nuclear fusion at a level sufficient to halt the contraction [394]. For solar chemical composition, Mmin ≈ 0.075 M and Mmin increases if stellar metallicity is lower than solar [81]. Currently, among observed stars, the lowest dynamically determined mass has component C of a triple system AB Dor: (0.090 ± 0.005) M [115].

The maximum mass of the star is set by the proximity of the luminosity to the Eddington limit and pulsational instability. For solar chemical composition, this limit is close to 1000 M [38, 880].Footnote 3 But conditions of stellar formation, apparently, define a much lower mass limit. Currently, components of a Porb = 3.77 day eclipsing binary NGC3603-A1 have maximum dynamically-measured masses: (116±31) M and (89±16) M [683]. Since both stars are slightly-evolved main-sequence objects (WN6h), subject to severe-stellar-wind mass loss, their inferred initial masses could be higher: \(148_{- 27}^{+ 40}{M_ \odot}\) and \(106_{- 15}^{+ 92}{M_ \odot}\) [125]. Indirect evidence, based on photometry and spectral analysis suggests the possible existence of (200–300) M stars; see [813] for references and discussion.

For the metal-free stars that formed first in the Universe (the so-called Population III stars), the upper mass limit is rather uncertain. For example, it can be below 100 M [4, 745], because of the competition of accretion onto first formed compact core and nuclear burning and influence of UV-irradiation from nearby protostars. On the other hand, masses of Population III stars could be much higher due to the absence of effective coolants in the primordial gas. Then, their masses could be limited by pulsations, though, the upper mass limit remains undefined [703] and masses up to ≃ 1000 M are often inferred in theoretical studies.

The initial mass function (IMF) of main-sequence stars can be approximated by a power-law dN/dMMβ [669] and most simply may be presented taking β ≈ 1.3 for 0.07 ≲ M/M ≲ 0.5 and β ≈ 2.3 for M/M ≳ 0.5 [830]. Note, there are also claims that for the most massive stars (M ≳ 7 M) the IMF is much steeper — up to β = 3.8 ± 0.5, see [403] and references therein. Estimates of the current star-formation rate (SFR) in the Galaxy differ depending on the method, but most at present converge to ≃ 2M yr−1 [108, 353, 537]. This implies that in the past the Galactic SFR was much higher. For the most recent review of observations and models pertinent to the star-formation process, the origin of the initial mass function, and the clustering of stars, see [389].

The evolution of a single star and the nature of its compact remnant are determined by the main-sequence mass M0 and chemical composition. If M0 is lower than the minimum mass of stars that ignite carbon in the core, Mup, after exhausting the hydrogen and helium in its core, a carbon-oxygen WD forms. If M0 exceeds a certain limiting Mmas, the star proceeds to form an iron core, which collapses into a NS or a BH. In the stars of intermediate range between Mup and Mmas (called “super-AGB stars”Footnote 4) carbon ignites in a partially-degenerate core and converts its matter (completely or partially) into an oxygen-neon mixture. As conjectured by Paczyński and Ziolkowski [557], the TP-AGB stage of the evolution of starsFootnote 5 terminates when shells with positive specific binding energy εbind = uint + εgrav appear in the envelope of the star. In the latter expression, the internal energy term uint accounts for ideal gas, radiation, ionisation, dissociation and electron degeneracy, as suggested by Han et al. [266]. The upper limit on masses of model precursors of CO white dwarfs depends primarily on stellar metallicity, on the treatment of mixing in stellar interiors, on accepted rates of mass loss during the AGB and on more subtle details of the models, see, e.g., [278, 279, 338]. Modern studies suggest that for solar metallicity models it does not exceed 1.2 M [412].

As concerns more massive stars, if, due to He-burning in the shell, the mass of the core reaches ≈ 1.375 M, electron captures by 24Mg and 20Ne ensue and the core collapses to a neutron star producing an “electron-capture supernova” (ECSN) [488]). Otherwise, an ONe WD forms.Footnote 6 Figure 1 shows, as an example, the endpoints of stellar evolution for intermediate mass stars, computed by Siess [707, 708].Footnote 7 Note, results of computations very strongly depend on still uncertain rates of nuclear burning, the treatment of convection, the rate of assumed stellar-wind mass loss, which is poorly known from observations for the stars in this transition mass range, as well as on other subtle details of the stellar models (see discussion in [599, 325, 98, 749]). For instance, zero-age main sequence (ZAMS) masses of solar-composition progenitors of ECSN found by Poelarends et al. [599] are M0 ≈ (9.0–9.25) M.Footnote 8 In [295], the lower mass limit for core-collapse SN at solar metallicity is found to be equal to 9.5 M. According to Jones et al. [326], an MZAMS = 8.8 M star experiences ECSN, while a 9.5 M star evolves to Fe-core collapse. Takahashi et al. [749] find that a MZAMS = 10.4 M solar metallicity star explodes as ECSN, while an MZAMS = 11 M star experiences Fe-core collapse, consistent with Siess’ results.

Figure 1
figure 1

Endpoints of evolution of moderate-mass nonrotating single stars depending on initial mass and metallicity. Image reproduced with permission from [709], copyright by IAU.

ECSN eject only a small amount of Ni (< 0.015 M), and it was suggested that they may be identified with some subluminous type II-P supernovae [366]. Tominaga et al. [764] were able to simulate ECSN starting from first principles and to reproduce their multicolor light curves. They have shown that observed features of SN 1054 (see [741]) share the predicted characteristics of ECSN.

We should enter a caveat that in [412] it is claimed that opacity-related instability at the base of the convective envelope of (7–10) M stars may result in ejection of the envelope and, consequently, prevent ECSN.

Electron-capture SNe, if they really happen, presumably produce NS with short spin periods and, if in binaries, systems with short orbital periods and low eccentricities. Observational evidence for ECSN is provided by Be/X-ray binaries, which harbor two subpopulations, typical for post-core-collapse objects and for post-ECSN ones [369]. Low kicks implied for NSs formed via ECSN [597] may explain the dichotomous nature of Be/X-ray binaries. As well, thanks to low kicks, NS produced via ECSN may be more easily retained in globular clusters (e.g., [397]) or in regions of recent star formation.

The existence of the ONe-variety of WDs, predicted theoretically [24, 488, 301] and later confirmed by observations [774], is important in the general context of compact-star binary evolution, since their accretion-induced collapse (AIC) may result in the formation of low-mass neutron stars (almost) without natal kicks and may thus be relevant to the formation of NS binaries and low-mass X-ray binaries (see [21, 146, 798] and references therein). But since for the purpose of detection of gravitational waves they are not different from much more numerous CO WDs, we will, as a rule, not consider them below as a special class.

If M0Mmas (or 8–12 M), thermonuclear evolution proceeds until iron-peak elements are produced in the core. Iron cores are subject to instabilities (neutronization, nuclei photo-disintegration), that lead to gravitational collapse. The core collapse of massive stars results in the formation of a neutron star or, for very massive stars, a black holeFootnote 9 and is associated with the brightest astronomical phenomena such as supernova explosions (of type II, Ib, or Ib/c, according to the astronomical classification based on the spectra and light curves properties). If the pre-collapsing core retains significant rotation, powerful gamma-ray bursts lasting up to hundreds of seconds may be produced [850].

The boundaries between the masses of the progenitors of WDs or NSs and NSs or BHs are fairly uncertain (especially for BHs). Usually accepted typical masses of stellar remnants for single non-rotating solar-chemical-composition stars are summarized in Table 1.

Table 1 Types of compact remnants of single stars (the ranges of progenitor mass are shown for solar composition stars).

Note that the above-described mass ranges for different outcomes of stellar evolution were obtained by computing 1D-models of non-rotating stars. However, all star are rotating and some of them possess magnetic fields, thus, the problem is 2D (or 3D, if magnetic-field effects are considered). For a physical description of effects of rotation on stellar evolution, see [406, 561]. Realistic models with rotation should account for deviation from spherical symmetry, modification of gravity due to centrifugal force, variation of radiative flux with local effective gravity, transfer of angular momentum and transport of chemical species. Up to now, as a rule, models with rotation are computed in a 1D-approximation which, typically, makes use of the fact that in rotating stars mass is constant within isobar surfaces [174]. Regretfully, for low and intermediate-mass stars, model sequences of rotating stars covering evolution to advanced phases of AGB-evolution are absent. However, as noted by Domínguez et al. [156], rotation must strongly influence evolution at core helium exhaustion and the formation of the CO-core stage, when the latter experiences very strong contraction. With an increase of angular velocity, lower pressure is necessary to balance the gravity, hence the temperature of He-burning shells of rotating stars should be lower than in non-rotating stars of the same mass. This extends AGB-lifetime and results in an increase of the mass of CO-cores, i.e., the C-ignition limit may be shifted to lower masses compared to non-rotating stars. Most recently, grids of 1-D models of rotating stars were published, e.g., in [69, 234, 236].

The algorithm and first results of self-consistent calculations of rapidly-rotating 2D stellar models of stars in the early stages of evolution are described, e.g., in [639, 178]. We may note that these models are very important for deriving physical parameters of the stars from astro-seismological data.

For a more detailed introduction to the physics and evolution of stars, the reader is referred to the classical fundamental textbook [121] and to several more modern ones [612, 51, 53, 171, 297, 298]. Formation and physics of compact objects is described in more detail in the monographs [691, 53]. For recent studies and reviews of the evolution of massive stars and the mechanisms of corecollapse supernovae we refer to [406, 79, 320, 881, 212, 721].

1.2 Binary stars

A fundamental property of stars is their multiplicity. Among stars that complete their nuclear evolution in the Hubble time, the estimated binary fraction varies from ∼ (40–60)% for MM stars [168, 621] to almost 100% for more massive A/B and O-stars, e.g., [468, 382, 370, 469, 106, 671], (but, e.g., in [454, 670], a substantially lower binary fraction for massive stars is claimed).

Based on the summary of data on binary fraction B(M) provided in [381, 385, 671], van Haaften et al. [804] suggested an approximate formula

$$\mathcal{B}(M) \approx {1 \over 2} + {1 \over 4}\log (M)\quad \quad (0{.}08 \leq M/{M_ \odot} \leq 100),$$

considering all multiple systems as binaries.

The most crucial parameters of binaries include the component separation a and mass ratio q, since for close binaries they define the outcome of the Roche lobe overflow (in fact, the fate of the system). The most recent estimates for M-dwarfs and solar-type stars confirm earlier findings that the q-distribution does not strongly deviate from a flat one: dN/dqqβ, with β = 0.25 ± 0.29 [632]. This distribution is defined by the star-formation process and dynamical evolution in stellar clusters; see, e.g., [28, 569]. Distribution over a is flat in log between contact and ≃ 106 R [604].

We note, cautionarily, that all estimates of the binary fraction, mass-ratios of components and distributions over separations of components are plagued by numerous selection effects (see [380] for a thorough simulation of observations and modeling observational bias). A detailed summary of studies of multiplicity rates, distributions over orbital periods and mass ratios of components for different groups of stars may be found in [167].

In binary stars with sufficiently-large orbital separations (“wide binaries”) the presence of the secondary component does not influence significantly the evolution of the components. In “close binaries” the evolutionary expansion of stars leads to the overflow of the critical (Roche) lobe and mass exchange between the components RLOF. Consequently, the formation of compact remnants in close binaries differs from single stars (see Section 3 for more details).

As discussed above, the lower mass limit of NS progenitors is uncertain by several M. This limit is even more uncertain for the BH progenitors. For example, the presence of a magnetar (neutron star with an extremely large magnetic field) in the open cluster Westerlund 1 means that it descends from a star that is more massive than the currently-observed most-massive main-sequence cluster stars (because for a massive star the duration of the main-sequence stage is inversely proportional to its mass squared). The most massive main-sequence stars in this stellar cluster are found to have masses as high as 40 M, suggesting Mpre−BH ≳ 40 M [643]. On the other hand, it was speculated, based on the properties of X-ray sources, that in the initial mass range (20–50) M the mass of pre-BHs may vary depending on such poorly-known parameters as rotation or magnetic fields [176].

Binaries with compact remnants are primary potentially-detectable GW sources (see Figure 2). This figure plots the sensitivity of ground-based interferometers, LIGO, as well as the space laser interferometers LISA and eLISA, in the terms of dimensionless GW strain h measured over one year.Footnote 10 The strongest Galactic sources at all frequencies are the most compact NS binaries, WD binaries, and (still hypothetically) BHs. NS/BH binary systems are formed from initially massive binaries, while WD binaries descend from low-mass binaries.

Figure 2
figure 2

Sensitivity limits of GW detectors and the regions of the fh diagram occupied by some potential GW sources. (Courtesy G. Nelemans).

In this review we shall concentrate on the formation and evolution of compact star binaries most relevant to GW studies. The article is organized as follows. We start in Section 2 with a review of the main observational data on NS binaries, especially measurements of masses of NSs and BHs, which are most important for the estimate of the amplitude of the expected GW signal. We briefly discuss empirical methods to determine the NS binary coalescence rate. The basic principles of stellar binary evolution are discussed in Section 3. Then, in Section 4 we describe the evolution of massive stellar binaries. Next, we discuss the Galactic rate of formation of binaries with NSs and BHs in Section 5. Theoretical estimates of detection rates for mergers of relativistic stellar binaries are discussed in Section 6. Further, we proceed to the analysis of the formation of short-period binaries with WD components in Section 7, and consider observational data on WD binaries in Section 8. A model for the evolution of interacting double-degenerate systems is presented in Section 9. In Section 10 we describe gravitational waves from compact binaries with white dwarf components. Section 11 is devoted to the modeling of optical and X-ray emissions of AM CVn-stars. Our conclusions follow in Section 12.

2 Observations of Double Compact Stars

2.1 Compact binaries with neutron stars

NS binaries have been discovered when one of their components is observed as a radio pulsar [292]. The precise pulsar timing allows one to search for a periodic variation due to the binary motion. This technique is reviewed in detail by Lorimer [444]; applications of pulsar timing for general relativity tests are reviewed by Stairs [733]. Techniques and results of measurements of NS mass and radii in different types of binaries are summarized by Lattimer [411].

Basically, pulsar timing provides the following Keplerian orbital parameters of the binary system: the binary orbital period Pb as measured from periodic Doppler variations of the pulsar spin, the projected semi-major axis x = a sin i as measured from the semi-amplitude of the pulsar radial velocity curve (i is the binary inclination angle defined such that i = 0 for face-on systems), the orbital eccentricity e as measured from the shape of the pulsar radial velocity curve, and the longitude of periastron ω at a particular epoch T0. The first two parameters allow one to construct the mass function of the secondary companion,

$$f({M_{\rm{p}}},{M_{\rm{c}}}) = {{4{\pi ^2}{x^3}} \over {P_{\rm{b}}^2{T_ \odot}}} = {{{{({M_{\rm{c}}}\sin i)}^3}} \over {{{({M_{\rm{c}}} + {M_{\rm{p}}})}^2}}}{.}$$

In this expression, x is measured in light-seconds, TGM/c3 = 4.925490947 μs, and Mp and Mc denote masses of the pulsar and its companion, respectively. This function gives the strict lower limit on the mass of the unseen companion. However, assuming the pulsar mass to have the typical value of a NS mass (for example, confined between the lowest measured NS mass 1.25 M for PSR J0737−3039B [451] and the maximum measured NS mass of 2.1 M in the NS-WD binary PSR J0751+1807 [527]), one can estimate the mass of the secondary star even without knowing the binary inclination angle i.

Long-term pulsar timing allows measurements of several relativistic phenomena: the advance of periastron \({\dot \omega}\), the redshift parameter γ, the Shapiro delay within the binary system qualified through post-Keplerian parameters r, s, and the binary orbit decay b. From the post-Keplerian parameters the individual masses Mp, Mc and the binary inclination angle i can be calculated [74].

Of the post-Keplerian parameters of pulsar binaries, the periastron advance rate is usually measured most readily. Assuming it to be entirely due to general relativity, the total mass of the system can be evaluated:

$$\dot \omega = 3{\left({{{2\pi} \over {{P_{\rm{b}}}}}} \right)^{5/3}}{{T_ \odot ^{2/3}{{({M_{\rm{c}}} + {M_{\rm{p}}})}^{2/3}}} \over {(1 - {e^2})}}{.}$$

The high value of the derived total mass of a system (≳ 2.5 M) suggests the presence of another NS or even a BH.Footnote 11

If the masses of components, binary period, and eccentricity of a compact binary system are known, it is easy to calculate the time it takes for the binary companions to coalesce due to GW emission using the quadrupole formula for GW emission [579] (see Section 3.1.4 for more detail):

$${\tau _{{\rm{GW}}}} \approx 4.8 \times {10^{10}}{\rm{yr}}{\left({{{{P_{\rm{b}}}} \over {\rm{d}}}} \right)^{8/3}}{\left({{\mu \over {{M_ \odot}}}} \right)^{- 1}}{\left({{{{M_{\rm{c}}} + {M_{\rm{p}}}} \over {{M_\odot}}}} \right)^{- 2/3}}{(1 - {e^2})^{7/2}}.$$

Here μ = MpMc/(Mp + Mc) is the reduced mass of the binary. Some observed and derived parameters of known compact binaries with NSs are collected in Tables 2 and 3.

