"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

10 Gravitational Waves from Compact Binaries with White-Dwarf Components

We need to start this section with a caveat that until now virtually all studies on the detection of low-frequency GWR from compact binaries, with the single exception of [531*], have been made with the future launch of LISA in mind. At the time of writing only several months have elapsed since the decision to cancel LISA and the introduction of the eLISA project. Therefore, almost all estimates involving sensitivity of the detector are still “LISA-oriented” and only in some cases corrections for the reduced (by about an order of magnitude) eLISA sensitivity are available. Nevertheless, as the physics behind GWR emission is unchanged, we present in this section, as examples, “LISA-oriented” results, unless stated otherwise.

It was suggested initially that contact W UMa binaries will dominate the Galactic gravitational-wave background at low frequencies [487]. However, it was shown in [784, 467, 180*, 426*, 282*] that it will, most probably, be totally dominated by detached and semidetached WD binaries.

As soon as it was recognized that the birth rate of Galactic close WD binaries may be rather high, and even before detection of the first close detached DD was reported in 1988 [665], in 1987 Evans, Iben, and Smarr [180] accomplished an analytical study of the detectability of the signal from the Galactic ensemble of DDs, assuming certain average parameters for DDs. Their main findings can be formulated as follows. Let us assume that there exists a certain distribution of DDs over frequency of the signal f and the strain amplitude h: n (f, h). The weakest signal is hw. For the time span of observations τint, the frequency resolution bin of the detector is Δfint ≈ 1∕τint. Then, integration of n (f, h) over amplitude down to a certain limiting h and over Δf gives the mean number of sources per unit frequency resolution bin for a volume defined by h. If for a certain h n

∫ ∞ ( dn ) ----- Δ νint = 1, (72 ) hn df dh
then all sources with hn > h > hw overlap. If in a certain resolution bin hn > hw does not exist, individual sources may be resolved in this bin for a given integration time (if they are above the detector’s noise level). In the bins where binaries overlap, they produce “confusion noise”: an incoherent sum of signals; the frequency above which the resolution of individual sources becomes possible was referred to as the “confusion limit”, fc. Evans et al. found fc ≈ 10 mHz and 3 mHz for integration times 106 s and 108 s, respectively.

Independently, the effect of confusion of Galactic binaries was demonstrated by Lipunov, Postnov, and Prokhorov [427], who used simple analytical estimates of the GW confusion limited signal produced by unresolved binaries whose evolution is driven by GWs only; in this approximation, the expected level of the signal depends solely on the Galactic merger rate of WD binaries (see [252*] for more details). Later, analytic studies of the GW signal produced by stellar binaries at low frequencies were continued in [187, 282, 608, 281*]. A more detailed approach to the estimate of the GW foreground is possible using population synthesis models [426, 828*, 518*, 519*, 661, 864*, 865*]. Convenient analytical expressions allowing us to compare the results of different studies as a function of the assumed Galactic model and merger rate of dwarf binaries are provided by Nissanke et al. [531*].

Note that in early studies the combined signal of Galactic binaries was dubbed “background” and considered as a “noise”. However, Farmer and Phinney [188] stressed that this signal will in fact be a “foreground” for “backgrounds” produced in the early universe. But only circa 2008 – 2009 the term “foreground” became common in discussions of Galactic binaries. As summarized by Amaro-Seoane et al. [10], the overall level of the foreground is a measure of the total number of ultra-compact binaries; the spectral shape of the foreground contains information about the homogeneity of the sample, as models with specific types of binaries predict a very distinct shape; the geometrical distribution of the sources can be found by a probe like eLISA.

Giampieri and Polnarev [240], Farmer and Phinney [188] and Edlund et al. [170*] showed that, due to the concentration of sources in the Galactic center and the inhomogeneity of the LISA-antenna pattern, the foreground should be strongly modulated in a year of observations (Figure 29*), with time periods in which the foreground is by more than a factor two lower than during other periods. This is not the case for the signal from extra-galactic binaries, which should be almost isotropic. The characteristics of the modulation can provide information on the distribution of the sources in the Galaxy, as the different Galactic components (thin disk, thick disk, halo) contribute differently to the modulation. Additionally, during “low signal” periods, antenna will be able to observe objects that are away from the Galactic plane.

