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 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 () 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]²⁵ and it became the prototype for a subclass of binaries.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 , consistent with GR predictions [832]. Equally significant is the recent discovery of a 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 , which is consistent, within errors, with expectations from GR – [277]. The strain amplitude of gravitational waves from this object at a frequency of  2.6 mHz should be , 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. 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 – may acquire companions separated by 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 (, , ) 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,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 to is stable for binary mass ratios ; for it proceeds in the thermal time scale of the donor, ; for it proceeds in the dynamical time scale . The mass loss occurs in the dynamical time scale, , 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 ) 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 is even or rises to small (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 do not evolve past the core-hydrogen burning stage in the Hubble time.

Figure 9: Formation of close dwarf binaries and their descendants (scale and color-coding are arbitrary).

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 () 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.

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 , helium cores are degenerate, and if these stars overflow the Roche lobe prior to He core ignition, they produce helium white dwarfs. Binaries with non-degenerate He-core donors () 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 [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 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 and the WD has a low-mass () 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 , 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 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: 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 , hydrogen burns steadily. If , H-burning shells are thermally unstable; with decrease of the strength of flashes increases. For , 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 as a function of . 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 () 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 [836]. We shall consider these possible progenitors of SNe Ia in more detail below (Section 7.3).

If the WD belongs to the ONe-variety, it may experience an AIC into a neutron star due to electron captures on Ne and Mg, and a low-mass X-ray binary may be formed [83].

The final stages of the evolution of CVs are not clear (and, in fact, poorly investigated). It was hypothesized [659] that, when the donor mass decreases below several hundredths of , the disk-orbit tidal interaction becomes inefficient and the disk turns into a sink of orbital momentum. More orbital momentum is drained from the orbit than returned back, the mass loss rate increases and the donor can be tidally disrupted.

As noted by G. Dubus (private communication) at , which may be attained in the Hubble time, the conventional picture of mass exchange may become invalid: the circularization radius (the minimum radius of the accretion disk defined by the specific angular momentum of the matter at the point and its interaction with the donor) exceeds the outer radius of the disk (determined by tidal effects [554, 567, 809*]). The resonance effects come into play and gaps may appear in the outer disk, since orbital resonances confine the motion of disk particles to certain radius ranges, similar to the formation of gaps in the asteroid belt and Saturn rings. It is not clear how the disk behaves then and how it interacts with still inflowing matter.

7.2 White-dwarf binaries

The second common envelope stage may happen when the giant companion to the WD overfills its Roche lobe. Before proceeding to a discussion of later stages, we note the following. Usually, this second common envelope is thought to result, based merely on the final separation of the components as given by Eqs. (56*) and (59*) – in a merger, if at the end of common envelope event the radius of one of components becomes larger than its Roche-lobe radius, or in the formation of a detached system otherwise. However, as was speculated by Livio and Riess [437] after the first detection of SN Ia with hydrogen in the spectrum (SN 2002ic [264]), there is a very small probability () that WD merges with the core of an AGB star when part of the AGB envelope (common envelope) is still not ejected. These considerations are corroborated by results of 1D [313, 341] and 3-D simulations, [571] which show that only a fraction of the common envelope is ejected in the dynamical time scale, and the rest of the matter remains in the vicinity of the close pair formed by the core of the AGB star and the spiraling-in WD. As mentioned in Section 3.6, interaction with this matter may result in further shrinking of the system and ultimate merger of the components. If both cores are carbon-oxygen and the total mass of the merger product exceeds , it may explode as a SN Ia; see brief discussion of this “core-degenerate” [341] SNe Ia scenario in Section 7.3.

If the system avoids merging inside the CE and the donor had a degenerate core, a close WD binary or DD is formed (the left branch of the scenario shown in Figure 9*). In Figure 11* we show an indicative plot of possible combinations of components in double-degenerate systems [461] (it is assumed that the bulk chemical composition of WD is uniquely related to its mass).

Figure 11: Possible combinations of masses and chemical compositions of components in a close WD binary [461]. Solid curves are lines of constant chirp mass (see Sections 3.1.2 and 10). Image reproduced with permission from [461], copyright by IOP.

In Figure 12* we compare the most recent simulated distribution of the total masses of WD binaries vs. their with the distribution of for 46 observed WD binaries known at the time of writing of the quoted study [766]. The limiting stellar magnitude was assumed to be . It is difficult to judge the agreement with observations because the sample of observed WDs is quite random, but the “naked eye” estimate suggests that the model that uses -formalism fits observations better than the model that uses -formalism.

Figure 12: The total masses of binaries in the simulated population of WD binaries with as a function of system’s orbital period. In the left plot the outcome of the first common envelope stage is described by -formalism, Eq. (59*), and in the right plot – by -formalism, Eq. (57*). In both plots the latter equation describes both common envelope stages. The grey scale (the same for both plots) corresponds to the density of objects in the linear scale. Observed WD binaries are shown as filled circles (see the original paper for references). The Chandrasekhar mass limit is shown by the dotted line. To the left of the dashed line the systems merge within 13.5 Gyr. Image reproduced with permission from [766], copyright by ESO.

The fate of DDs is solely determined by GWR. The closest of them may be brought into contact by AML via GWR. For instance, a WD pair can merge in 10 Gyr if the post-CE separation of the components is about . We shall discuss the population of observed detached DDs in Section 8. If one of the DD components fills its Roche lobe, the outcome of the contact depends on the chemical composition of the stars and their masses. There are several possible endpoints: merger leading to a supernova (the left and the right-most branches at the bottom of Figure 9*, see Section 7.3), stable mass exchange with the formation of an AM CVn system (as indicated in the figure, see Section 9), direct formation of a single massive WD (the central branch, see Section 7.3.2 and discussion of Figure 18*) or the formation of an R CrB type star with an extended helium envelope [825*, 306, 885, 480], which will also end its evolution as a single WD (not shown in Figure 9*). White dwarfs produced by the merger process may be hidden among single WDs without traces of H or He, like those with oxygen-rich atmospheres [222].