Table 2 Observed parameters of neutron star binaries.
Table 3 Derived parameters of neutron star binaries.

2.2 How frequent are NS binary coalescences?

As is seen in Table 3, only six NS binary systems presently known will merge over a time interval shorter than ≈ 10 Gyr: J0737−3039A, B1534+12, J1756−2251, J1906+0746, B1913+16, and B2127+11C. Of these six systems, one (PSR B2127+11C) is located in the globular cluster M15. This system may have a different formation history, so usually it is not included in the analysis of the coalescence rate of Galactic compact binaries. The formation and evolution of relativistic binaries in dense stellar systems is reviewed elsewhere [39]. For a general review of pulsars in globular clusters see also [82].

Let us try to estimate the NS-binary merger rate from pulsar binary statistics, which is free from many uncertainties of stellar evolution. Usually, the estimate is based on the following extrapolation [505, 586]. Suppose, we observe i classes of Galactic pulsar binaries. Taking into account various selection effects of pulsar surveys (see, e.g., [504, 361]), the Galactic number of pulsars Ni in each class can be evaluated. To compute the Galactic merger rate of NS binaries, we need to know the time since the birth of the NS observed as a pulsar in the given binary system. This time is the sum of the observed characteristic pulsar age τc and the time required for the binary system to merge due to GW orbit decay τGW. With the exception of PSR J0737−3039B and the recently discovered PSR J1906+0746, pulsars that we observe in NS binary systems are old recycled pulsars that were spun-up by accretion from the secondary companion to the period of several tens of ms (see Table 2). Thus, their characteristic ages can be estimated as the time since termination of spin-up by accretion (for the younger pulsar PSR J0737−3039B this time can also be computed as the dynamical age of the pulsar, P/(2), which gives essentially the same result).

Then the merger rate \({{\mathcal R}_i}\) can be calculated as \({{\mathcal R}_i} \sim {N_i}/({\tau _c} + {\tau _{{\rm{GW}}}})\) (summed over all pulsar binaries). The detailed analysis [361] indicates that the Galactic merger rate of NS binaries is mostly determined by pulsars with faint radio luminosity and short orbital periods. Presently, it is the nearby (600 pc) pulsar-binary system PSR J0737−3039 with a short orbital period of 2.4 hr [77] that mostly determines the empirical estimate of the merger rate. According to Kim et al. [362], “the most likely values of DNS merger rate lie in the range 3–190 per Myr depending on different pulsar models”. This estimate has recently been revised in [363] based on the analysis of binary pulsars in the Galactic disk: PSR 1913+16, PSR 1534+12, and pulsar binary PSR J0737−3039A and J0737−3039B, giving the Galactic NS + NS coalescence rate \({{\mathcal R}_G} = 21_{- 14 - 17}^{+ 28 + 40}\) per Myr (95% and 90% confidence level, respectively). The estimates by population synthesis codes are still plagued by uncertainties in the statistics of binaries, in modeling binary evolution and supernovae. The most optimistic “theoretical” predictions amount to ≃ 300 Myr−1 [788, 34, 157].

An independent estimate of the NS-binary merger rate can also be obtained using another astrophysical argument, originally suggested by Bailes [20]. 1) Take the formation rate of single pulsars in the Galaxy \({{\mathcal R}_{{\rm{PSR}}}} \sim 1/50{\rm{y}}{{\rm{r}}^{-1}}\) (e.g., [190]; see also the discussion on the NS formation rate of different types in [351]), which appears to be correct to within a factor of two. 2) Take the fraction of NS binary systems in which one component is a normal (not recycled) pulsar and which are close enough to merge within the Hubble time, fDNS ∼ (a few) × 1/2000). Assuming a steady state, this fraction yields the formation rate of coalescing NS binaries in the Galaxy \({\mathcal R} \sim {\mathcal R_{{\rm{PSR}}}}{f_{{\rm{DNS}}}} \sim ({\rm{a}}\,{\rm{few}}\,10{\rm{s}}) \times {\rm{My}}{{\rm{r}}^{- 1}}\), in good correspondence with other empirical estimates [363]. Clearly, the Bailes limit ignores the fact that the NS formation rate can actually be higher than the pulsar formation rate [351] and possible selection effects related to the evolution of stars in binary systems, but the agreement with other estimates seems to be encouraging.

Extrapolation beyond the Galaxy is usually done by scaling the Galactic merger rate to the volume over which the merger events can be detected for a given GW detector’s sensitivity. The scaling factor widely used is the ratio between the B-band luminosity density in the local Universe, correlating with the star-formation rate (SFR), and the B-band luminosity of the galaxy [586, 333, 376]. For this purpose one can also use the direct ratio of the galactic star formation rate SFRG ≃ 2 M yr−1 [108, 353, 537] to the (dust-corrected) star formation rate in the local Universe SFRloc ≃ 0.03 M yr−1 Mpc−3 [577, 682]. These estimates yield the (Euclidean) relation

$${\mathcal{R}_{\rm{V}}} \simeq 0{.}01{\mathcal{R}_{\rm{G}}}[{\rm{Mp}}{{\rm{c}}^{- 3}}]{.}$$

This estimate is very close to the local density of equivalent Milky-Way type galaxies found in [376]: 0.016 per cubic Mpc.

Bear in mind that since NS-binary coalescences can be strongly delayed compared to star formation, the estimate of their rate in the nearby Universe based on present-day star formation density may be underevaluated. Additionally, these estimates inevitably suffer from many other astrophysical uncertainties. For example, a careful study of local SFR from analysis of an almost complete sample of nearby galaxies within 11 Mpc using different SFR indicators and supernova rate measurements during 13 years of observations [63] results in the local SFR density SFRloc ≃ 0.008 M yr−1 Mpc−3. However, Horiuchi et al. [288] argue that estimates of the local SFR can be strongly affected by stellar rotation. Using new stellar evolutionary tracks [173], they derived the local SFR density SFRloc ≃ 0.017 M yr−1 Mpc3 and noted that the estimate of the local SFR from core-collapse supernova counts is higher by a factor of 2–3.

Therefore, the actual value of the scaling factor from the galactic merger rate is presently uncertain to within a factor of at least two. The mean SFR density steeply increases with distance (e.g., as ∼ (1 + z)3.4±0.4 [127]) and can be higher in individual galaxies. Thus, for the conservative galactic NS + NS merger rate \({{\mathcal R}_G} \sim {10^{- 5}}{\rm{y}}{{\rm{r}}^{- 1}}\) using scaling relation (5) we obtain a few NS-binary coalescences per year of observations within the assumed advanced GW detectors horizon Dhor = 400 Mpc.

However, one should differentiate between the possible merging rate within some volume and the detection rate of certain types of compact binaries from this volume (see Section 6 below and [2] for more detail) — the detection rate by one detector can be lower by a factor of (2.26)3. The correction factor takes into account the averaging over all sky locations and orientations.

The latest results of the search for GWs from coalescing binary systems within 40 Mpc volume using LIGO and Virgo observations between July 7, 2009, and October 20, 2010 [3] established an observational upper limit to the 1.35 M + 1.35 M NS binary coalescence rate of < 1.3 × 10−4 yr−1 Mpc−3. Adopting the scaling factor from the measured local SFR density [63], the corresponding galactic upper limit is \({{\mathcal R}_G} < 0.05{\rm{y}}{{\rm{r}}^{- 1}}\). This is still too high to put interesting astrophysical bounds, but even upper limits from the advanced LIGO detector are expected to be very constraining.

2.3 Black holes in binary systems

Black holes (BH) in binary systems remain on the top of astrophysical studies. Most of the experimental knowledge on BH physics has so far been obtained through electromagnetic channels (see [618] for a review), but the fundamental features of BHs will be studied through GW experiments [673]. Stellar-mass black holes result from gravitational collapse of the cores of the most massive stars [851] (see [216] for the recent progress in the physics of gravitational collapse).

Stellar-mass BHs can be observed in close binary systems at the stage of mass accretion from the secondary companion as bright X-ray sources [690]. X-ray studies of BHs in binary systems (which are usually referred to as ‘Black Hole Candidates’, BHC) are reviewed in [633]. There is strong observational evidence of the launch of relativistic jets from the inner parts of accretion disks in BHC, which can be observed in the range from radio to gamma-rays (the ‘galactic microquasar’ phenomenon, [486, 197, 756]).

Optical spectroscopy and X-ray observations of BHC allow measurements of masses and, under certain assumptions, spins of BHs [476]. The measured mass distribution of stellar mass BHs is centered on ∼ 8 M and appears to be separated from NS masses [550] by a gap (the absence of compact star masses in the ∼ 2–5 M range) [22, 100, 549, 189, 580] (see, however, the discussion of possible systematic errors leading to overestimation of dynamical BH masses in X-ray transients in [386]). The mass gap, if real, can be indicative of the supernova-explosion-mechanism details (see, for example, [610, 37, 213]). It is also possible that the gap is due to “failed core-collapse supernovae” from red supergiants with masses 16.5 M < M < 25 M, which would produce black holes with a mass equal to the mass of the helium core (5–8 M) before the collapse, as suggested by Kochanek [371]. Adopting the latter hypothesis would increase by ∼ 20% the stellar-mass BH formation rate. However, it is unclear how binary interaction can change the latter possibility. We stress here once again that this important issue remains highly uncertain due to the lack of “first-principles” calculations of stellar-core collapses.

Most of more than 20 galactic BHC appear as X-ray transients with a rich phenomenology of outbursts and spectral/time variability [633], and only a few (Cyg X-1, LMC X-3, LMC X-1 and SS 433) are persistent X-ray sources. Optical companions in BH transients are low-mass stars (main-sequence or evolved) filling Roche lobes, and the transient activity is apparently related to accretion-disk instability [409]. Their evolution is driven by orbital angular momentum loss due to GW emission and magnetic stellar wind [619]. Persistent BHC, in contrast, have early-type massive optical companions and belong to the class of High-Mass X-ray Binaries (HMXB) (see [755, 797] and references therein).

Like in the case of pulsar timing for NSs, to estimate the mass of the BH companion in a close binary system one should measure the radial velocity curve of the companion (usually optical) star from spectroscopic observations. The radial velocity curve (i.e., the dependence of the radial velocity of the companion on the binary phase) has a form that depends on the orbital eccentricity e, and the amplitude K. In the Newtonian limit for two point masses, the binary mass function can be expressed through the observed quantities Pb (the binary orbital period), orbital eccentricity e and semi-amplitude of the radial velocity curve K as

$$f({M_v}) = {{{M_x}{{\sin}^3}i} \over {{{({M_x} + {M_v})}^2}}} = {{{P_b}} \over {2\pi G}}K_v^3{(1 - {e^2})^{3/2}} \approx 1{.}038 \times {10^{- 7}}[{M_ \odot}]{\left({{{{K_v}} \over {{\rm{km}}/{\rm{s}}}}} \right)^3}\left({{{{P_b}} \over {1{\rm{d}}}}} \right){(1 - {e^2})^{3/2}}{.}$$

From here one readily finds the mass of the unseen (X-ray) companion to be

$${M_x} = f({M_v}){\left({1 + {{{M_v}} \over {{M_x}}}} \right)^2}{1 \over {{{\sin}^3}i}}{.}$$

Unless the mass ratio Mv/Mx and orbital inclination angle i are known from independent measurements (for example, from the analysis of optical light curve and the duration of X-ray eclipse), the mass function of the optical component gives the lower limit of the BH mass: Mxf(Mv).

When using the optical mass function to estimate the BH mass as described above, one should always check the validity of approximations used in deriving Eq. (6) (see a detailed discussion of different effects related to the non-point-like shape of the optical companion in, e.g., [99]). For example, the optical O-B or Wolf-Rayet (WR) stars in HMXBs have a powerful high-velocity stellar wind, which can affect the dynamical BH-mass estimate based on spectroscopic measurements (see the discussion in [452] for BH + WR binaries).

Parameters of known Galactic and extragalactic HMXB with black holes are summarized in Table 4.

Table 4 Observed parameters of HMXB with black holes.

No PSR + BH system has been observed so far, despite optimistic expectations from the early population synthesis calculations [431] and recent examination of the possible dynamical formation of such a binary in the Galactic center [191]. Therefore, it is not possible to obtain direct experimental constraints on the NS + BH coalescence rate based on observations of real systems. Recently, the BH candidate with Be-star MWC 656 in a wide 60-day orbit was reported [93] (see Table 4). It is a very weak X-ray source [498], but can be the counterpart of the gamma-ray source AGL J2241+4454. The BH mass estimation in this case depends on the spectral classification and mass of the Be star. This system can be the progenitor of the long-sought BH + NS binary. Whether or not this binary system can become a merging BH + NS binary depends on the details of the common envelope phase, which is thought to happen after the bright accretion stage in this system, when the Be-star will evolve to the giant stage.

2.4 A model-independent upper limit on the BH-BH/BH-NS coalescence rate

Even without using the population synthesis tool, one can search for NS + BH or BH + BH progenitors among known BH in HMXB. This program has been pursued in [75, 32], [33]. In these papers, the authors examined the future evolution of two bright HMXB IC10 X-1/NGC300 X-1 found in nearby low-metallicity galaxies. Both binaries consist of massive WR-stars (about 20 M) and BH in close orbits (orbital periods about 30 hours). Masses of WR-stars seem to be high enough to produce the second BH, so these system may be immediate progenitors of coalescing BH-BH systems. Analysis of the evolution of the best-known Galactic HMXB Cyg X-1 [32], which can be a NS + BH progenitor, led to the conclusion that the galactic formation rate of coalescing NS + BH is likely to be very low, less than 1 per 100 Myrs. This estimate is rather pessimistic, even for advanced LIGO detectors. Implications of the growing class of short-period BH + WR binaries (see Table 4) for the (BH + BH)/(BH + NS) merger rate are discussed in [452].

While for NS-binary systems it is possible to obtain the upper limit for the coalescence rate based on observed pulsar binary statistics (the Bailes limit, see Section 2.2 above), it is not so easy for BH + NS or BH + BH systems due to the (present-day) lack of their observational candidates. Still, a crude estimate can be found from the following considerations. A rough upper limit on the coalescence rate of BH + NS(BH) binaries is set by the observed formation rate of high-mass X-ray binaries, their direct progenitors. The present-day formation rate of galactic HMXBs is about \({{\mathcal R}_{{\rm{HMXB}}}} \sim {N_{{\rm{HMXB}}}}/{t_{{\rm{HMXB}}}} \sim {10^{- 3}}\) per year (here we conservatively assumed NHMXB = 100 and tHMXB = 105 yr). This estimate can be made more precise considering that only very close compact binary systems can coalesce in the Hubble time, which requires a common envelope stage after the HMXB stage to occur (see Figure 7). The CE stage is likely to happen in sufficiently close binaries after the bright X-ray accretion stage.Footnote 12 The analysis of observations of different X-ray-source populations in the galaxies suggests ([243] and references therein) that only a few per cent of all BHs formed in a galaxy can pass through a bright accretion stage in HMXBs. For a galactic NS formation rate of once per several decades, a minimum BH progenitor mass of 20 M and the Salpeter initial mass function, this yields an estimate of a few ×10−5 per year. The probability of a given progenitor HMXB becoming a merging NS + BH or BH + BH binary is very model-dependent. For example, recent studies of a unique galactic microquasar SS 433 [58, 101] suggest the BH mass in such a system to be at least ∼ 5 M and the optical star mass to be above 15 M. In SS 433, the optical star fills the Roche lobe and forms a supercritical accretion disk [183]. The mass transfer rate onto the compact star is estimated to be about 10−4 M yr−1. According to the canonical HMXB evolution calculations, a common envelope must have been formed in such a binary on a short time scale (thousands of years) (see Section 3.5 and [319]), but the observed stability of binary-system parameters in SS433 over 30 years [132] shows that this is not the case. This example clearly illustrates the uncertainty in our understanding of HMXB evolution with BHs.

Thus, we conclude that in the most optimistic case where the (NS + BH)/(BH + BH) merger rate is equal to the formation rate of their HMXB progenitors, the upper limit for Galactic \({{\mathcal R}_{{\rm{NS + BH}}}}\) is a few × 10−5 yr−1. The detection or non-detection of such binaries within ∼ 1000 Mpc distance by the advanced LIGO detectors can therefore very strongly constrain our knowledge of the evolution of HMXB systems.