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Figure 29: Simulated time-series of the DD Galactic foreground signal of 3 years of data. “Noise” is the instrumental LISA noise. Based on computations in [519*]. Image reproduced with permission from [170*], copyright by ASP.

Referring the reader for comprehensive review of eLISA science to [10], we briefly note that measurements of individual binaries provide the following information. If the system is detached, the evolution of the signal is dominated by gravitational radiation and, in principle, three variables may be measured:

5∕3 2∕3 − 1 5∕3 11∕3 11 ˙f h ∝ ℳ f D , ˙f ∝ ℳ f , f¨= 3--f, (73 )
where h is the strain amplitude, f is the frequency, ℳ = (m1m2 )3∕5∕(m1 + m2)1∕5 is the chirp mass, and D is the distance to the source. Thus, the determination of h, f, and f˙ (which is expected to be possible for 25% of eLISA sources [10]) provides ℳ and D. The value of ¨f (which may be possible for a few high-SNR systems) gives information on deviations from the “pure” GW signal due to tidal effects and mass-transfer interactions. An estimate of the effects of tidal interaction is especially interesting, since it is expected that this effect will be typical for pre-merger systems in which it may lead to strong heating, raising the luminosity of each component to ≃ 20L ⊙, which gives a chance of optical detection on the time scale of a human life; moreover, it can even affect conditions for thermonuclear burning in the inner and outer layers of WD [829, 304, 827, 218, 219].

Nelemans et al. [518*] constructed a population synthesis model of the gravitational-wave signal from the Galactic disk population of binaries containing two compact objects. The model included detached DDs, semidetached DDs,32 detached systems of NSs and BHs with WD companions, NS and BH binaries. For details of the model we refer the reader to the original paper and references therein. Table 8 shows a number of systems with different combinations of components in the Nelemans et al. model.33 Note that these numbers strongly depend on assumptions in the population synthesis code, especially on the normalization of the stellar birth rate, star formation history, distributions of binaries over initial masses of components, their mass-ratios and orbital separations, the treatment of stellar evolution, common envelope formalism, etc. For binaries with relativistic components (i.e., descending from massive stars) an additional uncertainty is brought in by assumptions on stellar wind mass loss and natal kicks. The factor of uncertainty in the estimated number of systems of a specific type may easily exceed 10 (cf. [265, 518*, 791, 294, 519*, 661, 864*]). Thus, these numbers should be taken with some caution.

Table 8 immediately shows that detached DDs, as expected, dominate the population of compact binaries, as found by other authors, as well. While Table 8 gives total numbers, examples of contributions from different formation channels and spatial components of the Galaxy may be found, e.g., in [661, 864*, 865], with a caveat that models in these paper are different from the model in [518*]. Population-synthesis studies confirmed the expectation that a contribution to the signal in the LISA/eLISA frequency range from thick disk, halo, and bulge binaries should not be significant, since the binary systems there are predominantly old and either merged or evolved to too long periods as IDD [660, 864*, 661]. It was also shown that the contribution from extra-galactic binaries to the signal in the LISA/eLISA range will be ∼ 1% of the foreground [188]. The list of models computed before 2012 and an algorithm allowing one to compare the results of different calculations to the extent of their dependence on Galactic model and normalization may be found in [531*].

Population synthesis computations yield the ensemble of Galactic binaries at a given epoch with their specific parameters M1, M2, and a and the Galactic location. Figure 30* shows examples of the relationship between the frequency of emitted radiation and amplitude of the signals from the “typical” double degenerate system that evolves into contact and merges, for an initially detached double degenerate system that stably exchanges matter after the contact, i.e., an AM CVn-type star and its progenitor, and for an UCXB and its progenitor. For the AM CVn system effective spin-orbital coupling is assumed [512, 464*], Figure 22*. The difference in the properties and evolution of DD and IDD (merger vs. bounce and continuation of evolution with decreasing mass) is reflected in the distributions and magnitude of their strains shown in Figure 31* [170*].

Note that there is a peculiar difference between WD pairs that merge and pairs that experience a stable mass exchange. The pairs that coalesce stop emitting GWs on a relatively short time-scale (on the order of the period of the last stable orbit, typically a few minutes) [443]. Thus, if we would be lucky to observe a chirping WD and a sudden disappearance of the signal, this will manifest a merger. However, the chance to detect such an event is small since the Galactic rate of merging WD binaries is −2 −1 ∼ 10 yr only.