If the donor has a non-degenerate He-core (ZAMS mass ) and the system does not merge, after the second CE-stage a helium star + WD system can arise (the “Non-degenerate He-core” branch of evolution shown to the right in Figure 9*). From the point of GW detection, low-mass He-stars ( for ) can be important. The lifetime of He-remnants of the close binary components can be as long as that of their main-sequence progenitors [301*]:

( is the mass in solar units). These stars expand only slightly at the core-helium burning stage. If the separation of components is sufficiently small (), AML via GWR may bring He-star to RLOF, while He is still burning in its core. If , a stable mass exchange is possible [674*, 819]. A detailed study of the evolution of semidetached low-mass He-stars with WD companions may be found in [870], see also Section 9. These stars overfill their Roche lobes at orbital periods ranging from 16 to 50 min and first evolve to shorter periods with a typical of several . The mass loss quenches the central nuclear burning, and the helium star becomes “semi-degenerate”. An AM CVn-type system may be formed in this way, see Section 9) for more details. One cannot exclude that a Chandrasekhar mass may be accumulated by the WD in this channel of evolution, but the probability of such a scenario seems to be very low,  1% of the inferred galactic rate of SNe Ia [727*]. If the He-star completes the core He-burning before RLOF, it becomes a CO-WD. In Figure 9* it “jumps” into the “double degenerate” branch of evolution.

If the mass of the helium remnant-star is , it can hardly overfill the Roche lobe in the core He-burning stage, but after exhaustion of He in the core it expands and may refill the Roche lobe [301*]. Mass-loss rates of these “helium-giants” occur in the thermal time scale of their envelopes: . They may lose several and, if companion WD is initially sufficiently massive, the latter may accumulate  [860*, 819]. If the “massive” He-star completes core He-burning before RLOF or its companion does not explode, the He-star turns into a CO-WD. In Figure 9* it “jumps” into the “double degenerate” branch of evolution.

7.3 Type Ia supernovae

The most intriguing possible outcomes of the evolution of compact binaries with WDs are explosions of type Ia SNe. Presently, the most stringent constraints on the nature of the exploding object in SN Ia are placed by the recent SN 2011fe. We reproduce from the paper by Bloom et al. [57*] Figure 13* showing the constraints on the parameters of the exploded object in SN 2011fe, which were obtained by excluding certain regions of progenitors space on observational grounds. The blank space in Figure 13* leaves only a compact object (WD or NS) as the progenitor to this SN, but the possible NS phase transition to a quark star is highly unlikely.

Figure 13: Constraints on mass, effective temperature, radius and average density of the primary star of SN 2011fe. The shaded red region is excluded by non-detection of an optical quiescent counterpart in the Hubble Space Telescope imaging. The shaded green region is excluded from considerations of the non-detection of a shock breakout at early times. The blue region is excluded by the non-detection of a quiescent counterpart in the Chandra X-ray imaging. The location of the H, He, and C main-sequence is shown, with the symbol size scaled for different primary masses. Several observed WDs and NSs are shown. The primary radius in units of is shown for mass . Image reproduced with permission from [57], copyright by AAS.

The exploding star radius estimate in Figure 13* is derived by assuming that SN 2011fe was discovered about 11 hr after the explosion, as inferred by Nugent et al. [539], from a simple shock-breakout model, which gives a measure of the stellar radius. However, as noted by Piro and Nakar [593], since the early luminosity of SN Ia is powered by radioactive decay of ⁵⁶Ni, if there is no Ni mixed in the outermost layers of the ejecta, there can be a “dark phase” lasting from a few hours to several days, where the only source of emission is radiative cooling of shock-heated gas that is too dim to be detected. Duration of the “dark stage” depends on radial distribution of ⁵⁶Ni. Piro and Nakar suggested that the explosion of SN 2011fe occurred earlier than estimated by Nugent et al., and bounds on the radius of the exploding object are less stringent than those suggested by Bloom et al. However, even a several times larger limit hardly leaves room for any exploding object different from a WD. A comprehensive analysis of data on SN 2011fe obtained until mid-2013 and their implications for the problem of progenitors of SN Ia can be found in the review paper by Chomiuk [107]. Early time observations of SN 2011fe can be explained in the frame of “pulsational-delayed detonations” paradigm for SNe Ia explosions [145].

The theory of stellar evolution should be able to explain the formation of SNe Ia progenitors, which fit the basic observational constraints: the explosion energy, the frequency of explosions in the galaxies with different morphology, the observed light curves and spectra, chemical yields, etc.

Until very recently, the most popular paradigms for SNe Ia used the accumulation of the Chandrasekhar mass by a CO WD via accretion in semidetached binaries or in a merging WD binary with the total mass exceeding the Chandrasekhar limit. We discuss them below and then proceed to some other ideas actively debated at present.

7.3.1 Single-degenerate scenario

The formation of an WD prone to a thermonuclear explosion seems to be a natural outcome of the accretion of hydrogen in a semidetached binary. The masses of CO WDs are limited by (see Section 1). It is easily envisioned that, if hydrogen burning on the surface of an accreting CO WD is stable or occurs in mild flashes, as well as the subsequent helium burning, and the WD is not eroded, the matter can be accumulated on the WD surface to increase the WD mass to . The WD mass growth via the hydrogen burning (“CV, SSS” branch of evolution in Figure 9*) is considered to be equivalent to the direct accretion of helium at the rate determined by hydrogen burning [532]. Strictly speaking, the latter assumption is not completely accurate, since H-burning modifies the thermal state of the He-layer that accumulates beneath the H-burning shell, and in the case of the direct accumulation of helium on the WD surface, the WD evolution can be different. The efficiency of matter retention on the WD surface can also differ [94]. However, this issue has not been systematically studied as yet.