3 Basic Principles of the Evolution of Binary Stars

Beautiful early general reviews of the topic can be found, e.g., in [45, 795] and more later ones, e.g., in [755, 171, 799]. Here we restrict ourselves to recalling several facts concerning binary evolution that are most relevant to the formation and evolution of compact binaries. We will not discuss possible dynamical effects on binary evolution (like the Kozai-Lidov mechanism of eccentricity change in hierarchical triple systems, see e.g., [693]). Readers with experience in the field can skip this section.

3.1 Keplerian binary system and radiation back reaction

We start with some basic facts about Keplerian motion in a binary system and the simplest case of the evolution of two point masses due to gravitational radiation losses. The stars are highly condensed objects, so their treatment as point masses is usually adequate for the description of their interaction in the binary. Furthermore, Newtonian gravitation theory is sufficient for this purpose as long as the orbital velocities are small compared to the speed of light c. The systematic change of the orbit caused by the emission of gravitational waves will be considered in a separate paragraph below.

3.1.1 Keplerian motion

Let us consider two point masses M1 and M2 orbiting each other under the force of gravity. It is well known (see [404]) that this problem is equivalent to the problem of a single body with mass μ moving in an external gravitational potential. The value of the external potential is determined by the total mass of the system

$$M = {M_1} + {M_2}{.}$$

The reduced mass μ is

$$\mu = {{{M_1}{M_2}} \over M}{.}$$

The body μ moves in an elliptic orbit with eccentricity e and major semi-axis a. The orbital period P and orbital frequency Ω = 2π/P are related to M and a by Kepler’s third law

$${\Omega ^2} = {\left({{{2\pi} \over P}} \right)^2} = {{GM} \over {{a^3}}}{.}$$

This relationship is valid for any eccentricity e.

Individual bodies M1 and M2 move around the barycenter of the system in elliptic orbits with the same eccentricity e. The major semi-axes ai of the two ellipses are inversely proportional to the masses

$${{{a_1}} \over {{a_2}}} = {{{M_2}} \over {{M_1}}},$$

and satisfy the relationship a = a1 + a2. The position vectors of the bodies from the system’s barycenter are \({\vec r_1} = {M_2}\vec r/({M_1} + {M_2})\) and \({\vec r_1} = - {M_1}\vec r/({M_1} + {M_2})\), where \(\vec r = {\vec r_1} - {\vec r_2}\) is the relative position vector. Therefore, the velocities of the bodies with respect to the system’s barycentre are related by

$$- {{{{\vec V}_1}} \over {{{\vec V}_2}}} = {{{M_2}} \over {{M_1}}},$$

and the relative velocity is \(\vec V = {\vec V_1} - {\vec V_2}\).

The total conserved energy of the binary system is

$$E = {{{M_1}\vec V_1^2} \over 2} + {{{M_2}\vec V_2^2} \over 2} - {{G{M_1}{M_2}} \over r} = {{\mu {{\vec V}^2}} \over 2} - {{G{M_1}{M_2}} \over r} = - {{G{M_1}{M_2}} \over {2a}},$$

where r is the distance between the bodies. The orbital angular momentum vector is perpendicular to the orbital plane and can be written as

$${\vec J_{{\rm{orb}}}} = {M_1}{\vec V_1} \times {\vec r_1} + {M_2}{\vec V_2} \times {\vec r_2} = \mu \vec V \times \vec r{.}$$

The absolute value of the orbital angular momentum is

$$\vert {\vec J_{{\rm{orb}}}}\vert = \mu \sqrt {GMa(1 - {e^2}){.}}$$

For circular binaries with e = 0 the distance between orbiting bodies does not depend on time,

$$r(t,e = 0) = a,$$

and is usually referred to as orbital separation. In this case, the velocities of the bodies, as well as their relative velocity, are also time-independent,

$$V \equiv \vert \vec V\vert = \Omega a = \sqrt {GM/a} ,$$

and the orbital angular momentum becomes

$$\vert {\vec J_{{\rm{orb}}}}\vert = \mu Va = \mu \Omega {a^2}{.}$$

3.1.2 Gravitational radiation from a binary

The plane of the orbit is determined by the orbital angular momentum vector \({\vec J_{orb}}\) The line of sight is defined by a unit vector \(\vec n\). The binary inclination angle i is defined by the relation \(\cos i = (\vec n,{\vec J_{orb}}/{J_{orb}})\) such that i = 90° corresponds to a system visible edge-on.

Let us start from two point masses M1 and M2 in a circular orbit. In the quadrupole approximation [405], the two polarization amplitudes of GWs at a distance r from the source are given by

$${h_ +} = {{{G^{5/3}}} \over {{c^4}}}{1 \over r}2(1 + {\cos ^2}i){(\pi fM)^{2/3}}\mu \cos (2\pi ft),$$
$${h_ \times} = \pm {{{G^{5/3}}} \over {{c^4}}}{1 \over r}4\cos i{(\pi fM)^{2/3}}\mu \sin (2\pi ft){.}$$

Here f = Ω/π is the frequency of the emitted GWs (twice the orbital frequency). Note that for a fixed distance r and a given frequency f, the GW amplitudes are fully determined by \(\mu {M^{2/3}} = {{\mathcal M}^{5/3}}\), where the combination

$$\mathcal{M} \equiv {\mu ^{3/5}}{M^{2/5}}$$

is called the “chirp mass” of the binary. After averaging over the orbital period (so that the squares of periodic functions are replaced by 1/2) and the orientations of the binary orbital plane, one arrives at the averaged (characteristic) GW amplitude

$$h(f,\mathcal{M},r) = {(\langle h_ + ^2\rangle + \langle h_ \times ^2\rangle)^{1/2}} = {\left({{{32} \over 5}} \right)^{1/2}}{{{G^{5/3}}} \over {{c^4}}}{{{\mathcal{M}^{5/3}}} \over r}{(\pi f)^{2/3}}{.}$$

3.1.3 Energy and angular momentum loss

In the approximation and under the choice of coordinates that we are working with, it is sufficient to use the Landau-Lifshitz gravitational pseudo-tensor [405] when calculating the gravitational waves energy and flux. (This calculation can be justified with the help of a fully satisfactory gravitational energy-momentum tensor that can be derived in the field theory formulation of general relativity [17]). The energy dE carried by a gravitational wave along its direction of propagation per area dA per time dt is given by

$${{dE} \over {dAdt}} \equiv F = {{{c^3}} \over {16\pi G}}\left[ {{{\left({{{\partial {h_ +}} \over {\partial t}}} \right)}^2} + {{\left({{{\partial {h_ \times}} \over {\partial t}}} \right)}^2}} \right]{.}$$

The energy output dE/dt from a localized source in all directions is given by the integral

$${{dE} \over {dt}} = \int {F(\theta ,\phi){r^2}d\Omega {.}}$$


$${\left({{{\partial {h_ +}} \over {\partial t}}} \right)^2} + {\left({{{\partial {h_ \times}} \over {\partial t}}} \right)^2} = 4{\pi ^2}{f^2}{h^2}(\theta ,\phi)$$

and introducing

$${h^2} = {1 \over {4\pi}}\int {{h^2}(\theta ,\phi)d\Omega ,}$$

we write Eq. (22) in the form

$${{dE} \over {dt}} = {{{c^3}} \over G}{(\pi f)^2}{h^2}{r^2}{.}$$

Specifically for a binary system in a circular orbit, one finds the energy loss from the system (sign minus) with the help of Eqs. (23) and (20):

$${{dE} \over {dt}} = - \left({{{32} \over 5}} \right){{{G^{7/3}}} \over {{c^5}}}{(\mathcal{M}\pi f)^{10/3}}{.}$$

This expression is exactly the same that can be obtained directly from the quadrupole formula [405],

$${{dE} \over {dt}} = - {{32} \over 5}{{{G^4}} \over {{c^5}}}{{M_1^2M_2^2M} \over {{a^5}}}{.}$$

rewritten using the definition of the chirp mass and Kepler’s law. Since energy and angular momentum are continuously carried away by gravitational radiation, the two masses in orbit spiral towards each other, thus increasing their orbital frequency Ω. The GW frequency f = Ω/π and the GW amplitude h are also increasing functions of time. The rate of the frequency change isFootnote 13

$$\dot f - \left({{{96} \over 5}} \right){{{G^{5/3}}} \over {{c^5}}}{\pi ^{8/3}}{\mathcal{M}^{5/3}}{f^{11/3}}{.}$$

In spectral representation, the flux of energy per unit area per unit frequency interval is given by the right-hand-side of the expression

$${{dE} \over {dA\;df}} = {{{c^3}} \over G}{{\pi {f^2}} \over 2}\left({{{\left\vert {\tilde h{{(f)}_ +}} \right\vert}^2} + {{\left\vert {\tilde h{{(f)}_ +}} \right\vert}^2}} \right) \equiv {{{c^3}} \over G}{{\pi {f^2}} \over 2}S_h^2(f),$$

where we have introduced the spectral density \(S_h^2(f)\) of the gravitational wave field h. In the case of a binary system, the quantity Sh is calculable from Eqs. (18 and 19):

$${S_h} = {{{G^{5/3}}} \over {{c^3}}}{\pi \over {12}}{{{\mathcal{M}^{5/3}}} \over {{r^2}}} - {1 \over {{{(\pi f)}^{7/3}}}}{.}$$

3.1.4 Binary coalescence time

A binary system in a circular orbit loses energy according to Eq. (24). For orbits with non-zero eccentricity e, the right-hand-side of this formula should be multiplied by the factor

$$f(e) = \left({1 + {{73} \over {24}}{e^2} + {{37} \over {96}}{e^4}} \right){(1 - {e^2})^{- 7/2}}$$

(see [579]). The initial binary separation a0 decreases and, assuming Eq. (25) is always valid, the binary should vanish in a time

$${t_0} = {{{c^5}} \over {{G^3}}}{{5a_0^4} \over {256{M^2}\mu}} = {{5{c^2}} \over {256}}{{{{({P_0}/2\pi)}^{8/3}}} \over {{{(G\mathcal{M})}^{5/3}}}} \approx (9{.}8 \times {10^6}\;{\rm{yr}}){\left({{{{P_0}} \over {1\;{\rm{h}}}}} \right)^{8/3}}{\left({{{\mathcal{M}} \over {{M_ \odot}}}} \right)^{- 5/3}}{.}$$

As noted above, gravitational radiation from the binary depends on the chirp mass \({\mathcal M}\), which can also be written as \({\mathcal M} \equiv M{\eta ^{3/5}}\), where η is the dimensionless ratio η = μ/M. Since η ≤ 1/4, one has \({\mathcal M} \lesssim 0.435M\). For example, for two NSs with equal masses M1 = M2 = 1.4 M, the chirp mass is \({\mathcal M} \approx 1.22{M_ \odot}\). This explains the choice of normalization in Eq. (29).

The coalescence time for a binary star with an initially eccentric orbit with e0 ≠ 0 and initial separation a0 is shorter than the coalescence time for an object with a circular orbit and the same a0 [579]:

$${t_c}({e_0}) = {t_0}f({e_0}),$$

where the correction factor f(e0) is

$$f({e_0}) = {{48} \over {19}}{{{{(1 - e_0^2)}^4}} \over {e_0^{48/19}{{\left({1 + {{121} \over {304}}e_0^2} \right)}^{3480/2299}}}}\int\nolimits_0^{{e_0}} {{{{{\left({1 + {{121} \over {304}}{e^2}} \right)}^{1181/2299}}} \over {{{(1 - {e^2})}^{3/2}}}}{e^{29/19}}de{.}}$$

To merge in a time interval shorter than 10 Gyr the binary should have a small enough initial orbital period \({P_0} \leq {P_{{\rm{cr}}}}({e_0},{\mathcal M})\) and, accordingly, a small enough initial semi-major axis \({a_0} \leq {a_{{\rm{cr}}}}({e_0},{\mathcal M})\). The critical orbital period is plotted as a function of the initial eccentricity e0 in Figure 3. The lines are plotted for three typical sets of masses: two neutron stars with equal masses (1.4 M + 1.4 M), a black hole and a neutron star (1.0 M + 1.4 M), and two black holes with equal masses (10 M + 10 M). Note that in order to get a significantly shorter coalescence time, the initial binary eccentricity should be e0 ≥ 0.6.

Figure 3
figure 3

The maximum initial orbital period (in hours) of two point masses that will coalesce due to gravitational wave emission in a time interval shorter than 1010 yr, as a function of the initial eccentricity e0. The lines are calculated for 10 M + 10 M (BH + BH), 10 M + 1.4 M (BH + NS), and 1.4 M +1.4 M (NS + NS).

3.1.5 Magnetic stellar wind

In the case of low-mass binary evolution, there is another important physical mechanism responsible for the removal of orbital angular momentum, in addition to the GW emission discussed above. This is the magnetic stellar wind (MSW), or magnetic braking, which is thought to be effective for main-sequence G-M dwarfs with convective envelopes, i.e., approximately, in the mass interval 0.3–1.2 M. The upper mass limit corresponds to the disappearance of a deep convective zone, while the lower mass limit stands for fully convective stars. In both cases a dynamo mechanism, responsible for enhanced magnetic activity, is thought to become ineffective. The idea behind angular momentum loss (AML) by magnetically-coupled stellar wind is that the stellar wind is compelled by magnetic field to corotate with the star to rather large distances, where it carries away large specific angular momentum [680]. Thus, it appears possible to take away substantial angular momentum without evolutionarily significant mass-loss in the wind. In the quantitative form, the concept of angular momentum loss by MSW as a driver of the evolution of compact binaries was introduced by Verbunt and Zwaan [810] when it became evident that momentum loss by GWs is unable to explain the observed mass-transfer rates in cataclysmic variables (CVs) and low-mass X-ray binaries, as well as the deficit of cataclysmic variables with orbital periods between 2 and 3 hr (the “period gap”).Footnote 14 Verbunt and Zwaan based their reasoning on observations of the spin-down of rotation of single G-dwarfs in stellar clusters with age, which is expressed by the phenomenological dependence of the equatorial rotational velocity V on age: V = λt−1/2 (“Skumanich law” [718]). In the latter formula λ is an empirically-derived coefficient ∼ 1. Applying this to a binary component and assuming tidal locking between the stellar axial rotation and orbital motion, one arrives at the rate of angular momentum loss via MSW

$${\dot J_{{\rm{MSW}}}} = - 0{.}5 \times {10^{- 28}}[{\rm{g}}\;{\rm{c}}{{\rm{m}}^2}\;{{\rm{s}}^{- 2}}]{\lambda ^{- 2}}{k^2}{M_2}R_2^4{\omega ^3},$$

where M2 and R2 are the mass and radius of the optical component of the system, respectively, ω = Ωorb is the spin frequency of the star’s rotation equal to the binary orbital frequency, and k2 ∼ 0.1 is the gyration radius of the optical component squared.

Radii of stars filling their Roche lobes should be proportional to binary separations, Roa, which means that the time scale of orbital angular momentum removal by MSW is \({\tau _{{\rm{MSW}}}} \equiv {({\dot J_{{\rm{MSW}}}}/{J_{{\rm{orb}}}})^{- 1}} \propto a\). This should be compared with AML by GWs with τGWa4. Clearly, MSW (if it operates) is more efficient at removing angular momentum from a binary system at larger separations (orbital periods), and at small orbital periods GWs always dominate. Magnetic braking is especially important in CVs and in LMXBs with orbital periods exceeding several hours and is the driving mechanism for mass accretion onto the compact component. Indeed, the Doppler tomography reconstruction of the Roche-lobe-filling low-mass K7 secondary star in a well-studied LMXB Cen X-4 revealed the presence of cool spots on its surface. The latter provide evidence for the action of magnetic fields at the surface of the star, thus supporting magnetic braking as the driving mechanism of mass exchange in this binary [689].

Equation (32) with assumed λ = 1 and k2 = 0.07 [627] is often considered as a “standard”. However, Eqs. (25) and (32) do not allow one to reproduce some observed features of CVs, see, e.g., [627, 600, 434, 771, 368], but the reasons for this discrepant behavior are not clear as yet. Recently, Knigge et al. [368], based on the study of properties of donors in CVs, suggested that a better description of the evolution of CVs is provided by scaling Eq. (32) by a factor of (0.66 ± 0.05)(R2/R)−1 above the period gap and by scaling Eq. (25) by a factor 2.47 ± 0.22 below the gap (where AML by MSW is not acting). Note that these “semi-empirical”, as called by the their authors, numerical factors must be taken with some caution, since they are based on the fitting of the PorbM2 relation for observed stars by a single evolutionary sequence with the initially-unevolved donor with M2,0 = 1 M and MWD,0 = 0.6 M.

