"The Evolution of Compact Binary Star Systems"
Konstantin A. Postnov and Lev R. Yungelson 
1 Introduction
1.1 Formation of stars and end products of their evolution
1.2 Binary stars
2 Observations of Double Compact Stars
2.1 Compact binaries with neutron stars
2.2 How frequent are NS binary coalescences?
2.3 Black holes in binary systems
2.4 A model-independent upper limit on the BH-BH/BH-NS coalescence rate
3 Basic Principles of the Evolution of Binary Stars
3.1 Keplerian binary system and radiation back reaction
3.2 Mass exchange in close binaries
3.3 Mass transfer modes and mass and angular momentum loss in binary systems
3.4 Supernova explosion
3.5 Kick velocity of neutron stars
3.6 Common envelope stage
3.7 Other notes on the CE problem
4 Evolutionary Scenario for Compact Binaries with Neutron Star or Black Hole Components
4.1 Compact binaries with neutron stars
4.2 Black-hole–formation parameters
5 Formation of Double Compact Binaries
5.1 Analytical estimates
5.2 Population synthesis results
6 Detection Rates
7 Short-Period Binaries with White-Dwarf Components
7.1 Formation of compact binaries with white dwarfs
7.2 White-dwarf binaries
7.3 Type Ia supernovae
7.4 Ultra-compact X-ray binaries
8 Observations of Double-Degenerate Systems
8.1 Detached white dwarf and subdwarf binaries
9 Evolution of Interacting Double-Degenerate Systems
9.1 “Double-degenerate family” of AM CVn stars
9.2 “Helium-star family” of AM CVn stars
9.3 Final stages of evolution of interacting double-degenerate systems
10 Gravitational Waves from Compact Binaries with White-Dwarf Components
11 AM CVn-Type Stars as Sources of Optical and X-Ray Emission
12 Conclusions

8 Observations of Double-Degenerate Systems

Interrelations between observations and theoretical interpretations are different for different groups of compact binaries. Cataclysmic variables like novae have been observed for centuries, their lower-amplitude cousins (including AM CVn-type stars) for decades, see [821] for a comprehensive historical review. Their origin and evolution found theoretical explanation after effects of common envelopes and orbital angular momentum loss, in particular via gravitational waves radiation and magnetic braking, were recognized in the late 1960 – 1980s [551*, 553, 784*, 837, 810]. As of February 1, 2006 when the atlas and catalogue of CVs by Downes et al. [161, 162] were frozen, they contained about 1600 objects, but the number of known and candidate CVs is continually growing, see, e.g., [163]. Orbital periods were measured for about 1100 CVs; see the online catalogue by Kolb and Ritter [646, 645]. In particular, very recent discoveries [90*, 89, 395, 415] brought the number of confirmed and candidate AM CVn stars and related helium-rich CVs to over 40.

Ultracompact X-ray binaries were discovered with the advent of the X-ray astronomy era in the late 1960s and can be found already in the first published catalogs of X-ray sources (see, for instance, [239]). Their detailed optical study became possible only with 8 m-class telescopes. Currently, the known population of UCXB consists of 13 systems with reliably-measured orbital periods, three candidates with uncertain orbital periods, five without period measurement but very low optical-to-X-ray ratios, and eight faint but persistent X-ray sources [309]. The place of UCXBs in the scenarios of the evolution of close binaries, their origin and evolution were studied already before optical identification [785*, 626*, 778*]. Here we briefly consider information on the known population of detached close white dwarf and subdwarf binaries with WD companions. Interacting double-degenerate stars will be discussed in Sections 9 and 11.

8.1 Detached white dwarf and subdwarf binaries

Unlike CVs and UCXBs, the existence of close detached white dwarfs (double degenerates, DDs) was first deduced from the analysis of scenarios for the evolution of close binaries [824, 784*, 785, 300, 825]. It was also suggested that DDs may be precursors of SNe Ia. This theoretical prediction stimulated optical surveys for close DDs, and the first such binary was detected in 1988 by Saffer, Liebert, and Olszewski [665*]. However, a series of surveys for DDs performed over a decade [648, 66, 206, 462, 666] resulted in only about a dozen of definite detections [460].

