1 Introduction

Close binary stars consisting of two compact stellar remnants (white dwarfs (WDs), neutron stars (NSs), or black holes (BHs)) are considered as primary targets of the forthcoming field of gravitational wave (GW) astronomy since their orbital evolution is entirely controlled by emission of gravitational waves and leads to ultimate coalescence (merger) of the components. Close compact binaries can thus serve as testbeds for theories of gravity. The double NS(BH) mergers should be the brightest GW events in the 10 – 1000 Hz frequency band of the existing GW detectors like LIGO [16], VIRGO [5], or GEO600 [347]. Such mergers 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). Double WDs, especially interacting binary WDs observed as AM CVn-stars, are potential GW sources within the frequency band of the space GW interferometers like LISA [97Jump To The Next Citation Point] or future detectors [69]. The double WD mergers also stay among the primary candidate mechanisms for type Ia supernova (SN Ia) explosions, which are crucial in modern cosmological studies.

Compact binaries are the end products of the evolution of binary stars, and the main purpose of the present review is to describe the astrophysical knowledge on their formation and evolution. We shall discuss the present situation with the main parameters determining their evolution and the rates of coalescence of double NSs/BHs and WDs.

About 6% of the baryonic matter in the Universe is confined in stars [115]. The typical mass of a stationary star is close to the solar value M ≈ 2 × 1033 g ⊙. The minimum mass of a stationary star at the main sequence (MS) is set by the condition of stable hydrogen burning in its core Mmin ≈ 0.08M ⊙ [205]. The maximum mass of solar composition stars inferred observationally is close to 150 M ⊙ [105]; for very low metallicity stars it is derived by the linear analysis of pulsational stability and is close to 300M ⊙ [15]. Stars and stellar systems are formed due to the development of the gravitational (Jeans) instability in turbulized molecular clouds. The minimum protostellar mass is dictated by the opacity conditions in the collapsing fragments and is found to lie in the range 0.01– 0.1 M ⊙ in both analytical [344] and numerical calculations (see, e.g., [75]). It is established from observations that the mass distribution of main-sequence stars has a power-law shape [366257Jump To The Next Citation Point], dN ∕dM ∼ M −β, with β = − 1.2 for 0.08 ≲ M ∕M ⊙ ≲ 0.5, β = − 2.2 for 0.5 ≲ M ∕M ⊙ ≲ 1.0, and β = − 2.2 to − 3.2 for 1.0 ≲ M ∕M ⊙ ≲ 150 [201202].

The evolution of a single star is determined by its initial mass at the main sequence M0 and the chemical composition. If M0 ≲ 8– 12M ⊙, the carbon-oxygen (CO) (or oxygen-neon (ONe) at the upper end of the range) stellar core becomes degenerate and the evolution of the star ends up with the formation of a CO or ONe white dwarf. The formation of a WD is accompanied by the loss of stellar envelope by stellar wind in the red giant and asymptotic giant branch stages of evolution and ejection of a planetary nebula. The boundary between the masses of progenitors of WDs and NSs is not well defined and is, probably, between 8 and 12 M ⊙ (cf. [163Jump To The Next Citation Point159349161117325350Jump To The Next Citation Point121378]).

At the upper boundary of the mass range of white dwarf progenitors, formation of ONe WDs is possible. The masses of stars that produce ONe WDs are still highly uncertain. However, strong observational evidence for their existence stems from the analysis of nova ejecta [405]. This variety of WDs is important in principle, because accretion induced collapse (AIC) of them may result in formation of neutron stars (see [29378Jump To The Next Citation Point] and references therein), but since for the purpose of detection of gravitational waves they are not different from the much more numerous CO-WDs, we will, as a rule, not consider them below as a special class.

If M0 ≳ (10– 12)M ⊙, thermonuclear evolution proceeds until iron-peak elements are produced in the core. Iron cores are subjected to instabilities (neutronization, nuclei photodesintegration, or pair creation for the most massive stars) 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 hole 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 [456].

The boundaries between the masses of progenitors of WDs or NSs and NSs or BHs are fairly uncertain (especially for BHs). Typically accepted masses of stellar remnants for nonrotating solar chemical composition stars are summarized in Table 1.

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

Initial mass [M ⊙] remnant type mean remnant mass [M ⊙]

0.95 < M < 8– 10 WD 0.6
8 –10 < M < 25 –30 NS 1.35
25 –30 < M < 150 BH ∼ 10

For a more detailed introduction into the physics and evolution of stars the reader is referred to the classical textbook [68]. Formation and physics of compact objects is described in more detail in monographs [37636]. For a recent review of the evolution of massive stars and the mechanisms of core-collapse supernovae we refer to [457112Jump To The Next Citation Point199].