The simplest reason for deviation of AML by MSW from Eq. (25) may be the unjustified extrapolation of stellar-rotation rates over several orders of magnitude — from slowly-rotating single field stars to rapidly-spinning components of close binaries. For the shortest orbital periods of binaries meant to evolve due to AML via GW, the presence of circumbinary discs may enhance their orbital angular momentum loss [841].

3.2 Mass exchange in close binaries

As mentioned in the introductory Section 1, all binaries may be considered either as “close” or as “wide”. In the former case, mass exchange between the components can occur. This process can be accompanied by mass and angular momentum loss from the system.

The shape of the stellar surface is determined by the shape of the equipotential level surface Φ = const. Conventionally, the total potential, which includes gravitational and centrifugal forces is approximated by the Roche potential (see, e.g., [375]), which is defined under the following assumptions:

  • the gravitational field of two components is approximated by that of two point masses;

  • the binary orbit is circular;

  • the components of the system corotate with the binary orbital period.

Let us consider a Cartesian reference frame (x,y,z) rotating with the binary, with the origin at the primary M1; the x-axis is directed along the line of centers; the y-axis is aligned with the orbital motion of the primary component; the z-axis is perpendicular to the orbital plane. The total potential at a given point (x,y,z) is then

$$\Phi = - {{G{M_1}} \over {{{[{x^2} + {y^2} + {z^2}]}^{1/2}}}} - {{G{M_2}} \over {{{[{{(x - a)}^2} + {y^2} + {z^2}]}^{1/2}}}} - {1 \over 2}\Omega _{{\rm{orb}}}^2[{(x - \mu a)^2} + {y^2}],$$

where μ = M2/(M1 + M2), Ωorb = 2π/Porb.

A 3-D representation of a dimensionless Roche potential in a co-rotating frame for a binary with mass ratio of components q = 2 is shown in Figure 4.

Figure 4
figure 4

A 3-D representation of a dimensionless Roche potential in the co-rotating frame for a binary with a mass ratio of components q = 2.

For close binary evolution the most important is the innermost level surface that encloses both components. It defines the “critical”, or “Roche” lobes of the components.Footnote 15 Inside these lobes the matter is bound to the respective component. In the libration point L1 (the inner Lagrangian point) the net force exerted onto a test particle corotating with the binary vanishes, so the particle can escape from the surface of the star and be captured by the companion. The matter flows along the surface of the Roche-lobe filling companion in the direction of L1 and escapes from the surface of the contact component as a highly inhomogeneous stellar wind. Next, the wind forms a (supersonic) stream directed at a certain angle respective to the line connecting the centers of the components. Depending on the size of the second companion, the stream may hit the latter (called “direct impact”) or form a disk orbiting the companion [448].

Though it is evident that mass exchange is a complex 3D gas-dynamical process that must also take into account radiation transfer and, in some cases, even nuclear reactions, virtually all computations of the evolution of non-compact close binaries have been performed in 1-D approximations. The Roche-lobe overflow (RLOF) is conventionally considered to begin when the radius of an initially more massive and hence faster evolving star (primary component) becomes equal to the radius of a sphere with volume equal to that of the Roche lobe. For the latter, an expression precise to better than 1% for an arbitrary mass ratio of components q was suggested by Eggleton [172]:

$${{{R_l}} \over a} \approx {{0{.}49{q^{2/3}}} \over {0{.}6{q^{2/3}} + \ln (1 + {q^{1/3}})}}{.}$$

For practical purposes, such as analytical estimates, a more convenient expression is that suggested by Kopal [375] (usually called the “Paczyński formula”, who introduced it into close binary modeling):

$${{{R_l}} \over a} \approx 0{.}4623{\left({{q \over {1 + q}}} \right)^{1/3}},$$

which is accurate to ≲ 2% for 0 < q ≲ 0.8.

In close binaries, the zero-age main sequence (ZAMS) mass ceases to be the sole parameter determining stellar evolution. The nature of compact remnants of close binary components also depends on their evolutionary stage at RLOF, i.e., on the component separation and their mass-ratio q. Evolution of a star may be considered as consumption of nuclear fuel accompanied by an increase of its radius. Following the pioneering work of Kippenhahn and his collaborators on the evolution of close binaries in the late 1960s, the following basic cases of mass exchange are usually considered: A — RLOF at the core hydrogen-burning stage; B — RLOF at the hydrogen-shell burning stage; C — RLOF after exhaustion of He in the stellar core. Also, more “fine” gradations exist: case AB — RLOF at the late stages of core H-burning, which continues as Case B after a short break upon exhaustion of H in the core; case BB — RLOF by the star, which first filled its Roche lobe in case B, contracted under the Roche lobe after the loss of the hydrogen envelope, and resumed the mass loss due to the envelope expansion at the helium-shell burning stage. Further, one may consider the modes of mass-exchange, depending, e.g., on the nature of the envelope of the donor (radiative vs. convective), its relative mass, reaction of accretor etc., see, e.g., [171, 826] and Section 3.3.

In cases A and B of mass exchange the remnants of stars with initial masses lower than (2.3–2.8) M are degenerate He WD — a type of objects not produced by single stars.Footnote 16

In cases A and B of mass exchange the remnants of stars with initial mass ≳ 2.5 M are helium stars.

If the mass of the helium remnant star does not exceed ≃ 0.8 M, after exhaustion of the He in the core it does not expand, but transforms directly into a WD of the same mass [300]. Helium stars with masses between ≃ 0.8M and ≃ (2.3–2.8) M expand in the helium shell burning stage, re-fill their Roche lobes, lose the remnants of the helium envelopes and also transform into CO WD.

Stellar radius may be taken as a proxy for the evolutionary state of a star. We plot in Figure 5 the types of stellar remnant as a function of both initial mass and the radius of a star at the instant of RLOF. In Figure 6 initial-final mass relations for components of close binaries are shown. Clearly, relations presented in these two figures are only approximate, reflecting current uncertainty in the theory of stellar evolution (in this particular case, relations used in the population synthesis code IBiS [791] are shown.

Figure 5
figure 5

Descendants of components of close binaries depending on the radius of the star at RLOF. The upper solid line separates close and wide binaries (after [293]). The boundary between progenitors of He-and CO-WDs is uncertain by several 0.1 M, the boundary between CO and ONe varieties of WDs and WD and NSs — by ∼ 2 M. The boundary between progenitors of NS and BH is shown at 40 M after [643], while it may be possible that it really is between 20 M and 50 M (see Section 1 for discussion and references.)

Figure 6
figure 6

Relation between ZAMS masses of stars Mi and their masses at TAMS (solid line), masses of helium stars (dashed line), masses of He WD (dash-dotted line), masses of CO and ONe WD (dotted line). For stars with Mi ≲ 5 M we plot the upper limit of WD masses for case B of mass exchange. After [791].

3.3 Mass transfer modes and mass and angular momentum loss in binary systems

GW emission is the sole factor responsible for the change of orbital parameters of a detached pair of compact (degenerate) stars. However, it was recognized quite early on that in the stages of evolution preceding the formation of compact objects, the mode of mass transfer between the components and the loss of matter and orbital angular momentum by the system as a whole play a dominant dynamical role and define the observed features of the binaries, e.g., [556, 781, 631, 471, 603, 137].

Strictly speaking, as mentioned previously, these processes should be treated hydrodynamically and they require complicated numerical calculations. However, binary evolution can also be described semi-qualitatively, using a simplified description in terms of point-like bodies. The change of their integrated physical quantities, such as masses, orbital angular momentum, etc., governs the evolution of the orbit. This description turns out to be successful in reproducing the results of more rigorous numerical calculations (see, e.g., [705] for more details and references). In this approach, the key role is allocated to the total orbital angular momentum Jorb of the binary.

Let star 2 lose matter at a rate 2 < 0 and let β (0 ≤ β ≤ 1) be the fraction of the ejected matter that leaves the system (the rest falls on the first star), i.e., 1 = −(1 − β)2 ≥ 0. Consider circular orbits with orbital angular momentum given by Eq. (17). Differentiate both parts of Eq. (17) by time t and exclude dΩ/dt with the help of Kepler’s third law (10). This gives the rate of change of the orbital separation:

$${{\dot a} \over a} = - 2\left({1 + (\beta - 1){{{M_2}} \over {{M_1}}} - {\beta \over 2}{{{M_2}} \over M}} \right){{{{\dot M}_2}} \over {{M_2}}} + 2{{{{\dot J}_{{\rm{orb}}}}} \over {{J_{{\rm{orb}}}}}}{.}$$

In Eq. (36) ȧ and are not independent variables if the donor fills its Roche lobe. One defines the mass transfer as conservative if both β = 0 and \({\dot J_{{\rm{orb}}}} = 0\). The mass transfer is called non-conservative if at least one of these conditions is violated.

It is important to distinguish some specific cases (modes) of mass transfer:

  1. 1.

    conservative mass transfer,

  2. 2.

    non-conservative Jeans mode of mass loss (or fast wind mode),

  3. 3.

    non-conservative isotropic re-emission,

  4. 4.

    sudden mass loss from one of the components during supernova explosion, and

  5. 5.

    common-envelope stage.

As specific cases of angular momentum loss we consider GW emission (see Section 3.1.3 and 3.1.4) and the magnetically-coupled stellar wind (see Section 3.1.5), which drive the orbital evolution for short-period binaries. For non-conservative modes, one can also consider some less important cases, such as, for instance, the formation of a circumbinary ring by the matter leaving the system (see, e.g., [722, 732]). Here, we will not go into details of such sub-cases.

3.3.1 Conservative accretion

In the case of conservative accretion, matter from M2 is fully deposited onto M1. The transfer process preserves the total mass (β = 0) and the orbital angular momentum of the system. It follows from Eq. (36) that

$${M_1}{M_2}\sqrt a = {\rm{const}}{\rm{{.},}}$$

so that the initial and final binary separations are related as

$${{{a_{\rm{f}}}} \over {{a_{\rm{i}}}}} = {\left({{{{M_{1{\rm{i}}}}{M_{2{\rm{i}}}}} \over {{M_{1{\rm{f}}}}{M_{2{\rm{f}}}}}}} \right)^2}{.}$$

The orbit shrinks when the more massive component loses matter, and the orbit widens in the opposite situation. During such a mass exchange, the orbital separation passes through a minimum, if the masses become equal in the course of mass transfer.

3.3.2 The Jeans (fast wind) mode

In this mode the matter ejected by the donor completely escapes from the system, that is, β = 1. Escape of the matter can occur either via fast isotropic spherically-symmetric wind or in the form of bipolar jets moving from the system at high velocity. Escaping matter does not interact with another component. The matter escapes with the specific angular momentum of the mass-losing star J2 = (M1/M)Jorb (we neglect a possible proper rotation of the star, see [795]). For the loss of orbital momentum \({\dot J_{{\rm{orb}}}}\) it is reasonable to take

$${\dot J_{{\rm{orb}}}}{{{{\dot M}_2}} \over {{M_2}}}{J_2}{.}$$

In the case β = 1, Eq. (36) can be written as

$${{\dot \Omega {a^2}} \over {\Omega {a^2}}} = {{{{\dot J}_{{\rm{orb}}}}} \over {{J_{{\rm{orb}}}}}} - {{{M_1}{{\dot M}_2}} \over {M{M_2}}}{.}$$

Then Eq. (39) in conjunction with Eq. (38) gives Ωa2 = const, that is, \(\sqrt {GaM} = const\). Thus, as a result of the Jeans mode of mass loss, the change in orbital separation is

$${{{a_{\rm{f}}}} \over {{a_{\rm{i}}}}} = {{{M_{\rm{i}}}} \over {{M_{\rm{f}}}}}{.}$$

Since the total mass decreases, the orbit always widens.

3.3.3 Isotropic re-emission

The matter lost by star 2 can first accrete onto star 1, and then, a fraction β of the accreted matter, can be expelled from the system. This happens, for instance, when a massive star transfers matter onto a compact star on the thermal timescale (usually < 106 years). Accretion luminosity may exceed the Eddington luminosity limit, and the radiation pressure pushes the infalling matter away from the system, in a manner similar to the spectacular example of the SS 433 binary system. Other examples may be systems with helium stars transferring mass onto relativistic objects [442, 238]. In this mode of mass-transfer, the binary orbital momentum carried away by the expelled matter is determined by the orbital momentum of the accreting star rather than by the orbital momentum of the mass-losing star, since mass loss happens in the vicinity of the accretor. The assumption that all matter in excess of accretion rate can be expelled from the system, thus avoiding the formation of a common envelope, will only hold if the liberated accretion energy of the matter falling from the Roche lobe radius of the accretor star to its surface is sufficient to expel the matter from the Roche-lobe surface around the accretor, i.e., \({\dot M_d}\lesssim {\dot M_{{{\rm{d}}_{{\rm{max}}}}}} = {\dot M_{{\rm{Edd}}}}({r_{{\rm{L,a}}}}/{r_{\rm{a}}})\), where ra is the radius of the accretor [364].

The orbital momentum loss can be written as

$${\dot J_{{\rm{orb}}}} = \beta {{{{\dot M}_2}} \over {{M_1}}}{J_1},$$

where J1 = (M2/M)Jorb is the orbital momentum of the star M1. In the limiting case when all the mass initially accreted by M1 is later expelled from the system, β = 1, Eq. (41) simplifies to

$${{{{\dot J}_{{\rm{orb}}}}} \over {{J_{{\rm{orb}}}}}} = {{{{\dot M}_2}{M_2}} \over {{M_1}M}}{.}$$

After substitution of this formula into Eq. (36) and integration over time, one arrives at

$${{{a_{\rm{f}}}} \over {{a_{\rm{i}}}}} = {{{M_{\rm{i}}}} \over {{M_{\rm{f}}}}}{\left({{{{M_{{\rm{2i}}}}} \over {{M_{{\rm{2f}}}}}}} \right)^2}\exp \left({- 2{{{M_{{\rm{2i}}}} - {M_{{\rm{2f}}}}} \over {{M_{\rm{1}}}}}} \right).$$

The exponential term makes this mode of mass transfer very sensitive to the components’ mass ratio. If M1/M2 ≪ 1, the separation a between the stars may decrease so much that the approximation of point masses becomes invalid. Tidal orbital instability (Darwin instability) may set in, and the compact star’s may start spiraling toward the companion star centre (the common envelope stage; see Section 3.6). On the other hand, “isotropic reemission” may stabilize mass-exchange if M1/M2 > 1 [877].

Mass loss may be considered as occurring in the “isotropic re-emission” mode in situations in which hot white dwarf components of cataclysmic variables lose mass by optically-thick winds [346] or when time-averaged mass loss from novae is considered [867].

3.4 Supernova explosion

Supernovae explosions in binary systems occur on a timescale much shorter than the orbital period, so the loss of mass is practically instantaneous. This case can be treated analytically (see, e.g., [55, 59, 753]).

Clearly, even a spherically-symmetric sudden mass loss due to an SN explosion will be asymmetric in the reference frame of the center of mass of the binary system, leading to system recoil (‘Blaauw-Boersma’ recoil). In general, the loss of matter and radiation is non-spherical, so that the remnant of the supernova explosion (neutron star or black hole) acquires some recoil velocity called kick velocity \(\vec w\). In a binary, the kick velocity should be added to the orbital velocity of the pre-supernova star.

The usual treatment proceeds as follows. Let us consider a pre-SN binary with initial masses M1 and M2. The stars move in a circular orbit with orbital separation ai and relative velocity \({\vec V_i}\). The star M1 explodes leaving a compact remnant of mass Mc. The total mass of the binary decreases by ΔM = M1 − Mc. Unless the binary is disrupted, it will end up in a new orbit with eccentricity e, semi-major axis af, and angle θ between the orbital planes before and after the explosion. In general, the new barycenter will also receive some velocity, but we neglect this motion. The goal is to evaluate the parameters af, e, and θ.