Figure 14: Limits on different burning regimes of helium accreted onto a CO WD as a function of the WD mass and accretion rate [590], Piersanti, Tornambé & Yungelson (in prep.). Above the “RG Configuration” line, accreted He forms an extended red-giant–like envelope. Below , accreted He detonates and the mass is lost dynamically. In the strong-flashes regime, He-layer expands and mass is lost due to RLOF and interaction with the companion. Helium accumulation efficiency for this regime is shown in Figure 15*, and the critical He masses for WD detonation are shown in Figure 28*. Shaded region shows the domain of stable burning of accreted hydrogen.

Figure 15: The accumulation efficiency of helium as a function of the accretion rate for CO WD models of 0.6, 0.7, 0.81, 0.92, (top to bottom), as calculated by Piersanti et al. (in prep.). Dotted and dashed lines represent the accumulation efficiency for CO WDs with initial masses 0.9 and , respectively, after Kato & Hachisu [347].

Another version of this scenario, which is sometimes discussed, involves a WD accreting matter from the stellar wind of the companion in wide binary systems (in symbiotic stars). However, the following considerations immediately suggest low efficiency of this channel. In symbiotic binaries, the components evolve independently, hence the typical mass of the WD accretor should be small: . The efficiency of accretion from the stellar wind of the optical companion is on the order of 10% [62, 141]. Thus, there is little chance to accumulate , which is also confirmed by numerical simulations [876, 447, 871].

Note that the “single-degenerate”(SD) channel to SN Ia encounters severe problems. Stable burning of hydrogen or burning in mild flashes on the surface of a massive WD may occur within a narrow range of accretion rates (Figure 14*). Mass-loss rates by MS-stars in cataclysmic variables driven by the angular momentum loss via magnetic stellar wind are below several and result in Nova explosions, which limit the growth of or even cause their erosion [613*, 859*, 175]. Donors in the Hertzsprung gap or on the early red-giant branch may provide higher if they lose mass in a thermal time scale . Masses of companions should be commensurable to the masses of WD by virtue of the mass-exchange stability conditions. However, the mass loss in these stars in the initial phase of mass exchange proceeds on a time scale that is even shorter than the thermal one [555] and can exceed (Figure 10*) by 1 – 2 orders of magnitude and even be higher than .

It is possible that the precursor of a SD-SN Ia is observed, e.g., in the supersoft X-ray source WX Cen [620]. If for this system [541], reflects the real secular decrease of the orbital period, then the donor is more massive than the accretor, and the time scale of mass exchange is thermal. The estimated mass of the WD in this system is , and there is a chance that the mass of WD will grow up to . It is sometimes claimed, see, e.g., [349] and references therein, that recurrent novae may also be precursors of SNe Ia. These stars are presumably cataclysmic variables with high-mass () WD-components, accreting at rates slightly lower than the (see Figure 10*). In this burning regime some mass accumulation is possible, e.g., [613*, 859*]. However, since it is expected that the accumulation of very little mass is necessary for a massive dwarf to initiate a thermonuclear runaway (), the appropriate “theoretical” novae rate will far exceed the observed rate [729]. Apparently, the mass function of CO WD in cataclysmic variables is such, that the number of massive WD is vanishingly small.

As a remedy for the deficit of the candidate SD SNe Ia and lack, at that time, of obvious candidates for SN Ia from merging dwarfs, Hachisu et al. [257] suggested that the excess of matter over can be blown out from the system by optically-thick winds, similar to the winds blowing from novae after eruptions, which were introduced into consideration by the same authors [346].²⁹ The efficiency of accretion then becomes . The angular momentum is lost in the “isotropic reemission” mode, and this also raises the upper limit of the components ratio, thus allowing a stable mass exchange (see the discussion in [257, 873]). The strong wind stops when the mass-loss rate drops below . However, even in this regime of steady accretion WD must lose mass via a strong wind, since its surface is hot () and, hence, the efficiency of mass capture by the accretor is . It drops further in the flaring regime, when the expanding envelope of the WD may exceed the corresponding Roche lobe radius [303*]. Finally, at all accumulated mass is lost in novae explosions [613, 859].

We note parenthetically that it was suggested that WD losing mass in the regime of optically-thick winds (also called “Rapidly Accreting White Dwarfs”, RAWD) may be observable as low-luminosity Wolf–Rayet stars, possibly as WR nuclei of planetary nebulae or as V Sge type cataclysmic binaries with numerous emission lines of highly ionized species in their spectra, see [414] and references therein. In principle, RAWDs can exist and can be observed in nearby elliptical galaxies, such as M32 [97]. However, a dedicated survey for RAWDs in the central core of the Small Magellanic Cloud [414] did not reveal a single candidate system. On the other hand, since the appearance of RAWDs has never been modeled in detail, the nondetection of them, in fact, does not constrain the SD model of SNe Ia as yet.

Kato and Hachisu also introduced into the model of SD SNe Ia progenitors the “stripping effect”. It assumes that a strong hot wind from the WD irradiates the companion and induces an additional mass loss. The combined effect of the additional mass and angular momentum losses rises further the masses of WD companions, which still allow a stable mass exchange: to about and for MS and red-giant companions, respectively (see [258, 262] for the latest versions of the model). However, the “stripping effect” needs a very fine tuning and is hardly feasible.