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Figure 19: Known close binaries with two WD components, or a WD and a sd component. Red circles mark double-line WDs found by SPY. Green diamonds are single-line WDs found by SPY. Blue asterisks mark double-line WD discovered in surveys other than SPY. Magenta squares are sd + WD systems from SPY. Black crosses and small squares are single-line WD and sd found by different authors. Filled black circles are extremely low-mass WD (ELM) for which, typically, only one spectrum is observed. For single-line systems from SPY we assume inclination of the orbit ∘ i = 60, for other single-line systems we present lower limits of the total mass and indicate this by arrows. Green circles are double-lined ELM WD suggested to be definite precursors of AM CVn stars Several remarkable systems are labeled (see text for details). The “merger” line is plotted assuming equal masses of the components. This is an update of the plot provided by R. Napiwotzki for the previous version of this review.

The major effort to discover close DDs was undertaken by the “ESO Supernovae Ia Progenitors surveY (SPY)” project (PI – R. Napiwotzky): a systematic radial velocity survey for DDs with the UVES spectrograph at the ESO VLT (see [501, 373*, 502] for the design of the project and [503] for the latest summary of its results). The project was aimed at discovering DDs as potential progenitors of SN Ia, but brought, as a by-product, an immense wealth of data on physical parameters, kinematics etc. of white dwarfs and subdwarfs, see e.g., [373, 372, 186, 230, 573, 814, 339, 229]. Theoretical models predicted that it can be necessary to search for binarity of up to 1000 field WDs with stellar magnitude V ≲ 16 – 17 to find a WD with a mass close to M Ch [520]. More than 1000 white dwarfs and pre-white dwarfs were observed (practically all white dwarfs brighter than V ≈ 16.5, available for observations from the ESO site in Chile). SPY increased the number of detected DDs tremendously to more than 150. Their system parameters are continuously determined from follow-up observations. As well, among the objects detected by SPY there appeared a large number of subdwarf stars with massive WD companions, which are of substantial relevance to the SN Ia problem (see papers by S. Geier and his coauthors).

Figure 19* shows orbital periods and total masses of the currently known (February 2014) close DDs and compares them with the Chandrasekhar mass and the critical periods necessary for the binary components to merge in 10 Gyr for a given M tot assuming equal masses of components. In fact, only one DD with Mtot close to MCh was detected by SPY (see the figure). On the other hand, more than a hundred new close WD binaries were discovered by SPY (most of them are still awaiting follow-up observations for the determination of parameters).

In the meantime, several other interesting objects were discovered, including, PN G135.9+55.9 = TS01, first detected as the nucleus of a planetary nebula (PN) [769]. Later, its binarity was discovered. A study of its parameters led to a total mass estimate possibly exceeding MCh [740, 768]. Another remarkable system is V458 Vul (Nova Vul 2007 No. 1) – the nucleus of a planetary nebula with a massive WD companion, which exploded as a nova in 2007. Masses of components are estimated to be 0.58M ⊙ and ≳ M ⊙ [650, 120]. A high mass of PN in this system (∼ 0.2)M ⊙ points to its formation via ejection of a common envelope about 14 000 yr ago [833]. Components of the system will merge due to the orbital angular momentum loss via GWR in ∼ 107 yr. By this time, the current nucleus of PN will turn into a WD. Mass exchange after contact is expected to be unstable (see Figure 17*); this system is currently considered to be the most likely precursor of SN Ia.

Another possible SN Ia precursor is KPD 1930+2752 – an sdB star with an unseen massive WD companion. The estimated total mass of the system ranges from 1.36 M ⊙ to 1.48M ⊙ [231]. In ≃ 200 Myr prior to the merger, the present sdB star can turn into a WD, and then a mass-transfer phase will ensue. Its stability depends on the efficiency of the tidal coupling (Figure 17*). If the sdB-star still remains in the core He-burning stage at RLOF, an AM CVn system will be formed.