It is convenient to use an instantaneous reference frame centered on M2 right at the time of explosion. The x-axis is the line from M2 to M1, the y-axis points in the direction of \({\vec V_i}\), and the z-axis is perpendicular to the orbital plane. In this frame, the pre-SN relative velocity is \({\vec V_i} = (0,{V_i},0)\), where \({V_i} = \sqrt {G({M_1} + {M_2})/{a_i}}\) (see Eq. (16)). The initial total orbital momentum is \({\vec J_i} = {\mu _i}{a_i}(0,0 - {V_i})\). The explosion is considered to be instantaneous. Right after the explosion, the position vector of the exploded star M1 has not changed: \(\vec r = ({a_i},0,0)\). However, other quantities have changed: \({\vec V_{\rm{f}}} = ({w_x},{V_i} + {w_y},{w_z})\) and \({\vec J_{\rm{f}}} = {\mu _{\rm{f}}}{a_{\rm{i}}}(0,{w_z}, - ({V_i} + {w_y}))\), where \(\vec w = ({w_x},{w_y},{w_z})\) is the kick velocity and μf = McM2/(Mc + M2) is the reduced mass of the system after explosion. The parameters af and e are found by equating the total energy and the absolute value of the orbital momentum of the initial circular orbit to those of the resulting elliptical orbit (see Eqs. (13, 17, 15)):

$${\mu _{\rm{f}}}{{V_{\rm{f}}^2} \over 2} - {{G{M_{\rm{c}}}{M_2}} \over {{a_{\rm{i}}}}} = - {{G{M_{\rm{c}}}{M_2}} \over {2{a_{\rm{f}}}}},$$
$${\mu _{\rm{f}}}{a_{\rm{i}}}\sqrt {w_z^2 + {{({V_{\rm{i}}} + {w_y})}^2}} = {\mu _{\rm{f}}}\sqrt {G({M_{\rm{c}}} + {M_2}){a_{\rm{f}}}(1 - {e^2}){.}}$$

For the resulting af and e one finds

$${{{a_{\rm{f}}}} \over {{a_{\rm{i}}}}} = {\left[ {2 - \chi \left({{{w_x^2 + w_z^2 + {{({V_{\rm{i}}} + {w_y})}^2}} \over {V_{\rm{i}}^2}}} \right)} \right]^{- 1}}$$


$$1 - {e^2} = \chi {{{a_{\rm{i}}}} \over {{a_{\rm{f}}}}}\left({{{w_z^2 + {{({V_{\rm{i}}} + {w_y})}^2}} \over {V_{\rm{i}}^2}}} \right),$$

where χ ≡ (M1 + M2)/(Mc + M2) 1. The angle θ is defined by

$$\cos \theta = {{{{\vec J}_{\rm{f}}}\cdot{{\vec J}_{\rm{i}}}} \over {\vert {{\vec J}_{\rm{f}}}\Vert {{\vec J}_{\rm{i}}}\vert}},$$

which results in

$$\cos \theta = {{{V_{\rm{i}}} + {w_y}} \over {\sqrt {w_z^2 + {{({V_{\rm{i}}} + {w_y})}^2}}}}{.}$$

The condition for disruption of the binary system depends on the absolute value Vf of the final velocity, and on the parameter χ. The binary disrupts if its total energy defined by the left-hand-side of Eq. (44) becomes non-negative or, equivalently, if its eccentricity defined by Eq. (47) becomes e ≥ 1. From either of these requirements one derives the condition for disruption:

$${{{V_{\rm{f}}}} \over {{V_{\rm{i}}}}} \geq \sqrt {{2 \over \chi}} {.}$$

The system remains bound if the opposite inequality is satisfied. Eq. (49) can also be written in terms of the escape (parabolic) velocity Ve defined by the requirement

$${\mu _{\rm{f}}}{{V_{\rm{e}}^2} \over 2} - {{G{M_{\rm{c}}}{M_2}} \over {{a_{\rm{i}}}}} = 0{.}$$

Since χ = M/(M − ΔM) and \(V_{\rm{e}}^2 = 2G(M - \Delta M)/{a_{\rm{i}}} = 2V_{\rm{i}}^2/\chi\), one can write Eq. (49) in the form

$${V_{\rm{f}}} \geq {V_{\rm{e}}}{.}$$

The condition of disruption simplifies in the case of a spherically-symmetric SN explosion, that is, when there is no kick velocity, \(\vec w = 0\), and, therefore, Vf = Vi. In this case, Eq. (49) reads χ ≥ 2, which is equivalent to ΔM ≥ M/2. Thus, the system unbinds if more than half of the mass of the binary is lost. In other words, the resulting eccentricity

$$e = {{{M_1} - {M_{\rm{c}}}} \over {{M_{\rm{c}}} + {M_2}}}$$

following from Eqs. (46) and (47), and \(\vec w = 0\) becomes larger than 1, if ΔM > M/2.

So far, we have considered an originally circular orbit. If the pre-SN star moves in an originally eccentric orbit, the condition of disruption of the system under symmetric explosion reads

$$\Delta M = {M_1} - {M_{\rm{c}}} > {1 \over 2}{r \over {{a_{\rm{i}}}}},$$

where r is the distance between the components at the moment of explosion.

3.5 Kick velocity of neutron stars

The kick imparted to a NS at birth is one of the major problems in the theory of stellar evolution. By itself, it is an additional parameter, the introduction of which has been motivated first of all by high space velocities of radio pulsars inferred from the measurements of their proper motions and distances. Pulsars were recognized as a high-velocity Galactic population soon after their discovery in 1968 [256]. Shklovskii [704] put forward the idea that high pulsar velocities may result from asymmetric supernova explosions. Since then this hypothesis has been tested by pulsar observations, but no definite conclusions on its magnitude and direction have as yet been obtained.

Indeed, the distance to a pulsar is usually derived from the dispersion measure evaluation and crucially depends on the assumed model of electron density distribution in the Galaxy. In the middle of the 1990s, Lyne and Lorimer [449] derived a very high mean space velocity of pulsars with known proper motion of about 450 km s−1. This value was difficult to adopt without invoking an additional natal kick velocity of NSs.

The high mean space velocity of pulsars, consistent with earlier results by Lyne and Lorimer, was confirmed by the analysis of a larger sample of pulsars [284]. The recovered distribution of 3D velocities is well fit by a Maxwellian distribution with the mean value w0 = 400 ± 40 km s−1 and a 1D rms σ = 265 km s−1.

Possible physical reasons for natal NS kicks due to hydrodynamic effects in core-collapse super-novae are summarized in [402, 401]. Large kick velocities (∼ 500 km s−1 and even more) imparted to nascent NSs are generally confirmed by detailed numerical simulations (see, e.g., [536, 846]). Neutrino effects in the strong magnetic field of a young NS may also be essential in explaining kicks up to ∼ 100 km s−1 [109, 160, 398]. Astrophysical arguments favoring a kick velocity are also summarized in [754]. To get around the theoretical difficulty of insufficient rotation of pre-supernova cores in single stars to produce rapidly-spinning young pulsars, Spruit and Phinney [731] proposed that random off-center kicks can lead to a net spin-up of proto-NSs. In this model, correlations between pulsar space velocity and rotation are possible and can be tested in further observations.

Here we should note that the existence of some kick follows not only from the measurements of radio pulsar space velocities, but also from the analysis of binary systems with NSs. The impact of a kick velocity ∼ 100 km s−1 explains the precessing pulsar binary orbit in PSR J0045−7319 [345]. The evidence of the kick velocity is seen in the inclined, with respect to the orbital plane, circumstellar disk around the Be star SS 2883 — an optical component of a binary PSR B1259−63 [617]. Evidence for ∼ 150 km s−1 natal kicks has also been inferred from the statistics of the observed short GRB distributions relative to their host galaxies [31].

Long-term pulse profile changes interpreted as geodetic precession are observed in the relativistic pulsar binaries PSR 1913+16 [831], PSR B1534+12 [734], PSR J1141−6545 [289], and PSR J0737−3039B [78]. These observations indicate that in order to produce the misalignment between the orbital angular momentum and the neutron star spin, a component of the kick velocity perpendicular to the orbital plane is required [835, 839, 840]. This idea seems to gain observational support from recent thorough polarization measurements [324] suggesting alignment of the rotational axes with the pulsar’s space velocity. Detailed discussion of the spin-velocity alignment in young pulsars and implications for the SN kick mechanisms can be found in Noutsos et al. [538]. Such an alignment acquired at birth may indicate the kick velocity directed preferably along the rotation of the proto-NS. For the first SN explosion in a close binary system this would imply that the kick is mostly perpendicular to the orbital plane. Implications of this effect for the formation and coalescence rates of NS binaries were discussed by Kuranov et al. [396].

It is worth noting that the analysis of the formation of the relativistic pulsar binaryPSR J0737−3039 [595] may suggest, from the observed low eccentricity of the system e ≃ 0.09, that a small (if any) kick velocity may be acquired if the formation of the second NS in the system is associated with the collapse of an ONeMg WD due to electron captures. The symmetric nature of electron-capture supernovae was discussed in [597] and seems to be an interesting issue requiring further studies (see, e.g., [582, 397] for the analysis of the formation of NSs in globular clusters in the frame of this hypothesis). Note that electron-capture SNe are expected to be weak events, irrespective of whether they are associated with the core collapse of a star that retained some original envelope or with the AIC of a WD [644, 366, 146]. In the case of AIC, rapid rotation of a collapsing object along with flux freezing and dynamo action can grow the WD’s magnetic field to magnetar strengths during collapse. Further, magnetar generates outflow of the matter and formation of a pulsar wind nebula, which may be observed as a radio-source for a few month [592].

We also note the hypothesis of Pfahl et al. [583], based on observations of high-mass X-ray binaries with long orbital periods (≳ 30 d) and low eccentricities (e < 0.2), that rapidly rotating precollapse cores may produce neutron stars with relatively small kicks, and vice versa for slowly rotating cores. Then, large kicks would be a feature of stars that retained deep convective envelopes long enough to allow a strong magnetic torque, generated by differential rotation between the core and the envelope, to spin the core down to the very slow rotation rate of the envelope. A low kick velocity imparted to the second (younger) neutron star (< 50 km s−1) was inferred from the analysis of large-eccentricity pulsar binary PSR J1811−1736 [119]. The large orbital period of this pulsar binary (18.8 d) may then suggest an evolutionary scenario with inefficient (if any) common envelope stage [148], i.e., the absence of a deep convective shell in the supernova progenitor (a He-star). This conclusion can be regarded as supportive to ideas put forward by Pfahl et al. [583]. A careful investigation of radio profiles of double PSR J0737−3039A/B [198] and Fermi detection of gamma-ray emission from the recycled 22-ms pulsar [254] imply its spin axis to be almost aligned with the orbital angular momentum, which lends further credence to the hypothesis that the second supernova explosion in this system was very symmetric.

Small kicks in the case of e-capture in the O-Ne core leading to NS formation are justified by hydrodynamical considerations. Indeed, already in 1996, Burrows and Hayes [80] noted that large scale convective motions in O and Si burning stages preceding the formation of a Fe core may produce inhomogeneities in the envelope of the protostellar core. They in turn may result in asymmetric neutrino transport, which impart kicks up to 500 km s−1. Such a violent burning does not precede the formation of O-Ne cores and then small or even zero kicks can be expected.

In the case of asymmetric core-collapse supernova explosion, it is natural to expect some kick during BH formation as well [430, 214, 607, 609, 518, 34, 872]. The similarity of NS and BH distribution in the Galaxy suggesting BH kicks was noted in [327]. Evidence for a moderate (100–200 km s−1) BH kick has been derived from the kinematics of several BH X-ray transients (microquasars): XTE J1118+180 [208], GRO 1655−40 [838], MAXI J1659−152 [400]. However, no kicks or only small ones seem to be required to explain the formation of other BH candidates, such as Cyg X-1 [485], [845], X-Nova Sco [514], V404 Cyg [484]. Population synthesis modeling of the Galactic distribution of BH binaries supports the need for (possibly, bimodal) natal BH kicks [634]. Janka [321] argued that the similarity of BH kick distribution with NS kick distributions as inferred from the analysis by Repetto et al. [634] favors BH kicks being due to gravitational interaction with asymmetric mass ejection (the “gravitational tug-boat mechanism”), and disfavors neutrino-induced kicks (in the last case, by momentum conservation, BH kicks are expected to be reduced by the NS to the BH mass ratio relative to the NS kicks). Facing current uncertainties in SN explosions and BH formation mechanisms, it is not excluded that low-kick BHs can be formed without associated SN explosions due to neutrino asymmetry, while high-velocity Galactic BHs in LMXBs analyzed by Repetto et al. [634] can be formed by the gravitational tug-boat mechanism suggested by Janka [321].

To summarize, the kick velocity remains to be one of the important unknown parameters of binary evolution with NSs and BHs, and further phenomenological input here is of great importance. Large kick velocities will significantly affect the spatial distribution of coalescing compact binaries (e.g., [352]) and BH kicks are extremely important for BH spin misalignment in coalescing BH-BH binary systems (e.g., Gerosa et al. [237] and references therein). Further constraining this parameter with various observations and a theoretical understanding of the possible asymmetry of core-collapse supernovae seem to be of paramount importance for the formation and evolution of close compact binaries.

3.5.1 Effect of the kick velocity on the evolution of a binary system

The collapse of a star to a BH, or its explosion leading to the formation of a NS, are normally considered as instantaneous. This assumption is well justified in binary systems, since typical orbital velocities before the explosion do not exceed a few hundred km/s, while most of the mass is expelled with velocities of about several thousand km/s. The exploding star M1 leaves the remnant Mc, and the binary loses a part of its mass: ΔM = M1Mc. The relative velocity of stars before the event is

$${V_{\rm{i}}} = \sqrt {G({M_1} + {M_2})/{a_{\rm{i}}}} {.}$$

Right after the event, the relative velocity is

$${\vec V_{\rm{f}}} = {\vec V_{\rm{i}}} + \vec w{.}$$

Depending on the direction of the kick velocity vector \(\vec w\), the absolute value of \({\vec V_{\rm{f}}}\) varies in the interval from the smallest Vf = ∣Viw∣ to the largest Vf = Vi + w. The system gets disrupted if Vf satisfies the condition (see Section 3.4)

$${V_{\rm{f}}} \geq {V_{\rm{i}}}\sqrt {{2 \over \chi}} ,$$

where χ ≡ (M1 + M2)/(Mc + M2).

Let us start from the limiting case when the mass loss is practically zero (ΔM = 0, χ = 1), while a non-zero kick velocity can still be present. This situation can be relevant to BH formation. It follows from Eq. (54) that, for relatively small kicks, \(w < (\sqrt 2 - 1){V_{\rm{i}}}\), the system always (independent of the direction of \(\vec w\)) remains bound, while for \(w > (\sqrt 2 + 1){V_{\rm{i}}}\) the system always decays. By averaging over equally probable orientations of \(\vec w\) with a fixed amplitude w, one can show that in the particular case w = Vi the system disrupts or survives with equal probabilities. If Vf < Vi, the semi-major axis of the system becomes smaller than the original binary separation, af < ai (see Eq. (46)). This means that the system becomes harder than before, i.e. it has a greater negative total energy than the original binary. If \({V_i} < {V_{\rm{f}}} < \sqrt 2 {V_i}\), the system remains bound, but af > ai. For small and moderate kicks wVi, the probabilities for the system to become more or less bound are approximately equal.

In general, the binary system loses some fraction of its mass ΔM. In the absence of the kick, the system remains bound if ΔM < M/2 and gets disrupted if ΔMM/2 (see Section 3.4). Clearly, a “properly” oriented kick velocity (directed against the vector \({\vec V_{\rm{i}}}\)) can keep the system bound, even if it would have been disrupted without the kick. And, on the other hand, an “unfortunate” direction of \(\vec w\) can disrupt the system, which otherwise would stay bound.

Consider, first, the case ΔM < M/2. The parameter χ varies in the interval from 1 to 2, and the escape velocity Ve varies in the interval from \(\sqrt {2{V_i}}\) to Vi. It follows from Eq. (50) that the binary always remains bound if w < VeVi, and always unbinds if w > Ve + Vi. This is a generalization of the formulas derived above for the limiting case ΔM = 0. Obviously, for a given w, the probability for the system to disrupt or become softer increases when ΔM becomes larger. Now turn to the case ΔM > M/2. The escape velocity of the compact star becomes Ve < Vi. The binary is always disrupted if the kick velocity is too large or too small: w > Vi +Ve or w < Vi − Ve. However, for all intermediate values of w, the system can remain bound, and sometimes even more hard than before, if the direction of \(\vec w\) happened to be approximately opposite to \({\vec V_{\rm{i}}}\). A detailed calculation of the probabilities for the binary survival or disruption requires integration over the kick velocity distribution function \(f(\vec w)\) (see, e.g., [68]).

3.6 Common envelope stage

3.6.1 Formation of the common envelope

Common envelopes (CE) are definitely the most important (and not as yet solved) problem in the evolution of close binaries. In the theory of stellar evolution, it was recognized quite early that there are several situations when formation of an envelope that engulfs an entire system seems to be inevitable. It can happen when the mass transfer rate from the mass-losing star is so high that the companion cannot accommodate all the accreting matter [41, 868, 631, 523, 823, 614, 590]. Another instance is encountered when it is impossible to keep synchronous rotation of a red giant and orbital revolution of a compact companion [730, 8]. Because of tidal drag forces, the revolution period decreases, while the rotation period increases in order to reach synchronism (Darwin instability). If the total orbital momentum of the binary Jorb < 3JRG, the synchronism cannot be reached and the companion (low-mass star, white dwarf or neutron star) spirals into the envelope of the red giant. Yet another situation is the formation of an extended envelope, which enshrouds the system due to unstable nuclear burning at the surface of an accreting WD in a compact binary [738, 559, 534]. It is also possible that a compact remnant of a supernova explosion with “appropriately” directed kick velocity finds itself in an elliptic orbit whose minimum periastron distance af (1 − e) is smaller than the radius of the companion.