Further major complication of the SD SNe Ia model is associated with a mismatch of regimes of stable and unstable burning of H and He, noted by Iben and Tutukov [303]; see the paper by Bours et al. [64] for the latest study of the influence of H and He retention efficiency upon the rate of SNe Ia with SD-progenitors. In Figure 14* we show the limits of different burning regimes of helium accreted onto a CO WD as a function of the WD mass and accretion rate [590], Piersanti, Yungelson and Tornambé (in prep.) and overplot them by the limits of the domain of stable burning of accreted hydrogen. Figure 14* clearly suggests that, if even some He can be accumulated via H-burning in the regimes of steady burning or mild flashes, it can be almost entirely lost in strong flashes of He-burning (Figure 15*). In fact, the issue of He-accumulation is still not settled. In the latter Figure, we also show for comparison He-accumulation efficiencies computed by Kato & Hachisu [348] under different assumptions. The most important source of difference is, in our opinion, the ignorance of the interaction of the expanding envelope of the flaring WD with the companion in [348].

Results of the most recent calculations of the accretion of H up to the accumulation of a He-layer prone to explosion through thousands of small-scale outbursts, which do not erode the dwarf, are controversial. Idan et al. [307] found that accretion of hydrogen onto massive WD at leads to a strong flash removing all accumulated helium, instead of expected steady growth of accreted layer. In fact, since optically-thick wind is considered in the model, effective rate of He-accumulation via H-burning is only about and WDs in the model of Idan et al. are in the strong He-flashes regime. On the other hand, Newsham et al. [525] claim that WD with 0.70, 1.00, and accreting at similar can continuously grow in mass despite recurrent helium flashes and, hence, can be progenitors of SNe Ia. But we note that in the computations of Newsham et al. the accreted matter does not interact with the core material; this seems to be at odds with the general wisdom suggesting that diffusion or other mixing processes (e.g., shear mixing) should occur at the interface.

As already mentioned, Bours et al. [64] attempted to compare the rates of SNe Ia produced via the SD channel using several sets of retention efficiencies and values of the common envelope parameter or -parameter of Eq. (59*) employed by different groups in the population synthesis codes. For a Milky-Way like galaxy, none of the computations produced an SNe Ia rate larger than  10% of that inferred for the galaxy –  [87]. Claeys et al. [112] have shown that SD SNe Ia start to dominate SN Ia rate if common envelopes are inefficient (); the rate of SNe Ia also then becomes comparable to the rate in the field-galaxy dominated sample, but remains a factor of three lower than the rate in galaxy clusters. Only unrealistic unlimited accretion onto WDs allows one to obtain an SNe Ia rate compatible with the observed galaxy-cluster rate.

Yoon and Langer [861, 862] and Hachisu et al. [260] have shown that if accreting WDs rotate differentially, their mass can grow without explosion up to , unless they lose angular momentum through secular instabilities. Supernova explosions happen during the spin-down of such a WD (“spin-up/spin-down” scenario). If true, this is a possible way to accumulate enough mass for SN Ia in the SD-scenario and even to provide an explanation for some “super-Chandrasekhar” SN Ia. Conclusions of Yoon and Langer were questioned by Piro [591] who claims that baroclinic instabilities and/or the shear growth of small magnetic fields provide sufficient torque to bring the WD very close to rigid-body rotation in a time shorter than the typical accretion time scale; in the latter case the mass of the non-exploding WD is limited by [169], leaving no room for efficient mass accumulation. Thus, the “spin-up/spin-down” scenario is still highly questionable.

There is ample observational evidence that at least a fraction of SN Ia is not produced by semi-detached systems or systems with a detached giant (symbiotic stars). The major objection to the SD scenario comes from the fact that no hydrogen is observed in the SNe Ia spectra, while it is expected that up to several tenths of of H-rich matter may be stripped from the companion by the SN shell, see [563, 436, 692] and references therein. Hydrogen, if present, may be discovered both in very early and late optical spectra of SNe and in radio- and X-ray [60, 413]. Radio emission is produced by a synchrotron mechanism from the region of interaction between the high-velocity SN ejecta and the much more slowly moving circumstellar medium formed by the wind of the SN precursor or the wind of its companion [103, 102]. The peak monochromatic emission depends on the ratio of the rate of mass loss in the wind and its velocity and allows one to estimate the pre-explosion . Panagia et al. [565] find a upper limit of on a steady pre-supernova mass-loss rate for 27 SNe. This excludes relatively massive companions to WDs. On the other hand, such do not exclude systems, in which a fraction of mass lost by low-mass companions is accreted onto WDs and retained (see above).

Non-accreted material blown away from the system before the explosion would become circumstellar matter (CSM), hence the detection of CSM in SN Ia spectra would lend credence to the SD-model. Ionization and latter recombination of CSM would manifest itself as time-variable absorption features detectable (in principle) via multi-epoch high-spectral-resolution observations. However, it was noted that for confidential differentiation between CSM and ISM features for an individual SN, multi-epoch observations are needed; this condition was not always satisfied in early CSM detection reports. Currently, it is estimated that 18 ± 11% of events exhibit time-variability that can be associated with CSM [742]. However, immediately after the very first report of CSM-discovery in SN 2006X [572], Chugai [110] noted that for the wind densities typical for red giants, the expected optical depth of the wind in Na i lines is too small for their detection, while under the same conditions the optical depth of the Ca ii 3934 Å absorption line is sufficient for detection, and concluded that the Na i and Ca ii absorption lines detected in SN 2006X could not be formed in the red giant wind and are most likely related to clouds at distances exceeding the dust evaporation radius (). The problem of the interpretation of sodium-line–variability observations still remains open.