For the system with a NS, the mass exchange rate is limited by the critical Eddington value, and excess of matter is “re-ejected” from the system (see Section 3.3.3 and [877]). Note that for an AM CVn-type star it takes only ∼ 300 Myr after contact to evolve to log f = − 3, which explains their accumulation at lower f. For UCXBs this time interval is only ∼ 20 Myr.

Table 8: Current birth rates and merger rates per year for Galactic disk binaries containing two compact objects and their total number in the Galactic disk [518*].
Type Birth rate Merger rate Number
Detached DD 2.5 × 10–2 1.1 × 10–2 1.1 × 108
Semidetached DD 3.3 × 10–3 4.2 × 107
NS + WD 2.4 × 10–4 1.4 × 10–4 2.2 × 106
NS + NS 5.7 × 10–5 2.4 × 10–5 7.5 × 105
BH + WD 8.2 × 10–5 1.9 × 10–6 1.4 × 106
BH + NS 2.6 × 10–5 2.9 × 10–6 4.7 × 105
BH + BH 1.6 × 10–4 2.8 × 106

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Figure 30: Dependence of the dimensionless strain amplitude for a WD + WD detached system with initial masses of the components of 0.6M ⊙ + 0.6 M ⊙ (red line), a WD + WD system with 0.6M + 0.2M ⊙ ⊙ (blue line) and a NS + WD system with 1.4M + 0.2 M ⊙ ⊙ (green line). All systems have an initial separation of components 1 R ⊙ and are assumed to be at a distance of 1 Kpc (i.e., the actual strength of the signal has to be scaled with factor 1∕d, with d in Kpc). For the DD system, the line shows an evolution into contact, while for the other two systems the upper branches show pre-contact evolution and lower branches – a post-contact evolution with mass exchange. The total time-span of evolution covered by the tracks is 13.5 Gyr. Red dots mark the positions of systems with components’ mass ratio q = 0.02, below which the conventional picture of evolution with a mass exchange may be not valid. The red dashed line marks the position of the confusion limit as determined in [519*].
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Figure 31: The distribution of Galactic detached WD + WD binaries and interacting WD (AM CVn stars) as a function of the gravitational wave frequency and chirp mass. From [170*], based on computations in [519*].
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Figure 32: GWR foreground produced by detached and semidetached WD binaries as it was expected to be detected by LISA. The assumed integration time is 1 yr. The ‘noisy’ black line gives the total power spectrum, the white line shows the average. The dashed lines show the expected LISA sensitivity for a S∕N of 1 and 5 [408]. Semidetached WD binaries contribute to the peak between log f ≃ − 3.4 and − 3.0. Image reproduced with permission from [518*], copyright by ESO.
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Figure 33: The number of systems per bin on a logarithmic scale. Semidetached WD binaries contribute to the peak between log f ≃ − 3.4 and − 3.0. Image reproduced with permission from [518*], copyright by ESO.
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Figure 34: Fraction of bins that contain exactly one system (solid line), empty bins (dashed line), and bins that contain more than one system (dotted line) as a function of the signal frequency. From [518*], copyright by ESO.

In the model discussed, the systems are distributed randomly in the Galactic disk according to

− R∕H 2 − 3 ρ(R, z) = ρ0e sech (z∕β ) pc , (74 )
where H = 2.5 kpc [664] and β = 200 pc. The Sun is located at R ⊙ = 8.5 kpc and z⊙ = − 30 pc. Then it is possible to compute the strain amplitude for each system. The power spectrum of the signal from the population of binaries as it would be detected by a gravitational wave detector, may be simulated by computation of the distribution of binaries over Δf = 1∕T wide bins, with T being the total integration time. Figure 32* shows the resulting confusion limited foreground signal. In Figure 33* the number of systems per bin is plotted. The assumed integration time is T = 1 yr. Semidetached WD binaries, which are less numerous than their detached cousins and have lower strain amplitude, dominate the number of systems per bin in the frequency interval − 3.4 ≲ log f (Hz ) ≲ − 3.0 producing a peak there, see also Figure 33*.

Results presented below may be, to some extent, considered as an illustration of the signal detected by a space-born detector, since they are model-dependent and “LISA-oriented”, but they show, at least qualitatively, the features of the signal detected by any space-based GW antenna.