The common-envelope stage appears to be unavoidable on observational grounds. The evidence for a dramatic orbital angular momentum decrease in the course of evolution follows immediately from observations of cataclysmic variables, in which a white dwarf accretes matter from a small red dwarf main-sequence companion, close binary nuclei of planetary nebulae, low-mass X-ray binaries, and X-ray transients (neutron stars and black holes accreting matter from low-mass main-sequence dwarfs). At present, the typical separation of components in these systems is ∼ R, while the formation of compact stars requires progenitors with radii ∼ (100–1000) R.Footnote 17 Indirect evidence for the common envelope stage comes, for instance, from X-ray and FUV observations of a prototypical pre-cataclysmic binary V471 Tau showing anomalous C/N contamination of the K-dwarf companion to white dwarf [164, 714].

Let us consider the following mass-radius exponents that describe the response of a star to the mass loss in a binary system:

$${\zeta _L} = \left({{{\partial \ln {R_1}} \over {\partial \ln {M_1}}}} \right),\quad \quad {\zeta _{{\rm{th}}}} = {\left({{{\partial \ln {R_L}} \over {\partial \ln {M_1}}}} \right)_{{\rm{th}}}},\quad \quad {\zeta _{{\rm{ad}}}} = {\left({{{\partial \ln {R_L}} \over {\partial \ln {M_1}}}} \right)_{{\rm{ad}}}},$$

where ζL is the response of the Roche lobe to the mass loss, ζth — the thermal-equilibrium response, and ζad — the adiabatic hydrostatic response. If ζad > ζL > ζth, the star retains hydrostatic equilibrium, but does not remain in thermal equilibrium and mass loss occurs in the thermal timescale of the star. If ζL > ζad, the star cannot retain hydrostatic equilibrium and mass loss proceeds in the dynamical time scale [558]. If both ζad and ζth exceed ζL, mass loss occurs due to expansion of the star caused by nuclear burning or due to the shrinkage of the Roche lobe owed to the angular momentum loss.

A high rate of mass overflow onto a compact star from a normal star is always expected when the normal star goes off the main sequence and develops a deep convective envelope. The physical reason for this is that convection tends to make entropy constant along the radius, so the radial structure of convective stellar envelopes is well described by a polytrope (i.e., the equation of state can be written as P = 1+1/n) with an index n = 3/2. The polytropic approximation with n = 3/2 is also valid for degenerate white dwarfs with masses not too close to the Chandrasekhar limit. For a star in hydrostatic equilibrium, this results in the inverse mass-radius relation, RM−1/3, first found for white dwarfs. Removing mass from a star with a negative power of the mass-radius relation increases its radius. On the other hand, the Roche lobe of the more massive star should shrink in response to the conservative mass exchange between the components. This further increases the mass loss rate from the Roche-lobe filling star leading to dynamical mass loss and eventual formation of a common envelope. If the star is completely convective or completely degenerate, dynamically-unstable mass loss occurs if the mass ratio of components (donor to accretor) is ≳ 2/3.

In a more realistic case when the star is not completely convective and has a radiative core, certain insight may be gained by the analyses of composite polytropic models. Conditions for the onset of dynamical mass loss become less rigorous [555, 283, 722]: the contraction of a star replaces expansion if the relative mass of the core mc > 0.214 and in a binary with a mass ratio of components close to 1, a Roche-lobe filling star with a deep convective envelope may remain dynamically stable if mc > 0.458. As well, a stabilizing effect upon mass loss may have mass and momentum loss from the system, if it happens in a mode that results in an increase of the specific angular momentum of the binary (e.g., in the case of isotropic reemission by a more massive component of the system in CVs or UCXBs).

Criteria for thermal or dynamical mass loss upon RLOF and the formation of a common envelope need systematic exploration of the response of stars to mass removal in different evolutionary stages and at different rates. The response also depends on the mass of the star. While computational methods for such an analyses are elaborated [224, 225], calculations of respective grids of models with full-fledged evolutionary codes at the time of writing of this review were not completed.

3.6.2 “Alpha”-formalism

Formation of the common envelope and evolution of a binary inside the former is a 3D hydrodynamic process that may also include nuclear reactions; and current understanding of the process as a whole, as well as computer power, are not sufficient for a solution of the problem (see, for detailed discussion, [319]). Therefore, the outcome of this stage is, most commonly, evaluated in a simplified way, based on the balance of the binding energy of the stellar envelope and the binary orbital energy, following independent suggestions of van den Heuvel [794], Tutukov and Yungelson [783], and Webbink [825] and commonly named after the later author as “Webbink’s” or “α”-formalism. The orbital evolution of the compact star m inside the envelope of the donor-star M1 is driven by dynamic friction drag [553]. This leads to a gradual spiral-in process of the compact star. The above-mentioned energy condition may be written as

$${{G{M_1}({M_1} - {M_c})} \over {\lambda {R_L}}} = {\alpha _{{\rm{ce}}}}\left({{{Gm{M_c}} \over {2{a_{\rm{f}}}}} - {{G{M_1}m} \over {2{a_{\rm{i}}}}}} \right),$$

where ai and af are the initial and final orbital separations, αce is the common envelope parameter [783, 438] that describes the efficiency of expenditure of orbital energy on expulsion of the envelope and λ is a numerical coefficient that depends on the structure of the donor’s envelope, introduced by de Kool et al. [138], and RL is the Roche lobe radius of the normal star that can be approximated, e.g., by Eqs. (34) or (35). From Eq. (56) one derives

$${{{a_{\rm{f}}}} \over {{a_{\rm{i}}}}} = {{{M_c}} \over {{M_1}}}{\left({1 + {{2{a_{\rm{i}}}} \over {\lambda {\alpha _{{\rm{CE}}}}{R_{\rm{L}}}}}{{{M_1} - {M_c}} \over m}} \right)^{- 1}}\lesssim{{{M_c}} \over {{M_1}}}{m \over {\Delta M}},$$

where ΔM = M1Mc is the mass of the ejected envelope. For instance, the mass Mc of the helium core of a massive star can be approximated as [782]

$${M_{{\rm{He}}}} \approx 0{.}1{({M_1}/{M_ \odot})^{1{.}4}}{.}$$

Then, ΔMM and, e.g., in the case of mM the orbital separation during the common envelope stage can decrease by as much as a factor of 40 or even more.

This treatment does not take into account possible transformations of the binary components during the CE-stage. In addition, the pre-CE evolution of the binary components may be important for the onset and the outcome of the CE, as calculations of synchronization of red giant stars in close binaries carried out by Bear and Soker [29] indicate. The outcome of the common envelope stage is considered as a merger of components if af is such that the donor core comes into contact with the companion. Otherwise, it is assumed that the core and companion form a detached system with the orbital separation af. Note an important issue raised recently by Kashi and Soker [341]: while Eq. (57) can formally imply that the system becomes detached at the end of the CE stage, in fact, some matter of the envelope can not reach the escape velocity and remains bound to the system and forms a circumbinary disk. Angular momentum loss due to interaction with the disk may result in a further reduction of binary separation and merger of components. This may influence the features of such stellar populations as a close WD binary, hot subdwarfs, cataclysmic variables) and the rate of SN Ia (see Section 7).

In the above equation for the common envelope, the outcome of the CE stage depends on the product of two parameters: λ, which is the measure of the binding energy of the envelope to the core prior to the mass transfer in a binary system, and αCE, which is the common envelope efficiency itself. Evaluation of both parameters suffers from large physical uncertainties. For example, for λ, the most debatable issues are the accounting of the internal energy in the binding energy of the envelope and the definition of the core/envelope boundary itself. Some authors argue (see e.g., [315]) that enthalpy rather than internal energy should be included in the calculation of λ, which seems physically justifiable for convective envelopesFootnote 18. We refer the interested reader to the detailed discussion in [139, 319].

There exist several sets of fitting formulas for λ [856, 855] or binding energy of the envelopes [446] based on detailed evolutionary computations for a range of stellar models at different evolutionary stages. But we note, that in both studies the same specific evolutionary code ev [601] was used and the core/envelope interface at location in the star with hydrogen abundance in the hydrogen-burning shell X = 0.1 was assumed; in our opinion, the latter assumption is not justified neither by any physical assumptions nor by evolutionary computations for Roche-lobe filling stars. Even if the inaccuracy of the latter assumption is neglected, it is possible to use these formulas only in population synthesis codes based on the evolutionary tracks obtained by ev code.

Our test calculations do not confirm the recent claim [313] that core/envelope interface, which also defines the masses of the remnants in all cases of mass-exchange, is close to the radius of the sonic velocity maximum in the hydrogen-burning shell of the mass-losing star at RLOF. As concerns parameter αCE, the most critical issue is whether the sources other than orbital energy can contribute to the energy balance, which include, in particular, recombination energy [724], nuclear energy [299], or energy released by accretion onto the compact star. It should then be noted that sometimes imposed to Eq. (56) restriction αCE ≤ 1 can not reflect the whole complexity of the processes occurring in common envelopes. Although full-scale hydrodynamic calculations of a common-envelope evolution exist, (see, e.g., [747, 140, 636, 637, 571] and references therein), we stress again that the process is still very far from comprehension, especially, at the final stages.

Actually, the state of the common envelope problem is currently such that it is possible only to estimate the product αCEλ by modeling specific systems or well defined samples of objects corrected for observational selection effects, like it is done in [765].

3.6.3 “Gamma”-formalism

Nelemans et. al. [517] noted that the α-formalism failed to reproduce parameters of several close white dwarf binaries known circa 2000. In particular, for close helium white dwarf binaries with known masses of both components, one can reconstruct the evolution of the system “back” to the pair of main-sequence progenitors of components, since there is a unique relationship between the mass of a white dwarf and the radius of its red giant progenitor [631], which is almost independent of the total mass of the star. Formation of close white dwarf binaries should definitely involve a spiral-in phase in the common envelope during the second episode of mass loss (i.e., from the red giant to the white dwarf remnant of the original primary in the system). In observed systems, mass ratios of components tend to concentrate to Mbright/Mdim ≃ 1. This means, that, if the first stage of mass transfer occurred through a common envelope, the separation of components did not reduce much, contrary to what is expected from Eq. (57). The values of αCEλ, which appeared necessary for reproducing the observed systems, turned out to be negative, which means that the simple energy conservation law (57) is violated in this case. Instead, Nelemans et al. suggested that the first stage of mass exchange, between a giant and a main-sequence star with comparable masses (q ≥ 0.5), can be described by what they called “γ-formalism”, in which not the energy but the angular momentum J is balanced and conservation of energy is implicitly implied, though formally this requires αCEλ > 1:

$${{\delta J} \over J} = \gamma {{\Delta M} \over {{M_{{\rm{tot}}}}}}{.}$$

Here ΔM is the mass lost by the donor, Mtot is the total mass of the binary system before formation of the common envelope, and γ is a numerical coefficient. Similar conclusions were later reached by Nelemans and Tout [516] and van der Sluys et al. [803] who analyzed larger samples of binaries and used detailed fits to evolutionary models instead of simple “core-mass — stellar radius” relation applied by Nelemans et al. [517]. It turned out that combination of γ = 1.75 for the first stage of mass exchange with αCEλ = 2 for the second episode of mass exchange enables, after taking into account selection effects, a satisfactory model for population of close binary WD with known masses. In fact, findings of Nelemans et al. and van der Sluys et al. confirmed the need for a loss of mass and momentum from the system to explain observations of binaries mentioned in Section 3.3. (Note that γ = 1 corresponds to the loss of the angular momentum by a fast stellar wind, which always increases the orbital separation of the binary.) We stress that no physical process behind γ-formalism has been suggested as yet, and the model remains purely phenomenological and should be further investigated.Footnote 19

We should note that “γ-formalism” was introduced under the assumption that stars with deep convective envelopes (giants) always lose mass unstably, i.e., is high. As mentioned above, if the mass exchange is nonconservative, the associated angular momentum loss can increase the specific orbital angular momentum of the system \(\propto \sqrt a\) and hence the orbital separation a. Then the increasing Roche lobe can accommodate the expanding donor, and the formation of a common envelope can be avoided. In fact, this happens if part of the energy released by accretion is used to expel the matter from the vicinity of the accretor (see, e.g., computations presented by King and Ritter [365], Beer et al. [30], Woods et al. [849] who assumed that reemission occurs). It is not excluded that such nonconservative mass-exchange with mass and momentum loss from the system may be the process underlying the “γ-formalism”.

Nelemans and Tout [516], Zorotovic et al. [886] and De Marco et al. [139] attempted to estimate αCE using samples of close WD + M-star binaries. It turned out that their formation can be explained both if αCEλ > 0 in Eq. (56) or if γ ≈ 1.5 in Eq. (59). This is sometimes considered as an argument against the γ-formalism. But we note that in progenitors of WD + M-star binaries the companions to giants are low-mass small-radius objects, which are quite different from M ≥ M companions to WD in progenitors of WD + WD binaries and, therefore, the energetics of the evolution in CE in precursors of two types of systems can differ. In a recent population synthesis study of the post-common-envelope binaries aimed at comparing to SDSS stars and taking into account selection effects, Toonen and Nelemans [765] concluded that the best fit to the observations can be obtained for small universal αCEλ = 0.25 without the need to invoke the γ-formalism. However, they also note that for almost equal mass precursors of WD binaries the widening of the orbit in the course of mass transfer is needed (like in γ-formalism), while the formation of low mass-ratio WD+M-star systems requires diminishing of the orbital separation (as suggested by α-formalism).

3.7 Other notes on the CE problem

A particular case of common envelopes (“double spiral-in”) occurs when both components are evolved. It was suggested [520] that a common envelope is formed by the envelopes of both components of the system and the binding energy of them both should be taken into account, i.e., the outcome of CE must be found from a solution of an equation

$${E_{{\rm{bind}},1}} + {E_{{\rm{bind}},2}} = {\alpha _{{\rm{CE}}}}({E_{{\rm{orb}},i}} - {E_{{\rm{orb}},f}}){.}$$

Formulations of the common envelope equation different from Eq. (56) are found in the literature (see, e.g., [139] for a review); af/ai similar to the values produced by Eq. (56) are then obtained for different αceλ values.

Common envelope events are expected to be rare (≲ 0.1 yr−1 [791]) and short-lasting (∼ years). Thus, the binaries at the CE-stage itself are difficult to observe, despite energy being released at this stage comparable to the binding energy of a star, and the evidence for them comes from the very existence of compact object binaries, as described above in Section 3.6.1. Recently, it was suggested that luminous red transients — the objects with peak luminosity intermediate between the brightest Novae and SN Ia (see, e.g., [342]) — can be associated with CE [316, 792].

We recall also that the stability and timescale of mass-exchange in a binary depends on the mass ratio of components q, the structure of the envelope of the Roche-lobe filling star, and possible stabilizing effects of mass and momentum loss from the system [780, 283, 873, 261, 268, 318, 196, 76, 849]. We note especially recent claims [848, 570] that the local thermal timescale of the super-adiabatic layer existing over a convective envelope of giants may be shorter than the donor’s dynamic timescale. As a result, giants may adjust to a high mass loss and retain their radii or even contract (instead of dramatically expanding, as expected in the adiabatic approximation [224, 225]). As well, the response of stars to mass loss evolves in the course of the latter depending on whether the super-adiabatic layer may be removed. The recent discovery of young helium WD companions to blue stragglers in wide binaries (Porb from 120 to 3010 days) in open cluster NGC 188 [248] supports the idea that in certain cases RLOF in wide systems may be stable. These considerations, which need further systematic exploration, may change the “standard” paradigm of the stability of mass exchange formulated in this section.

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 M1 ≳ 8 M. 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]Footnote 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”.

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 (e = 0.3–0.9) and Porb ≥ 30 day and objects with e < 0.2 and Porb ≤ 30 day. 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 e < 0.27 and the measured or estimated masses of second-born neutron stars in most of these systems are close to 1.25 M (for recent confirmation of these observations see, e.g., [685, 369, 799, 198]). A neutron star would have 1.25 M mass if it is a product of an electron-capture supernova (ECSN), which was accompanied by the loss of binding energy equivalent to ≈ 0.2 M. 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 σ = (20–30) km s−1 is inferred instead of σ ∼ 200 km s−1).

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.

Figure 7
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.

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. 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 M1 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 tnuclM−2). The star burns out hydrogen in its central parts, so that a dense central helium core with mass MHe ≃ 0.1 (M/M)1.4 forms by the time the star leaves the main sequence. The expected number of such binaries in the Galaxy is about 104.