Figure 16: Upper panel: model light curve of a SN Ia having collided with a red giant companion separated by . The luminosity due to the collision is prominent at times days. The black dashed line shows the analytic prediction for the early phase luminosity. Lower panel: Signatures of interaction in the early broadband light curves of SN Ia for a red-giant companion at (green lines), a main-sequence companion at (blue lines), and a main-sequence companion at (red lines). The ultraviolet light curves are constructed by integrating the flux in the region 1000 – 3000 Å and converting to the AB magnitude system. For all light curves shown, the viewing angle is 0. Image reproduced with permission from [340*], copyright by AAS.

Recently, Kasen [340] suggested an important test of possible collisions of SN ejecta with companion stars, by observing emissions from the impact of shocks and SN debris due to dissipation of the kinetic energy and reheating of the gas, occurring in the immediate vicinity of the star, at . Immediately after the explosion, an X-ray outburst with at (0.1 – 2) keV lasting from minutes to hours is produced. Later, radiative diffusion from deeper layers of the shock-heated ejecta produces optical/UV emissions that exceed the radioactively-powered luminosity of the SN for a few days after the explosion. The properties of the emission provide a straightforward measure of the distance between the stars and the companion’s radius (in the case of RLOF). The light-curves modeled by Kasen for different companions are shown in Figure 16*. The effect is prominent only for viewing angles looking down upon the shocked region () and, thus, may be found in statistical studies. Bianco et al. [46] applied Kasen’s models to a sample of 87 spectroscopically-confirmed SNe Ia from the Supernova Legacy Survey [16] and ruled out the contribution from white dwarf – red giant binary systems (separation ) to SNe Ia explosions greater than 10% at the 2 level, and greater than 20% at the 3 level. As noted by Kasen, optical detection of main-sequence companions () will be challenging, requiring the measurement of subtle differences in the light curves at times  2 days. Hayden et al. [272] applied the SN Ia light-curve template to 108 objects with well-confirmed early-light curves from the Sloan Digital Sky Survey (SDSS-II [209]) and found no shock signatures. This study limited the mass of putative companions to exploding objects by about and “strongly disfavors red-giant companions”.

In the SD scenario, the former companion of the exploding WD is expected to survive. It must have a high spatial velocity – (100 – 300) km s^–1 [435] and a high luminosity, see Table 7.

In Table 7 we present a summary of the expected properties of companions compiled by Schaefer and Pagnotta [679] (see this paper for a brief review of attempts to find ex-companions). As for now, all attempts to find ex-companions failed (in the particular case of SNR 0509-67.5 in SMC studied by Schaefer and Pagnotta, the non-detection was at the 5- level).

A very “popular” object for the search of the former companion is Tycho’s SN, but even for this close remnant of a relatively recent event (AD 1572) all claimed detections were disproved, see, e.g., [355]. For another galactic SN Ia candidate – Kepler SN (AD 1604), the possible candidates with are excluded [354].

Badenes et al. [18] noted that most models for SN Ia in the SD-scenario predict optically thick outflows from the white dwarf surface with velocities above 200 km s^–1. Such outflows should excavate large low-density cavities around the progenitors. However, Badenes et al. found that such cavities are incompatible with the dynamics of the forward shock and the X-ray emission from the shocked ejecta in the known SNe Ia remnants in the Galaxy (Kepler, Tycho, SN 1006), LMC (0509-67.5, 0519-69.0, N103B), and in M31 (SN 1885). No sources corresponding to the pre-explosion objects have been found in the archival Chandra data as yet [529]. On the other hand, e.g., spectropolarimetric observations of type Ia SN 2012fr suggest a simple axial symmetry of the explosion, which, together with the observed features of Ca and Si in the spectrum, is inconsistent with the DD merger scenario [473].

It was speculated that in the case of the above-mentioned “spin-up/spin-down” scenario for semidetached progenitors of SNe Ia, if the remnant of the donor star becomes a WD, during the spin-down phase it can become too dim for detection and lose traces of hydrogen [153, 328, 152]. However, the viscous time scale of a WD is still too uncertain for definite conclusions [477].

As noted first by Canal et al. [84], combinations of masses of accretors and donors enabling stable mass transfer and H-burning on the WD surface exist only for  10⁹ years after the star-formation burst. This means that the current rate of SNe Ia with semidetached progenitors in early-type galaxies should be small, much lower than observed. This conclusion is apparently supported by the deficit of observed supersoft X-ray sources in the latter galaxies [244, 151]: while the number of supersoft sources necessary to be consistent with the rate of SNe Ia in MW or nearby galaxies is  1000, less than about 100 are observed. We note with caution that, although WDs that accrete H at the rates allowing its stable burning are conventionally, following [800], identified with SSS, this inference still needs to be confirmed by rigorous modeling of the spectra of accreting WDs. Another test of the contribution of accreting WDs to the rate of SNe Ia in early-type galaxies was suggested by Woods and Gilfanov [847]: since accreting WDs, even if bloated, have photospheric temperatures  10⁵ – 10⁶ K, they are powerful sources of ionising UV emission. Then, if a significant population of steadily-burning hydrogen WDs exists, strong emission in He ii 4686 recombination lines should be expected from the interstellar medium. Application of this test to the sample of about 11 500 co-added spectra of passively evolving galaxies allowed one to limit the contribution of SD-scenarios to the total SNe Ia rate by (5 – 10)% [323].