Figure 32* shows that there are many systems with a signal amplitude much higher than the average in the bins with f < fc, suggesting that even in the frequency range seized by the confusion noise some systems can be detectable above the noise level. The latter will affect the ability to detect extra-galactic massive black-hole binaries (Figure 2*) and to derive their parameters.

Population synthesis also shows that the notion of a unique “confusion limit” is an artefact of the assumption of a continuous distribution of systems over their parameters. For a discrete population of sources it appears that for a given integration time there is actually a range of frequencies where there are both empty resolution bins and bins containing more than one system (see Figure 34*). For this “statistical” notion of fc, Nelemans et al. [518*] get the first bin containing exactly one system at log fres (Hz ) ≈ − 2.84, while up to logf (Hz ) ≈ − 2.3 there are bins containing more than one system. Other authors find similar, but slightly more conservative limits on log fres: − 2.25 [864], − 2.44 [828], − 2.52 [661].

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Figure 35: Strain amplitude spectral density (in Hz–1/2) versus frequency for the verification binaries and the brightest binaries in the simulated Galactic population of ultra-compact binaries [519*]. The solid line shows the sensitivity of eLISA. The assumed integration time is 2 yrs. 100 simulated binaries with the largest strain amplitude are shown as red squares. Observed ultra-compact binaries are shown as blue squares, while the subsample of them that can serve as verification binaries is marked as green squares. Image reproduced with permission from [11], copyright by the authors.

Nelemans et al. [518*] also noted the following point. Previous studies of GW emission from the AM CVn systems have found that they hardly contribute to the GW foreground, despite that at f ≈ (0.3 − 1.0) mHz they outnumber the detached DDs. This happens because at these f their chirp mass ℳ is much smaller than that of a typical detached system. However, it was overlooked before that at higher frequencies, where the number of AM CVn systems is much smaller, their ℳ is similar to that of the detached systems from which they descend. This is also confirmed by the latest studies of GW foreground by Nissanke et al., see Figure 36*.

In [518*, 519*] the following objects were considered as resolved: single source per frequency bin with signal-to-instrument noise ratio (SNR) ≥ 1 for sources that can be resolved above the confusion limit f c or SNR ≥ 5 for systems with f < fc that are detectable above the noise level. This resulted in approximately equal numbers of resolvable detached double degenerates and interacting double degenerates — about 11 000 objects of every kind.

However, the above-mentioned criterion for resolution may be oversimplified. A more rigorous approach implies detection using an iterative identify-and-subtract process, where the signal-to-noise ratio of a source is evaluated with respect to noise from both the instrument and the partially-subtracted foreground [763, 531*]. This procedure was applied by Nissanke et al. [531*] to the same sample of binaries as in [518*, 519*], for both “optimistic”(Case 1 below) and “pessimistic” (Case 2) scenarios (see Sections 7 and 9) with different efficiency of the formation of double degenerates.

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Figure 36: The number density as a function of GW-strain amplitude h and frequency f for close detached WD (top) and AM CVn stars (bottom) in Case 1 and Case 2. Green dots denote individually-detected WD binary systems (for one year of observation with the 5-Mkm detector); red dots – AM CVn systems from the WD binary channel and blue dots – AM CVn systems from the He star channel. Image reproduced with permission from [531*], copyright by AAS.

Figure 36* shows the frequency-space density and GW foreground of WD and AM CVn binary systems for these two scenarios, as computed in [531*]. Qualitatively the results are in good agreement with [518*, 519*], despite the fact that the predicted number of systems is lower by a factor of five.

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Figure 37: The frequency-space density and GW foreground of DD and AM CVn systems for two scenarios. In each subplot, the bottom panel shows the power spectral density of the unsubtracted (blue) and partially subtracted (red) foreground, compared to instrumental noise (black). The open white circles indicate the frequency and amplitude of the “verification binaries”. The green dots show the individually-detectable DD systems; the red dots show the detectable AM CVn systems formed from DD progenitors and the blue dots show the detectable AM CVn systems that arise from the He-star–WD channel. The top panel shows histograms of the detected sources in the frequency space. Image reproduced with permission from [531*], copyright by AAS.

The frequency-space density and GW foreground of DD and AM CVn systems for two scenarios are shown in Figure 37* and appropriate numbers of systems are presented in Table 9.