  2. 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 \({\tau _{{\rm{KH}}}} \approx GM_1^2/{R_1}{L_1}\), 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. 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 (7–8) M [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. 4.

    Stars more massive than ≃ 8 M end their evolution by forming a NS. ZAMS mass range 8–12(±1) M, 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 (0.76 ± 0.16) × 10−2 yr−1 [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].

    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. 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. (4651). 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. 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 7 M 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 ∼ 0.1 M yr−1, 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. 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.

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

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 M1 ≲ (40–50) M [470, 807].

There also exists a population of NSs accompanied by low-mass [∼ (1–2) M] 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 Porb and eccentricities e. In the upper panel of Figure 8 we show, as an example, a model of the probability distribution of Galactic NS + NS binaries with Porb ≤ 104 day in Porbe 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 Porb ≤ 1 day 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
figure 8

Upper panel: the probability distribution for the orbital parameters of the NS + NS binaries with Porb ≤ 104 day at the moment of birth. The darkest shade corresponds to a birthrate of 1.2 × 10−5 yr−1. 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 Porb and e.

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 Mcr 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] − 40 M; Woosley et al. [852] − 60 M; Portegies Zwart, Verbunt, and Ergma [606], Ergma & van den Heuvel [176], Brown et al. [70] − 20 M. A simple physical argument usually put forward in the literature is that the mantle of the main-sequence star with M > Mcr ≈ 30 M 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 ZZ the mass-range of precursors of black holes is constrained to M ≈ (25–60) M, 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 MBH 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 ∼ (1.6–2.5) M, 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 — (4–17) M [542, 475, 633]. A continuous range of BH masses up to 10–15 M 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 (16.5 M < M < 25 M) as the progenitors of Type IIP supernovae may indicate BH formation starting from progenitor masses as low as 16 M.

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 (16–30 M) 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 Mbh by the fraction of the pre-supernova mass M* that collapses into the BH: kBH = MBH/M*. In fact, the pre-supernova mass M* is directly related to Mcr, 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 \(M_{{\rm{BH}}}^{\min} = {k_{{\rm{BH}}}}{M_\ast}\), where M* itself depends on Mcr. The parameter kBH can vary in a wide range.

The third parameter, similar to the case of NS formation, is the possible kick velocity wBH 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

$${{{w_{{\rm{BH}}}}} \over {{w_{{\rm{NS}}}}}} = {{{M_ \ast} - {M_{{\rm{BH}}}}} \over {{M_ \ast} - {M_{{\rm{OV}}}}}} = {{1 - {k_{{\rm{BH}}}}} \over {1 - {M_{{\rm{OV}}}}/{M_ \ast}}},$$

where MOV = 2.5 M is the maximum NS mass. When MBH is close to MOV, the ratio wBH/wNS approaches 1, and the low-mass black holes acquire kick velocities similar to those of neutron stars. When MBH is significantly larger than MOV, the parameter kBH = 1, and the BH kick velocity becomes vanishingly small. The allowance for a quite moderate wBH 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 J) 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 cos J < 0. 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].

5 Formation of Double Compact Binaries

5.1 Analytical estimates

A rough estimate of the formation rate of compact binaries can be obtained ignoring many details of binary evolution. To do this, we shall use the observed initial distribution of binary orbital parameters and assume the simplest conservative mass transfer (M1 + M2 = const) without kick velocity imparted to the nascent compact stellar remnants during SN explosions.

5.1.1 Initial binary distributions

From observations of binaries it is possible to derive their formation rate as a function of initial masses of components M1, M2 (with mass ratio q = M2/M1 1), orbital semi-major axis A, and eccentricity e. According to [604, 811], the present birth rate of binaries in our Galaxy can be written in factorized form as

$${{dN} \over {dA d{M_1}dq dt}} \approx 0{.}087{\left({{A \over {{R_ \odot}}}} \right)^{- 1}}{\left({{{{M_1}} \over {{M_ \odot}}}} \right)^{- 2{.}5}}f(q),$$

where f(q) is a poorly constrained distribution over the initial mass ratio of binary components.

One usually assumes a mass ratio distribution law in the form \(f(q) \sim {q^{- \alpha q}}\) where αq is a parameter, also derived observationally, see Section 1. Another often used form of the q-distribution was suggested by Kuiper [391]:

$$f(q) = 2/{(1 + q)^2}{.}$$

The range of A is 10 ≤ A/R 106. In deriving the above Eq. (62), Popova et al. [604] took into account selection effects to convert the “observed” distribution of stars into the true one. An almost flat logarithmic distribution of semimajor axes was also found in [5].

Taking Eq. (62) at face value, assuming 100% binarity, the mass range of the primary components M1 = 0.08 M to 100 M, a flat distribution over semimajor axes (contact at ZAMS) ≤ log(A/R) ≤ (border between close and wide binaries), f(q) = 1 for close binaries, and f(q) ∝ q−2.5 for 0.3 ≤ q ≤ 1 and f(q) = 2.14 for q < 0.3 for wide binaries with log(A/R) 6, as accepted in Tutukov and Yungelson’s BPS code IBiS [791], we get SFR ≈ 8M per yr, which is several times higher than modern estimates of the current Galactic SFR. However, if 8 M per year is used as a constant average SFR for 13.5 Gyr, we get the right mass of the Galactic disc. Clearly, Eq. (62) is rather approximate, since most of the stellar mass resides in low-mass stars for which IMF, f(q), f(A), binary fraction, etc., are poorly known. However, if we consider only solar chemical composition stars with M1 > 0.95 M (which can evolve off the main-sequence in the Hubble time), we get, under the “standard” assumptions in the IBiS-code, e.g., a WD formation rate of 0.65 per yr, which is reasonably consistent with observational estimates (see Liebert et al. [420]), the SN II + SN Ib/c rate about 1.5/100 per yr, which is consistent with the inferred Galactic rate [86]) or with the pulsar formation rate [816]. We also get the “proper” rate of WD + WD mergers with Superchandrasekhar total mass for SN Ia (a few per thousand years).

5.1.2 Constraints from conservative evolution

For this estimate we shall assume that the primary mass should be at least 10 M. Equation (62) tells us that the formation rate of such binaries is about 1 per 50 years. We shall restrict ourselves by considering only close binaries, in which mass transfer onto the secondary is possible. This narrows the binary separation interval to 10–1000 R (see Figure 5); the birth rate of close massive (M1 > 10 M) binaries is thus 1/50 × 2/5 yr−1 = 1/125 yr−1. The mass ratio q should not be very small to make the formation of the second NS possible. The lower limit for q is derived from the condition that after the first mass transfer stage, during which the mass of the secondary increases, M2 + ΔM ≥ 10 M. Here ΔM = M1MHe and the mass of the helium core left after the first mass transfer is MHe ≈ 0.1(M1/M)1.4. This yields

$${m_2} + ({m_1} - 0{.}1m_1^{1{.}4}) > 10,$$

where we used the notation m = M/M, or in terms of q:

$$q \geq 10/{m_1} + 0{.}1m_1^{0{.}4} - 1{.}$$

An upper limit for the mass ratio is obtained from the requirement that the binary system remains bound after the sudden mass loss in the second supernova explosion.Footnote 21 From Eq. (51) we obtain

$${{0{.}1{{[{m_2} + ({m_1} - 0{.}1m_1^{1{.}4})]}^{1{.}4}} - 1{.}4} \over {2{.}8}} < 1,$$

or in terms of q:

$$q \leq 14{.}4/{m_1} + 0{.}1m_1^{0{.}4} - 1{.}$$

Inserting m1 = 10 in the above two equations yields the appropriate mass ratio range 0.25 < q < 0.69, i.e., 20% of the binaries for Kuiper’s mass-ratio distribution. So we conclude that the birth rate of binaries that can potentially produce a NS binary system is ≲ 0.2 × 1/125 yr−1 ≃ 1/600 yr−1.

Of course, this is a very crude upper limit — we have not taken into account the evolution of the binary separation, ignored initial binary eccentricities, non-conservative mass loss, etc. However, it is not easy to treat all these factors without additional knowledge of numerous details and parameters of binary evolution (such as the physical state of the star at the moment of the Roche lobe overflow, the common envelope efficiency, etc.). All these factors should decrease the formation rate of NS binaries. The coalescence rate of compact binaries (which is ultimately of interest to us) will be even smaller — for the compact binary to merge within the Hubble time, the binary separation after the second supernova explosion should be less than ∼ 100 R (orbital periods shorter than ∼ 40 d) for arbitrary high orbital eccentricity e (see Figure 3). The model-dependent distribution of NS kick velocities provides another strong complication. We also stress that this upper limit was obtained assuming constant Galactic star-formation rate and a normalization of the binary formation by Eq. (62).

Further (semi-)analytical investigations of the parameter space of binaries leading to the formation of coalescing NS binaries are still possible but technically very difficult, and we shall not reproduce them here. The detailed semi-analytical approach to the problem of the formation of NSs in binaries and the evolution of compact binaries has been developed by Tutukov and Yungelson [787, 788].

5.2 Population synthesis results

A distinct approach to the analysis of stellar binary evolution is based on the population synthesis method — a Monte Carlo simulation of the evolution of a sample of binaries with different initial parameters. This approach was first applied to model various observational manifestations of magnetized NSs in massive binary systems [377, 378, 147] and generalized to binary systems of arbitrary mass in [428] (The Scenario Machine code). To achieve a sufficient statistical significance, such simulations usually involve a large number of binaries, typically on the order of a million. The total number of stars in the Galaxy is still four orders of magnitude larger, so this approach cannot guarantee that rare stages of the binary evolution will be adequately reproduced. Footnote 22

Presently, there are several population synthesis codes used for massive binary system studies, which take into account with different degrees of completeness various aspects of stellar binary evolution (e.g., the codes by Portegies Zwart, Nelemans et al. [607, 872], Bethe and Brown [44], Hurley, Tout, and Pols [294], Belczynski et al. [35], Yungelson and Tutukov [791], De Donder and Vanbeveren [133]). A review of applications of the population synthesis method to various types of astrophysical sources and further references can be found in [602, 869]. Some results of population-synthesis calculations of compact-binary mergers carried out by different groups are presented in Table 6.

Table 6 Examples of the estimates for Galactic merger rates of relativistic binaries calculated under different assumptions on the parameters entering population synthesis.

Actually, the authors of the studies mentioned in Table 6 make their simulations for a range of parameters. We list in the table the rates for the models the authors themselves consider as “standard” or “preferred” or “most probable”, calculated for solar metallicity (or give the ranges). Generally, for the NS + NS merger rate Table 6 shows the scatter within a factor ∼ 4, which may be considered quite reasonable, having in mind the uncertainties in input parameters. There are several outliers, [788], [815], and [479]. The high rate in [788] is due to the assumption that kicks to nascent neutron stars are absent. The low rate in [815] is due to the fact that these authors apply in the common envelope equation an evolutionary-stage-dependent structural constant λ. Their range for λ is 0.006–0.4, to be compared with the “standard” λ = 0.5 applied in most of the other studies. Low λ favors mergers in the first critical lobe overflow episode and later mergers of the first-born neutron stars with their non-relativistic companionsFootnote 23. A considerable scatter in the rates of mergers of systems with BH companions is due, mainly, to uncertainties in stellar wind mass loss for the most massive stars. For instance, the implementation of winds in the code used in [607, 518] resulted in the absence of merging BH + BH systems, while a rather low assumed in [815] produced a high merger rate of BH + BH systems. We note an extreme scatter of the estimates of the merger rate of NS + NS binaries in [157]: the lowest estimate is obtained assuming very tightly bound envelopes of stars (with parameter λ = 0.01), while the upper estimate — assuming completely mass-conservative evolution. The results of the Brussels group [134, 479] differ from StarTrack-code results [157] and other codes in predicting an insignificant BH-BH merging rate. This is basically due to assumed enhanced mass loss in the red supergiant stage (RSG) of massive star evolution. In this scenario, unlike, e.g., Voss and Tauris’ assumptions [815], the allowance for the enhanced mass loss at the Luminous Blue Variable (LBV) phase of evolution for stars with an initial mass ≳ 30–40 M leads to a significant orbital increase and hence the avoidance of the second Roche-lobe overflow and spiral-in process in the common envelope, which completely precludes the formation of close BH binary systems merging within the Hubble time. The Brussels code also takes into account the time evolution of Galactic metallicity enrichment by massive single and binary stars. A more detailed comparison of different population synthesis results of NS + NS, NS + BH and BH + BH formation and merging rates can be found in [2].

A word of caution. It is hardly possible to trace the detailed evolution of each binary, so approximate descriptions of evolutionary tracks of stars, their interaction, effects of supernovae, etc. are invoked. Thus, fundamental uncertainties of stellar evolution mentioned above are complemented by (i) uncertainties of the scenario and (ii) uncertainties in the normalization of the calculations to the real Galaxy (such as the fraction of binaries among all stars, the star formation history, etc.). The intrinsic uncertainties in the population synthesis results (for example, in the computed event rates of binary mergers etc.) are in the best case not less than ≃ (2–3). This should always be borne in mind when using population synthesis calculations. However, we emphasize again the fact that the NS binary merger rate, as inferred from pulsar binary statistics with account for pulsar binary observations [77, 336, 363], is very close to the population syntheses estimates assuming NS kicks of about (250–300) km s−1.

6 Detection Rates

From the point of view of detection, a gravitational-wave signal from merging close binaries is characterized by the signal-to-noise ratio S/N, which depends on the binary masses, the distance to the binary, the frequency, and the detector’s noise characteristics. A pedagogical derivation of the signal-to-noise ratio and its discussion for different detectors is given, for example, in Section 8 of the review [252].

Coalescing binaries emit gravitational wave signals with a well known time-dependence (wave-form) h(t) (see Section 3.1 above). This allows one to use the technique of matched filtering [758]. The signal-to-noise ratio S/N for a particular detector, which is characterized by the noise spectral density Sn(f) [Hz−1/2] or the dimensionless noise rms amplitude hrms at a given frequency f, depends mostly on the “chirp” mass of the binary system = (M1 + M2)−1/5(M1M2)3/5 = μ3/5M2/5 (here μ = M1M2/M is the reduced mass and M = M1 + M2 is the total mass of the system) and its (luminosity) distance r: S/N5/6/r [758, 200]. For a given type of coalescing binary (NS + NS, NS + BH or BH + BH), the signal-to-noise ratio will also depend on the frequency of the innermost stable circular orbit fISCO ∼ 1/M, as well as on the orientation of the binary with respect to the given detector and its angular sensitivity (see, e.g., Section 8 in [252] and [673] for more detail). Therefore, from the point of view of detection of the specific type of coalescing binaries at a prerequisite signal-to-noise ratio level, it is useful to determine the detector’s maximum (or “horizon”) distance Dhor, which is calculated for an optimally oriented (“ideal”) coalescing binary with a given chirp mass . For a secure detection, the S/N ratio is usually raised up to 7–8 to avoid false alarms over a period of a year (assuming Gaussian noise).Footnote 24 This requirement determines the maximum distance from which an event can be detected by a given interferometer [126, 2]. The horizon distance Dhor of LIGO I/VIRGO (advanced LIGO, expected) interferometers for relativistic binary inspirals with account of the actual noise curve attained in the S5 LIGO scientific run are given in [2]: 33(445) Mpc for NS + NS (1.4 M + 1.4 M, NS = 1.22 M), 70(927) Mpc for NS + BH (1.4 M + 10 M), and 161(2187) Mpc for BH + BH (10 M + 10 M, BH = 8.7 M). The distances increase for a network of detectors.

It is worth noting that the dependence of the S/N for different types of coalescing binaries on fISCO changes rather slightly (to within 10%) as S/N5/6, so, for example, the ratio of BH and NS detection horizons scales as S/N, i.e., Dhor,BH/Dhor,NS = (BH/NS)5/6 ≈ 5.14. This allows us to estimate the relative detection ratio for different types of coalescing binaries by a given detector (or a network of detectors). Indeed, at a fixed level of S/N, the detection volume is proportional to \(D_{{\rm{hor}}}^3\) and therefore it is proportional to 5/2. The detection rate \({\mathcal D}\) for binaries of a given class (NS + NS, NS + BH or BH + BH) is the product of their coalescence rate V and the detector’s horizon volume ∝ 5/2 for these binaries.

It is seen from Table 6 that the model Galactic rate G of NS + NS coalescences is typically higher than the rate of NS + BH and BH + BH coalescences. However, the BH mass is significantly larger than the NS mass. So a binary involving one or two black holes, placed at the same distance as a NS + NS binary, produces a significantly larger amplitude of gravitational waves. With the given sensitivity of the detector (fixed S/N ratio), a BH + BH binary can be seen at a greater distance than a NS + NS binary. Hence, the registration volume for such bright binaries is significantly larger than the registration volume for relatively weak binaries. The detection rate of a given detector depends on the interplay between the coalescence rate and the detector’s response to the sources of one or another kind.