7.3.2 Merger of white dwarfs and the double-degenerate scenario for SN Ia

A compelling scenario for the accumulation of involves two CO WDs in a close binary system that can merge in the Hubble time due to the orbital angular momentum loss via GWR [784*, 785*, 300*, 825*]. Theoretical models suggested that a proper candidate WD pair with total mass may be found among about 1000 field WDs with stellar magnitudes [520*]. This number of (super)Chandrasekhar merging pairs also appeared sufficient to obtain the observed rate of SNe Ia in a Milky-Way—like galaxy. It was speculated in [784*], that, due to the fact that degenerate dwarfs expand with mass loss (), if the mass ratio of components is , the merger of pairs of WDs occurs on the dynamical time scale (in a time comparable to a few orbital periods) and it was suggested that the lighter of the two dwarfs transforms into a “heavy disc” or “envelope” from which matter accretes onto the central object. This inference was confirmed by SPH calculations of Benz et al. [42] and many other studies. However, it was shown for 1D non-rotating models that the central C-ignition and SN Ia explosion are possible only for an accretion rate onto the central object  [533], while it was expected that in the merger products of dwarf binaries is close to because of high viscosity in the transition layer between the core and the disk [489]. For such a high , nuclear burning will start at the core edge and propagate inward. If this is not the case, an off-center ignition may be due to the Kelvin–Helmholtz contraction of the inner region of the envelope on a thermal timescale of 10³ – 10⁴ yr, which compresses the base of the envelope. Nuclear burning produces an ONeMg WD [699]. The latter will collapse without producing an SN Ia [311]. However, note that the analysis of the role of the angular momentum deposition into the central object and its thermal response to accretion in the “canonical” model led Piersanti, Tornambé and coauthors [589*, 767] to the conclusion that, as a result of the WD spin-up, instabilities associated with rotation, deformation of the WD, and AML by a distorted configuration via GWR, accretion occurs not as a continuous process but episodically and, hence, the resulting “effective” accretion rate onto the WD decreases to a few . At this , a close-to-center ignition of carbon becomes possible. However, as noted by Isern et al. [310], the difficulty of this scenario is the avoidance of mass loss from the system. The off-center C-ignition can also be avoided if, after the disruption of the secondary, several conditions are fulfilled: the local maximum temperature at the interface between the core (the former primary) and the envelope (the former secondary) must be lower than the critical limit for carbon ignition, the time scale of the neutrino cooling at the core/envelope interface is shorter than the angular momentum loss time scale, the mass-accretion rate from the disc must be lower than [863].

The problems of the SD scenario mentioned above, the discovery of several pairs of WDs with total mass close to MCh and merger time shorter than the Hubble one (see Section 8), and the discovery of several SNe Ia of super-Chandrasekhar mass [291, 280, 858, 866, 676, 711, 752, 677] lent more credence to the DD-scenario. However, the current formulation of this scenario is quite different from early versions due to several circumstances. First, the accurate posing of initial conditions for mass transfer and angular momentum conservation allowed one to follow the merger process for  100 orbits prior to the coalescence, to trace variation of the accretion rate during this phase, and to show that explosive events are possible before the coalescence or shortly after the merging [496, 166, 255*]. Second, it was realized that the stability of mass transfer also depends on the efficiency of spin-orbit coupling [464*]. In addition, if the circularization radius of the accretion stream is less than the accretor’s radius, a direct impact occurs. Otherwise, the accretion proceeds via disk. Analytical estimates of different regimes of mass exchange in WD binary systems are presented in Figure 17*.

Figure 17: Estimates of regimes of mass transfer in WD binaries. Instantaneous tidal coupling is assumed. For longer time scales of tidal coupling, the stability limit in the plot shifts down [464*]. The filled squares mark initial positions of the models studied in the quoted paper. Image reproduced with permission from [129], copyright by AAS.

Yet another very important, even crucial, fact concerns the role of the possible detonation of He. It was found by Livne and Glasner [439, 440] that accretion of He onto a CO WD at a rate close to results in the accumulation of a degenerate He layer, which detonates when its mass becomes . The helium burning sweeps around the WD and converging shock waves initiate a compression wave that can result in central carbon detonation. For a certain time, this model involving sub-Chandrasekhar mass accretors (nicknamed “edge-lit detonations”, ELD or “double-detonations”) that may occur in a MW-like galaxy at a rate of  10^–3 yr^–1, was considered as one of the alternative mechanisms for SD or DD scenarios for SNe Ia explosions. But it was shown by Höfflich et al [286, 285] that the behavior of light curves produced in this model did not resemble any of the known by mid-1990ies SNe Ia, and this mechanism was rejected. However, it was discovered later that there exists a rather numerous class of events, called SN Iax, observational properties of which are, to a certain extent, consistent with the model in which a CO WD accretes from a He-star, the burning starts in the center, the flame propagates to the surface, ejects some matter, but then damps out (called the“failed deflagration” scenario). Such SNe presumably do not consume the whole WD and leave bound remnants [203].

Interest in the “double-detonation” model re-surged in relation to the outcome of He accretion on a WD in interacting double degenerates [49*]. The studies by Fink et al. [199*] show that He detonation triggers core detonations in models ranging in core mass from 0.810 up to with corresponding shell masses from 0.126 down to . Similar estimates were obtained by Moll et al. [492]. It is possible that even the least massive CO WD () may experience double detonations if accreted layers of He are sufficiently massive, close to , as is shown by Sim et al. [713]. However, we note that the robustness of these results was questioned by Waldman et al. [818], since triggering a detonation must be investigated on scales much smaller than those used, e.g., by Fink et al. [199*]. Nevertheless, it is quite possible that the primary objects for the DD-scenario may be pairs of CO-accretors and helium or hybrid donors.