Table 9: Detectable sources (5 Mkm detector – two upper rows, 1 Mkm detector – two lower rows). Values are shown for SNR = 5 thr, one interferometric observable, and one year of observation, with approximate scalings as a function of −1∕2 −1∕2 ρeff = SNRthr (Tobs∕yr) (Nobs) [531].
AM CVn (He–WD)
case 1 11, 898 × (ρeff5)1.4 2, 093 × (ρ eff5)1.8 122 × (ρ eff5)2.8
case 2 13, 957 × (ρeff5)1.4 16 × (ρ eff5)1.7 72 × (ρ eff5)2.9
case 1 6, 034 × (ρeff5)1.2 719 × (ρ eff5)1.4 38 × (ρ eff5)2.6
case 2 6, 337 × (ρeff5)1.2 4 × (ρ eff5)1.4 25 × (ρ eff5)2.2

The estimates were made for two assumed arm-lengths of the detector. The number of detected sources and the level of residual confusion foreground may be scaled by power laws of a single “effective-SNR parameter” ρeff = SNRthr (Tobs∕yr)−1∕2(Nobs)− 1∕2, where SNRthr is the detection threshold, Tobs is the time of observations, and Nobs is the number of Time-Delayed Interferometry observables (TDI – a specific method to properly time-shift and linearly combine independent Doppler measurements, that is necessary in the case of unequal arm-lengths of the detector [150]; Nobs = 1 for a two-arms configuration of the detector).

For the LISA-like configuration, the number of detected DDs is comparable to earlier estimates for the “optimistic” case of [518*, 519*]: ≃ 11 000. The number of AM CVn stars that was comparable to the number of DDs, now strongly drops due to a higher required SNR defined with respect to both instrument and confusion noise and the usage of interferometric observables instead of the strain. For a 1 Mkm detector, the effect of changing detection criterion is much more profound – by a factor of two to four. The more rigorous detection criterion changes the number of AM CVn stars more than the number of DD, since the former have smaller chirp mass and strain amplitudes. Thus, the conclusion of early studies on the dominance of detached DD in the GW foreground formation remains valid.

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Figure 38: The histogram of apparent magnitudes of the WD binaries that are estimated to be individually detected by LISA. The black-and-white line shows the distribution in the V-band, the black line the distribution in the I-band and the grey histogram the distribution in the K-band. Galactic absorption is taken into account. The insert shows the bright-end tail of the V-band distribution. Image reproduced with permission from [509], copyright by IOP.

Peculiarly enough, as Figure 35* shows, the AM CVn-type systems appear, in fact, dominant among the “verification binaries” for LISA/eLISA: binaries that are well known from electromagnetic observations and whose radiation is estimated to be sufficiently strong to be detected; see the list of 30 promising candidates in [744] and references therein. The perspectives and strategy of discovering additional verification binaries were discussed by Littenberg et al. [433*], who used as an input catalog the model from [519*] with minor modifications. It was shown in [433] that for the “optimistic” scenario of the formation of AM CVn stars, 5 × 109 m arm-length configuration (full LISA), a 63% confidence interval of the sky-location posterior distribution area on the celestial sphere and limiting stellar magnitude of the sample m ≤ 24 the number of candidate double-degenerate verification binaries is ≈ 560. For the “pessimistic” scenario, short arm-length (9 1 × 10 m, similar to eLISA), 2 dΩ ≤ 1 deg, m ≤ 20 this number reduces to 9. If an additional detection requirement of being eclipsing is applied, these numbers are reduced by a factor of ∼ 3. The conclusion is that the chance for discovering new “verification binaries” is associated with the next generation of ground-based telescopes, like the 38-m European Extremely Large Telescope (E-ELT),34 the 30-m US telescope (TMT),35 the James Webb Space Telescope (JWST).36 The ESA GAIA mission will also be very helpful in localization of the sources. Resolved LISA sources are expected to be most numerous in the K-band, and a significant fraction of them will have K ≲ 29 (Figure 38*), within reach of the future facilities, whilst current instruments are mostly limited by K ≲ (20 –21 ) [509].

The most severe “astronomical” problems concerning “verification binaries” are their distances, which for most systems are only estimates, and poorly constrained component masses. Some features of “verification binaries” and prospects of detection of new objects in wide-field surveys are also discussed in [358].

The issue of “verification binaries” is related to the problem of detection of GWR sources in electromagnetic waves which is discussed in the next Section 11.

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