The binary system CD-30 11223 (GALEX J1411–3053) is the shortest Porb sdB + WD pair known to date [808]. Estimated masses of components are MsdB = 0.44 –0.48M ⊙ and MWD = 0.74 –0.77 M ⊙. The time before RLOF by the sdB star (≈ 30 Myr) is apparently too short for its transformation into a WD, and stable mass transfer onto the WD upon RLOF may be expected. Thus, this system is a potential precursor to an AM CVn star.

The most numerous population (58 objects) in Figure 19* is of extremely low mass white dwarfs (ELM) – helium white dwarfs with M ≲ 0.25 M ⊙ found in a targeted spectroscopic survey (see [72] for a summary of results; data for the plot taken from this paper and [360, 241]. Most of them have unseen heavier companions, most probably another WDs, while in some cases even NSs are suspected. Some may have brown-dwarf companions. About half of ELM WD will merge with their companions in less than the Hubble time. As Figure 20* shows, some of them, upon merger (which may take longer than the Hubble time) can start a stable mass transfer, i.e., can form AM CVn stars. The most remarkable of ELM WDs is SDSS J065133.338+284423.37 (J0651) [359, 73], the shortest known detached WD, for which relativistic decay of the orbital period is detected. It is expected that it will be possible to detect some ELM stars by space-born GW detectors [358].

Yet more important recently-discovered objects are the eclipsing ELM WD SDSS J075141.18–014120.9 (J0751) and ELM WD SDSS J174140.49+652638.7 (J1741) [360]. The masses of the components of J0751 are 0.168 M ⊙ and 0.97+0.06M ⊙ −0.01, the orbital period is Porb = 0.08 day. The binary components will merge in about 270 Myr. The parameters of J1741 are M1 = 0.19 ± 0.02 M ⊙, M2 ≥ 1.11 M ⊙, Porb = 0.06 day, the merging time is 160 Myr. With such a low mass ratio, one can almost certainly expect a stable RLOF (see Figure 22*) for J1741. For J0751, a stable mass exchange is also highly probable. This makes these two systems the first-discovered almost-certain progenitors of AM CVn stars and, possibly, He-novae (see Section 9).

The observed precursors of ELM WD can be hidden within EL CVn type binaries, which were recently identified as a separate class of objects – eclipsing binaries with short orbital periods (0.7 – 2.2) day, spectral type A primaries and very hot low-mass pre-He-WD [474]. Due to the large mass ratios of components, RLOF by the A-stars in these binaries will result in common envelopes. If components will not merge in CE, theses systems will turn into ELM WD binaries.

Evolutionary sequences for ELM WD were computed by Althaus et al. [9], who also provided interpolation formulas for fitting masses and cooling ages of ELM WD as functions of T eff and log g.

WD binary CSS 41177 is an eclipsing Porb = 2.78 hr system with helium WD components: M1 = 0.38 ± 0.02M ⊙, M2 = 0.32 ± 0.01 M ⊙ [65]. The merging time of components is 1.14 ± 0.05 Gyr and this system is the perfect candidate to form a single sdB star [786, 667], see Section 7.3.2.

The most massive WD binary with parameters estimated from astrometric data and optical and near-IR photometry is LHS 3236 [271]. It may be a pair of DA WD or a DA WD in a pair with a DC WD. Masses of its components are either 0.93 M ⊙ and 0.91M ⊙ or 0.98 M ⊙ and 0.69 M ⊙ (depending on types assigned to WD). However, a binary orbital period of 4.03 yr corresponds to the tremendously-long merging time ≃ 2 × 1013 Myr.

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Figure 20: Mass ratios of ELM WD. Vertical lines separate binaries which will, upon RLOF, exchange mass definitely stably (possible progenitors of AM CVn stars), WD for which stability of mass-exchange will depend on the efficiency of tidal interaction, and definitely unstable stars. The latter systems may be progenitors of SNe Ia, as discussed in Section 7.3.2. Courtesy T. Marsh [459].

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