Currently, systems with direct impact are the best studied (Figure 17*). As found by Guillochon et al. [255] and later explored in more detail by Dan et al. [130*, 128], in the direct impact systems He detonations in the torus of the accreted He with mass can occur several orbits before merging, if the nuclear timescale becomes shorter than the local dynamical timescale . If the subsequent shock compression is strong enough, the second detonation may follow in the CO core and lead to SN Ia. Another possibility is “surface detonation” in the layer of accreted matter immediately after merging with the subsequent central detonation.

In Figure 18* we show a “map” of the outcomes of mergers of white dwarfs after Dan et al. [128]. These authors systematically explored 225 WD models with different mass and chemical composition. Referring the interested reader for more details about the assumptions, computations and results to the original paper, we briefly consider the merger products shown in the Figure (moving from bottom left to upper right). Note that some of the results were well known earlier.

Figure 18: Outcomes of merger of WD binaries depending on the mass and chemical composition of the components. Systems in the hatched region are expected to experience He-detonations during mass transfer or at the time of merger. The numbers near the arrows indicate relevant timescales. Image reproduced with permission from Figure 1 of [128], copyright by the authors.

If the mass of the merger product of two He WDs exceeds , the He burns in flashes propagating inward and, finally, ignites in the core [667*]. The object turns into a low-mass compact He star. If the mass is larger than , an extended envelope forms after core He exhaustion [552*]. The final product is a CO WD with a relatively thick He mantle. Low-mass He stars formed via merger are identified with single sdB stars. Mergers of two hybrid or a CO WD and a He WD with the total mass lower than are suggested to produce sdO subdwarfs [743]. As a confirmation of the merger scenario, one can consider the discovery of a rapidly rotating sdB star SB 290 [228]. On the other hand, recently it was shown that the merger of two He WDs with mass as small as may result in the dynamical He-burning and, possibly, in detonation [287]. Incomplete He-burning is expected in the majority of WD explosions in this case, and ⁴⁰Ca, ⁴⁴Ti, or ⁴⁸Cr, rather than ⁵⁶Ni will be the predominant burning products.

The merger products of CO and He WDs with total mass exceeding form He giants. It is suggested that this is one of the ways to form R CrB stars [825*, 306]. Another path to R CrB stars may involve the merger of a CO WD and a hybrid WD [668, 566]. The fate of R CrB stars is determined by the “competition” between the core growth due to He-burning in the shell and mass loss from an extended envelope. In rare cases where CO core accumulates , a peculiar supernova explosion can be expected. Otherwise, a massive CO WD forms.

There is a small patch in Figure 18* occupied by CO WD binaries with total mass lower than . It is expected that the merger of their components will result in the formation of a massive single WD (see [322] and references therein). Several unusually massive WDs – GD 362 [223], RE J0317–853 [393], which have mass estimates close to the Chandrasekhar mass, are potential descendants of these pairs.

Helium detonations can result from the merging of WD pairs with parameters lying within the hatched region in Figure 18*. These include CO accretors and most massive helium and hybrid WD. Double-detonations are possible for most massive objects. The known problem of this mechanism of production of sub-Chandrasekhar SN Ia is that to make the modeled values consistent with observations of SN Ia, a fine tuning of He and C abundances in the accreting matter is required [387]. It is important to note that the newly-formed He WDs have thin hydrogen envelopes that may be stably transferred onto the companion prior to the merger. As found by Shen et al. [700], this material is ejected from the binary system and sweeps up the surrounding interstellar medium before the SN Ia explosion. The resulting circumstellar medium profiles closely match the ones found in some SN Ia.

As mentioned above, the merger of CO WD pairs with may result in SN Ia if certain conditions on the accretion rate at the core/envelope interface are fulfilled [589, 767, 863]. According to Dan et al. [128], the merging CO WDs detonate only if . Then they may be identified with super-Chandrasekhar SN Ia. However, we note that the consideration of abundances of elements synthesized in SNe Ia places certain constraints on the exploding object mass. For instance, the solar abundance of Mn is consistent with the assumption that about 50% of all SNe Ia are explosions of near-Chandrasekhar mass WDs [686].

Finally, it is possible that the merger involves a CO and an ONe WDs. Such (very rare) mergers can give rise to “hybrid SNe” which, in the presence of a He layer, will manifest themselves as a kind of SN Ib or, if the He-layer is absent, as Ia.

We should mention that there exist merger calculations leading to SN Ia for configurations that did not produce explosions in the simulations by Dan et al. [128].

Pakmor et al. [560] simulated a merger of and CO WD with surface layers of He and found that the merger is violent, and He detonation forms close to the surface of the core of the primary in the region compressed by a shock in the He layer after only 7 orbits, i.e., much earlier than in the study of Dan and coauthors, and also triggers detonation of C, not encountered for this combination of dwarfs in the study of Dan et al. Based on their result, Pakmor et al. claim that the merger of CO WDs with low-mass He envelopes or a CO and a He WD thus provide a unified model for normal and fast-declining SN Ia. Kromer et al. [388] were able to reproduce the early-time observables of a subluminous SN Ia with slowly-evolving light curve by violent merger of “naked” CO WD of and , which leads initially to the local C-ignition at the hottest (compressed and heated) region at the accretor surface; then detonation develops around the hot bubble and consumes the entire merged structure.

We note that Scalzo et al. [678] estimated that were ejected by 13 SN Ia studied by them. The correlation of ejected mass and ⁵⁶Ni in this set of SN Ia in most of the cases did not completely comply with the double-detonation model (e.g., [199]). The conclusion of Scalco et al. is that at least two formation channels, one for sub-Chandrasekhar mass SN Ia and another for Chandrasekhar mass and more massive SN Ia, are required. An interesting new channel to SN Ia discussed in this respect are direct collisions of WD, which are possible in triple systems consisting of two white dwarfs accompanied by a third star, thanks to Kozai resonances, which can reduce the time to the merger or collision [350, 399, 630]. Both sub-Chandrasekhar-mass and super-Chandrasekhar-mass SNe Ia could arise through this channel. This inference finds support in the discovery of doubly-peaked (“two-horn”) line profiles in several SN Ia [159]; such profiles are expected if two WD detonate [399].