Most stars in the Galaxy are found in multiple systems, with single stars (including our own Sun) being rather exceptions than a rule (see for example [88130]). In the 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 allows for a mass exchange between the components. In close binaries, the initial mass of the components at the zero-age main sequence (ZAMS) ceases to be the sole parameter determining their evolution. Consequently, the formation of compact remnants in binary stars differs from single stars. This is illustrated by Figure 1View Image which plots the type of the stellar remnant as a function of both initial mass and the radius of a star at the moment of the Roche-lobe overflow (RLOF). It is seen that wide binaries evolve as single stars, while for binaries with RLOF a new type of remnants appears – a helium WD, whose formation from a single star in the Hubble time is impossible1.

View Image

Figure 1: Descendants of components of close binaries depending on the radius of the star at RLOF. The boundary between progenitors of He and CO-WDs is uncertain by several 0.1 M ⊙, the boundary between WDs and NSs by ∼ 1M ⊙, while for the formation of BHs the lower mass limit may be even by ∼ 10M ⊙ higher than indicated.

Binaries with compact remnants are primary potential GW sources (see Figure 2View Image). This figure plots the sensitivity of ground-based interferometer LIGO, as well as the space laser interferometer LISA, in terms of dimensionless GW strain h measured over 1 year. The strongest Galactic sources at all frequencies are the most compact double NSs and BHs. Double WDs (including AM CVn-stars) and ultra-compact X-ray binaries (NS + WD) appear to be promising LISA sources.

View Image

Figure 2: Sensitivity limits of GW detectors and the regions of the fh diagram occupied by some of the potential GW sources. (Courtesy G. Nelemans.)

Double NS/BH systems result from the evolution of initially massive binaries, while double WDs are formed from the evolution of low-mass binaries. We shall consider them separately.

Binaries with NSs and BHs
Binary systems with components massive enough to produce NSs or BHs at the end of thermonuclear evolution may remain bound after two supernova explosions. Then, loss of energy and momentum by GWs controls entirely their evolution and gradual reduction of the binary separation may bring the components into contact. During the merger process ∼ 1052 erg are released as GWs [5960]. Such strong bursts of GWs can be reliably detected by the present-day ground-based GW detectors from distances up to several megaparsecs and are the most important targets for GW observatories such as LIGO, GEO, and VIRGO [124Jump To The Next Citation Point374].

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 [400]; see also [213] for a recent update of NS mass measurements.

The case is not so good with the rate of coalescence of relativistic binary stars. Unfortunately, there is no way to derive it from first principles – neither the formation rate of the progenitor binaries for compact double stars nor stellar evolution are known well enough. However, the situation is not completely hopeless, especially in the case of double NS systems. Natural appearance of rotating NSs with magnetic fields as radio pulsars allows searching for binary pulsars with secondary compact companion using powerful methods of modern radio astronomy (for example, in dedicated pulsar surveys such as the Parkes multi-beam pulsar survey [24699]).

Based on the observational statistics of the Galactic binary pulsars with another NS companion, one can evaluate the Galactic rate of binary NS formation and merging [314Jump To The Next Citation Point272Jump To The Next Citation Point191Jump To The Next Citation Point]. On the other hand, a direct simulation of binary star evolution 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 binary star formation and evolution. It is important and encouraging that both estimates (observational, as inferred from recent measurements of binary pulsars [49Jump To The Next Citation Point182Jump To The Next Citation Point], and theoretical from the population synthesis; see Section 6) now give very close estimates for the double NS star merger rate in the Galaxy of about one event per 10,000 years. No binary BH or NS + BH systems have been found so far, so merger rates of compact binaries with BHs have been evaluated as yet only from population synthesis studies.

Binaries with WDs
The interest in these binaries stems from several circumstances. First, they are considered as testbeds for gravitational wave physics. Second, with them SNe Ia are associated. SNe Ia are being used as the primary standard candle sources for the determination of the cosmological parameters Ω and Λ (see, e.g., [348308]). 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., [470Jump To The Next Citation Point245108]). The counts of distant SNe could be used to constrain cosmological parameters (see, e.g., [360]). Finally, close binaries with WDs are among the most promising verification binaries for LISA [390Jump To The Next Citation Point].

In this paper we shall concentrate on the formation and evolution of binary compact stars most relevant for GW studies. The paper is organized as follows. We start in Section 2 with a review of the main observational data on double NSs, 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 the empirical methods to determine double NS coalescence rate. The basic principles of binary stellar evolution are discussed in Section 3. Then, in Section 4 we describe the evolution of massive binary stars. We then discuss the Galactic rate of formation of binaries with NSs and BHs in Section 5. Theoretical estimates of detection rates for mergers of binary relativistic stars are discussed in Section 6. Further we proceed to the analysis of formation of short-period binaries with WD components in Section 7, and consider observational data on binary white dwarfs 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. Sections 11 and 12 are devoted, respectively, to the model of optical and X-ray emission of AM CVn-stars and to their subsample potentially observed both in electromagnetic and gravitational waves. Our conclusions follow in Section 13.

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