If we assign some characteristic (mean) chirp mass to different types of NS and BH binary systems, the expected ratio of their detection rates by a given detector is

$${{{\mathcal{D}_{{\rm{BH}}}}} \over {{\mathcal{D}_{{\rm{NS}}}}}} = {{{\mathcal{R}_{{\rm{BH}}}}} \over {{\mathcal{R}_{{\rm{NS}}}}}}{\left({{{{\mathcal{M}_{{\rm{BH}}}}} \over {{\mathcal{M}_{{\rm{NS}}}}}}} \right)^{5/2}},$$

where \({{\mathcal D}_{{\rm{BH}}}}\) and \({{\mathcal D}_{{\rm{NS}}}}\) refer to BH + BH and NS + NS pairs, respectively. Taking BH = 8.7 M (for 10 M + 10 M) and NS = 1.22 M (for 1.4 M + 1.4 M), Eq. (65) yields

$${{{\mathcal{D}_{{\rm{BH}}}}} \over {{\mathcal{D}_{{\rm{NS}}}}}} \approx 140{{{\mathcal{R}_{{\rm{BH}}}}} \over {{\mathcal{R}_{{\rm{NS}}}}}}{.}$$

As \({{{{\mathcal R}_{{\rm{BH}}}}} \over {{{\mathcal R}_{{\rm{NS}}}}}}\) is typically 0.1−0.01 (see Table 6), this relation suggests that the registration rate of BH mergers can be higher than that of NS mergers. Of course, this estimate is very rough, but it can serve as an indication of what one can expect from detailed calculations. We stress that the effect of an enhanced detection rate of BH binaries is independent of the desired S/N and other characteristics of the detector; it was discussed, for example, in [788, 429, 252, 157].

Unlike the ratio of the detection rates, the expected value of the detection rate of a specific type of compact coalescing binaries by a given detector (network of detectors) requires detailed evolutionary calculations and the knowledge of the actual detector’s noise curve, as discussed. To calculate a realistic detection rate of binary mergers the distribution of galaxies should be taken into account within the volume bounded by the detector’s horizon (see, for example, the earlier attempt to take into account only bright galaxies from Tully’s catalog of nearby galaxies in [425], and the use of the LEDA database of galaxies to estimate the detection rate of supernovae explosions [27]). In this context, a complete study of galaxies within 100 Mpc was done by Kopparapu et al. [376]. Based on their results, Abadie et al. [2] derived the approximate formula for the number of the equivalent Milky-Way-type galaxies within large volumes, which is applicable for distances ≳ 30 Mpc:

$${N_G}({D_{{\rm{hor}}}}) = {{4\pi} \over 3}{\left({{{{D_{{\rm{hor}}}}} \over {{\rm{Mpc}}}}} \right)^3}(0{.}0116){(2{.}26)^{- 3}}{.}$$

Here the factor 0.0116 is the local density of the equivalent Milky-Way-type galaxies derived in [376], and the factor 2.26 takes into account the reduction in the detector’s horizon value when averaging over all sky locations and orientations of the binaries. Then the expected detection rate becomes \({\mathcal D}({{\mathcal D}_{{\rm{hor}}}}) = {{\mathcal R}_G} \times {N_G}({D_{{\rm{hor}}}})\).

However, not only the mass and type of a given galaxy, but also the star formation rate and, better, the history of the star formation rate in that galaxy are needed to estimate the expected detection rate \({\mathcal D}\) (since the coalescence rate of compact binaries in the galaxies strongly evolves with time [432, 479, 158]).

So to assess the merger rate from a large volume based on galactic values, the best one can do at present appears to be using formulas like Eq. (67) or (5), given in Section 2.2. However, this adds another factor two of uncertainty to the estimates. Clearly, a more accurate treatment of the transition from galactic rates to larger volumes with an account of the galaxy distribution is very desirable.

To conclude, we will briefly comment on the possible electromagnetic counterparts of compact binary coalescences. It is an important issue, since the localization error boxes of NS + NS coalescences by GW detectors network only are expected to be, in the best case, about several square degrees (see the detailed analysis in [649], as well as [1] for a discussion of the likely evolution of sensitivity and sky localization of sources for the advanced detectors), which is still large for precise astronomical identification. Any associated electromagnetic signal can greatly help to pinpoint the source. NS binary mergings are the most likely progenitors of short gamma-ray bursts ([500, 204] and references therein). Indeed, recently, a short-hard GRB 130603B was found to be followed by a rapidly fading IR afterglow [43], which is most likely due to a ‘macronova’ or ‘kilonova’ produced by decaying radioactive heavy elements expelled during a NS binary merging [416, 658, 483, 290, 253]. Detection of electromagnetic counterparts to GW signals from coalescing binaries is an essential part of the strategy of the forthcoming advanced LIGO/VIRGO observations [706]. Different aspects of this multi-messenger GW astronomy are further discussed in papers [562, 530, 343, 1, 466], etc.

7 Short-Period Binaries with White-Dwarf Components

Binary systems with white-dwarf components that are interesting for general relativity and cosmology come in several flavors:

  • Detached white-dwarf binaries or “double degenerates” (DDs, we shall use both terms as synonyms below).

  • Cataclysmic variables (CVs) — a class of variable semidetached binaries containing a white dwarf and a companion star that is usually a red dwarf or a slightly evolved star, a subgiant.

  • A subclass of the former systems in which the Roche lobe is filled by another white dwarf or low-mass partially degenerate helium star (AM CVn-type stars or “interacting double-degenerates”, IDDs). They appear to be important “verification sources” for planned space-based low-frequency GW detectors.

  • Detached systems with a white dwarf accompanied by a low-mass non-degenerate helium star (sd + WD systems).

  • Ultracompact X-ray binaries (UCXBs) containing a NS and a Roche lobe overflowing WD or low-mass partially degenerate helium star.

As Figure 2 shows, compact stellar binaries emit gravitational waves within the sensitivity limits for space-based detectors if their orbital periods range from ∼ 20 s to ∼ 20 000 s. This means that, in principle, gravitational-wave radiation may play a pivotal role in the evolution of all AM CVn-stars, UCXBs, a considerable fraction of CVs, and some DD and sd + WD systems and they would be observable in GWs if detectors would be sensitive enough and confusion noise absent.

Though general relativity (GR) predicted that stellar binaries would be a source of gravitational waves as early as in the 1920s, this prediction became a matter of actual interest only with the discovery of the cataclysmic variable WZ Sge with orbital period Porb ≈ 81.5 min by Kraft, Mathews, and Greenstein in 1962 [384], who immediately recognized the significance of short-period stellar binaries as testbeds for gravitational wave physics. Another impetus to the study of binaries as sources of gravitational wave radiation (GWR) was imparted by the discovery of the ultra-short period variability of a faint blue star HZ 29 = AM CVn (Porb ≈ 18 min) by Smak in 1967 [719]. Smak [719] and Paczyński [551] speculated that the latter system is a close pair of white dwarfs, without specifying whether it is detached or semidetached. Faulkner et al. [195] inferred the status of AM CVn as a “semidetached white-dwarf-binary” nova. AM CVn was later classified as a cataclysmic variable after flickering typical for CVs was found for AM CVn by Warner and Robinson [822]Footnote 25 and it became the prototype for a subclass of binaries.Footnote 26

An evident milestone in GW studies was the discovery of a pulsar binary by Hulse and Taylor [757, 292] and the determination of the derivative of its Porb, consistent with GR predictions [832]. Equally significant is the recent discovery of a Porb = 12.75 min detached eclipsing white dwarf binary SDSS J065133.338+284423.37 (J0651, for simplicity) by Kilic, Brown et al. [359, 73]. The measured change of the orbital period of J0651 over 13 months of observations is (−9.8 ± 2.8) × 10−12 s s−1, which is consistent, within 3σ errors, with expectations from GR − (−8.2± 1.7) × 10−12 s s−1 [277]. The strain amplitude of gravitational waves from this object at a frequency of ∼ 2.6 mHz should be 1.2 × 10−22 Hz−1/2, which is about 10 000 times as high as that from the Hulse-Taylor binary pulsar. It is currently the second most powerful GW source known and it is expected to be discovered in the first weeks of operation of eLISA.Footnote 27

The origin of all above-mentioned classes of short-period binaries was understood after the notion of common envelopes and the formalism for their treatment were suggested in the 1970s (see Section 3.6). A spiral-in of components in common envelopes allowed to explain how white dwarfs — former cores of highly evolved stars with radii of ∼ 100 R — may acquire companions separated by ∼ R only. However, we recall that most studies of the formation of compact objects through common envelopes are based on a simple formalism of comparison of binding energy of the envelope with the orbital energy of the binary, supposed to be the sole source of energy for the loss of the envelope, as was discussed in Section 3.6.

Although we mentioned that some currently-accepted features of binary evolution may be subject to certain revisions in the future, one may expect that, for example, changes in the stability criteria for mass exchange will influence mainly the parameter space (M1, M2, a0) of progenitors of particular populations of stellar binaries, but not the evolutionary scenarios for their formation. If the objects of a specific class may form via different channels, the relative “weight” of the latter may change. While awaiting for detailed evolutionary calculations of different cases of mass exchange with “new” physics,Footnote 28 we present the currently-accepted scheme of the evolution of binaries leading to the formation of compact binary systems with WD components.

For stars with radiative envelopes, to the first approximation, the mass transfer from M1 to M2 is stable for binary mass ratios q = M2/M1 ≲ 1.2; for 1.2 ≲ q ≲ 2 it proceeds in the thermal time scale of the donor, tKH = GM2/RL; for q ≳ 2 it proceeds in the dynamical time scale \({t_{\rm{d}}} = \sqrt {{R^3}/GM}\). The mass loss occurs in the dynamical time scale, M/td, if the donor has a deep convective envelope or if it is degenerate and conditions for stable mass exchange are not satisfied. However, note that in the case of AGB stars, the stage when the photometric (i.e., measured at the optical depth τ = 2/3) stellar radius becomes equal to the Roche lobe radius, is preceded by RLOF by atmospheric layers of the star, and the dynamic stage of mass loss may be preceded by a quite long stage of stable mass loss from the radiative atmosphere of the donor [555, 594, 490]. It is currently commonly accepted, despite the lack of firm observational proof, that the distribution of binaries over q is even or rises to small q (see Section 5.1). Since the accretion rate is limited either by the rate that corresponds to the thermal time scale of the accretor or its Eddington accretion rate, both of which are typically lower than the mass-loss rate by the donor, the overwhelming majority (∼ 90%) of close binaries pass in their evolution through one to four common envelope stages.

An “initial donor mass — donor radius at RLOF” diagram showing descendants of stars after mass-loss in close binaries is presented above in Figure 5. We remind here that solar metallicity stars with M ≲ 0.95 M do not evolve past the core-hydrogen burning stage in the Hubble time.

7.1 Formation of compact binaries with white dwarfs

A flowchart schematically presenting the typical scenarios for formation of low-mass compact binaries with WD components and some endpoints of evolution is shown in Figure 9. Evolution of low and intermediate-mass close binaries (M1 ≤ (8–10) M) is generally much more complex than the evolution of more massive binaries (Figure 7). For this reason, not all possible scenarios are plotted, but only the most probable routes to SNe Ia and to systems that may emit potentially detectable gravitational waves. For simplicity, we consider only the most general case when the first RLOF results in the formation of a common envelope.

Figure 9
figure 9

Formation of close dwarf binaries and their descendants (scale and color-coding are arbitrary).

The overwhelming majority of stars in binaries fill their Roche lobes when they have He- or CO-cores, i.e., in cases B or C of mass exchange (Section 3.2). As noted in Section 3.6, the models of common envelopes are, in fact, absent. It is usually assumed that it proceeds in a dynamic or thermal time scale and is definitely so short that its duration can be neglected compared to other evolutionary stages.

7.1.1 Post-common envelope binaries

We recall, for convenience, that in stars with solar metallicity with a ZAMS mass below (2.3–2.8)M, helium cores are degenerate, and if these stars overflow the Roche lobe prior to He core ignition, they produce M ≲ 0.47 M helium white dwarfs. Binaries with non-degenerate He-core donors (M ≳ (2.3–2.8) M) first form a close He-star + MS-star pair that can be observed as a subdwarf (sdB or sdO) star with a MS companion [786]. The minimum mass of He-burning stars is close to 0.33 M [301]. The binary hypothesis for the origin of hot subdwarf stars was first suggested by Mengel et al. [478], but it envisioned a stable Roche lobe overflow. Apparently, there exist populations of close sdB/sdO stars formed via common envelopes and of wide systems with A-F type companions to subdwarfs, which may be post-stable-mass-exchange stars; similarly, the merger products and genuinely single stars can be found among sdB/sdO stars. For the overview of properties of sdB/sdO stellar binaries, models of the population and current state of the problem of the origin of these stars see, e.g., [667, 251, 668, 268, 267, 7, 879, 205, 273, 232, 510, 245, 227, 128, 233]. When a He-star with mass ≲ 2.3 M completes its evolution, a pair harboring a CO white dwarf and an MS-star appears. A large number of post-common envelope binaries or “WD + MS stars” is known (see, e.g., SDSS-sample and its analysis in [507, 887]). The most recent population synthesis model of this class of binaries was published by Toonen and Nelemans [765]. Of course, the WD + MS population is dominated by systems with low-mass MS-stars. Population synthesis studies suggest that about 1/3 of them never evolve further in the Hubble time (e.g., [869]), in a reasonable agreement with above-mentioned observations.

7.1.2 Cataclysmic variables

If after the first common-envelope stage the orbital separation of the binary a ≃ several R and the WD has a low-mass (≲ 1.4 M) main-sequence companion, the latter can overflow its Roche lobe during the hydrogen-burning stage or very shortly after it, because of loss of angular momentum by a magnetically-coupled stellar wind and/or GW radiation, see Sections 3.1.4 and 3.1.5. If the mass ratio of the components allows them to avoid merging in the CE (q ≲ 1.2), a cataclysmic variable (CV) can form. Variability of these stars can be due to thermal-viscous instability of accretion disks [546] and/or unstable burning of accreted hydrogen on the WD surface [481, 482, 656, 736, 737, 739]; see Warner’s monograph [821] for a comprehensive review of CVs in general, [409] for a review of the disk instability model and, e.g., [559, 296, 143, 175, 694, 613, 770, 859, 695, 535, 701, 696, 587, 588, 303, 242, 843, 307] for dependence of hydrogen burning regimes on the WD surface and characteristics of outbursts, depending on WD masses, their chemical composition (He, CO, ONe), temperature and accretion rate. Outbursts produced by thermonuclear burning of accreted hydrogen are identified with novae [736, 737] and are able to explain their different classes, see, e.g., [859]. As an example, we present in Figure 10 the dependence of limits of different burning regimes on mass and accretion rate for CO WD from Nomoto’s paper [535]. The stable burning limits found in this paper, by stability analysis of a steady-burning WD, within factor ≃ 2 agree with those obtained in a similar way, e.g., by Shen and Bildsten [696] and with the results of time-dependent calculations by Wolf et al. [843] (see Figure 9 in the latter paper.)

Figure 10
figure 10

Limits of different burning regimes of accreted hydrogen onto a CO WD as a function of mass of the WD and accretion rate [535]. If stableRG, hydrogen burns steadily. If stable, H-burning shells are thermally unstable; with decrease of the strength of flashes increases. For RG, hydrogen burns steadily but the excess of unburnt matter forms an extended red-giant-sized envelope. In the latter case white dwarfs may lose matter by wind or due to Roche lobe overflow. A dotted line shows the Eddington accretion rate Edd as a function of MWD. The dashed lines are the loci of the hydrogen envelope mass at hydrogen ignition. Image reproduced with permission from [535], copyright by AAS.

Binaries with WDs and similar systems with subgiants (M ≲ 2 M) that steadily burn accreted hydrogen are usually identified with supersoft X-ray sources (SSS) [800]. Note, however, that the actual fraction of the stable burning stage when these stars can be observed as SSS is still a matter of debate, see [259, 528]. Post-novae in the stage of residual hydrogen burning can also be observed as SSS [773, 276, 524]. In these systems, the duration of the SSS stage is debated as well, see, e.g., [843]. A review of SSS may be found, e.g., in [330]; the population synthesis models for SSS where computed, for instance, in [625, 329, 875, 871].

An accreting CO WD in a binary system can accumulate enough mass to explode as a SN Ia if hydrogen burns stably or in mild flashes. This is the “single-degenerate” (SD) scenario for SNe Ia, originally suggested by Schatzman as early as in 1963 [681] and “rediscovered” and elaborated numerically 10 years later by Whelan and Iben [