Having in mind still uncertain conditions for the initiation of C-burning, Moll et al. [491*] and Raskin et al. [628] considered SN Ia explosions initiated in the merger process of CO WD, either before disruption of the secondary (dubbed by them “peri-merger” detonations) or after complete disruption of the latter when the exploding object is surrounded by an envelope (a “disk”). An important conclusion of [491] is that in the case of “peri-merger” detonations, a model for configuration with masses generally reproduced normal SN Ia. Explosions of more massive configurations ( and ) yielded over-luminous SN Ia. Post-merger explosions produced peculiar SN Ia with features defined, to a great extent, by the disk structure; these SN Ia do not conform with Phillips [584] relation between the peak B-band magnitude and the decline rate of the light curve found for the “standard” SN Ia. In both cases, the observed brightness of events strongly depends on the viewing angle. The mergers of massive systems may be invoked to explain super-Chandrasekhar SNe Ia.

The difference in the results of simulations by different authors may be due to different initial conditions of models [130*, 629] and resolution of the codes used [130, 560]. Especially important is whether the binary components corotate at merger onset. These problems clearly warrant further examination.

Note, in addition, that the intermediate mass components of close binaries, before becoming white dwarfs, pass through the stage of a helium star. If the mass of the helium star exceeds , it expands to giant dimensions after exhaustion of He in the core and can overfill the Roche lobe and, under proper conditions, can stably transfer mass to the secondary companion [301]. For the range of mass-accretion rates expected for these stars, both the conditions for stable and unstable helium burning may be fulfilled. In the former case, the accumulation of and a SN Ia become possible, as it was shown explicitly by Yoon and Langer [860]. However, the probability of such a SN Ia is only [67].

Core-degenerate scenario.

As a subclass of merger scenarios for SN Ia, one may consider the “core-degenerate” (CD) scenario. In the original form this scenario involves the merger of a WD with the core of an AGB star [730] accompanied by an explosion in the presence of remainders of the AGB star, or a common envelope, or nearby shells of matter presumably ejected shortly before the explosion. It was aimed at explaining SN 2002ic with signatures of circumstellar matter in the spectrum [437, 111]. Only 16 objects of SN 2002ic class were known at the time of writing of this review [712]. Further development of the model resulted in a picture in which the merger product is a rapidly-rotating WD with mass close to or above . The fast rotation prevents immediate explosion, which happens only after a long spin-down stage, like in the spin-up/spin-down scenario [308]. The core-degenerate scenario is claimed to explain successfully, the basic features of SN 2011fe [723].

To summarize, the problem of progenitors of SNe Ia is still unsettled. Large uncertainties in the model parameters involved in the computation of the evolution leading to the SN Ia explosion and in calculations of the explosions themselves, do not allow us to exclude any type of progenitors. The existence of at least two families of progenitors is not excluded by observations (see, e.g., [458] for the latest discussion). A high proportion of “peculiar” SN Ia (36 ± 9)% [418] suggests a large spread in ignition conditions in the exploding objects that also may be attributed to the diversity of progenitors. Note that a high fraction of “peculiar” SNe Ia observed and their environmental dependence [641] cast certain doubts on their accuracy as standard candles for cosmology.

7.4 Ultra-compact X-ray binaries

The suggested channels for the formation of UCXBs in the field are, in fact, “hybrids” of scenarios presented in Figures 7* and 9*. In UCXB progenitors, the primary forms a neutron star directly via core collapse or via accretion-induced collapse (AIC) of a white dwarf, while the secondary is not massive enough to form a NS. Then, several scenarios similar to the scenarios for the systems with the first-formed white dwarf and driven by systemic AML via GWR are open. Usually, two main scenarios are considered in which either a white dwarf or a low-mass helium-star companion to the NS overflows the Roche lobe. A low-mass companion to a neutron star may also overfill the Roche lobe at the end of the main sequence and become a He-rich donor. The dominant scenario differs from one population synthesis study to another, depending, mainly, on assumptions on the initial binary parameters, common envelopes, the kick velocity of nascent neutron stars, conditions for the onset of mass transfer, retention efficiency of the matter by the NS, etc. In fact, in all cases evolution of the donors leads to their transformation into similar low-mass degenerate objects.

For the most recent studies of the origin, evolution and stability of UCXB and a review of earlier work we refer the reader to [805, 806, 804, 92, 275]. We specially note that in [36] it was found that the dominant accretors in UCXB can be black holes formed via AIC of a neutron star. However, this result can be a consequence of the choice of the upper limit for a neutron-star mass of , close to the lower bound of current theoretical estimates for the maximum mass of NS, see, e.g., [95].

For UCXBs, like for AM CVn stars, an analysis of details of the chemical composition of donors, which show up in their optical or X-ray spectra, seems to be a promising way for distinguishing between the possible progenitors [515, 521, 374]. For UCXBs, a different chemical composition of donors may manifest itself in properties of thermonuclear explosions (type I X-ray bursts) that occur on the surface of the accreting neutron stars with low magnetic fields [47]. We discuss this point in more detail in Section 9 both for UCXB and AM CVn stars.

In globular clusters, UCXBs are most likely formed by dynamical interactions, as first suggested by Fabian et al. [182]; see, e.g., [317, 441] and references therein for the latest studies on the topic.