7 Formation of Short-Period Binaries with a White-Dwarf Components

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

As Figure 2View Image shows, compact binary stars emit gravitational waves within the sensitivity limits for space-based detectors if their orbital periods are from ∼ 20 s to 20,000 s. This means that in principle all AM CVn-stars, UCXBs, a considerable fraction of all CVs with measured orbital periods, and some DD and SD + WD systems would be observable in GWs in the absence of confusion noise and sufficient sensitivity of detectors.

Though general relativity predicted that binary stars have to be the sources of gravitational waves in the 1920s, this prediction became a matter of actual interest only with the discovery of the Porb ≈ 81.5 min cataclysmic variable WZ Sge by Kraft, Mathews, and Greenstein in 1962 [200], who immediately recognized the significance of short-period binary stars as testbeds for gravitational waves 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 binary HZ 29 = AM CVn (Porb ≈ 18 min) by Smak in 1967 [380Jump To The Next Citation Point]. Smak [380] and Paczyński [301Jump To The Next Citation Point] speculated that the latter system is a close pair of white dwarfs, without specifying whether it is detached or semidetached. Faulkner et al. [101] inferred the status of AM CVn as a “double-white-dwarf semidetached” nova. AM CVn was later classified as a cataclysmic variable after flickering typical for CVs was found for AM CVn by Warner and Robinson [442]9 and became a prototype of a subclass of binaries10.

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 1970s (see Section 3.5). A spiral-in of components in common envelopes allowed to explain how white dwarfs – former cores of highly evolved stars with radii of ∼ 100R ⊙ – may acquire companions separated by ∼ R ⊙ only (for pioneering work see [302Jump To The Next Citation Point443Jump To The Next Citation Point417Jump To The Next Citation Point418Jump To The Next Citation Point162Jump To The Next Citation Point444Jump To The Next Citation Point]). We recall, however, 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, thought to be the sole source of energy for the loss of the envelope as described in Section 3.5. Though full-scale hydrodynamic calculations of a common-envelope evolution exist, for instance a series of papers by Taam and coauthors published over more than two decades (see [369368] and references therein), the process is still very far from comprehension.

We recall also that the stability and timescale of mass-exchange in a binary depends on the mass ratio of components q, the structure of the envelope of Roche-lobe filling star, and possible stabilizing effects of mass and momentum loss from the system [413146470Jump To The Next Citation Point129137173Jump To The Next Citation Point103Jump To The Next Citation Point48]. For stars with radiative envelopes, to the first approximation, mass exchange is stable if q ≲ 1.2; for 1.2 ≲ q ≲ 2 it proceeds in the thermal time scale of the donor; for q ≳ 2 it proceeds in the dynamical time scale. Mass loss occurs on a 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. It is currently commonly accepted, despite a firm observational proof is lacking, that the distribution of binaries over q is even or rises to small q (see Section 5). Since typically 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 lower than the mass-loss rate by the donor, the overwhelming majority (∼ 90%) of close binaries pass in their evolution through one to four stages of a common envelope.

An “initial donor mass – donor radius at RLOF” diagram showing descendants of stars after mass-loss in close binaries is presented in Figure 1View Image. We should remember here that solar metallicity stars with M ≲ 0.95 M ⊙ do not evolve past the core-hydrogen burning stage in Hubble time.

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Figure 5: Formation of close binary dwarfs and their descendants (scale and colour-coding are arbitrary).

Formation of compact binaries with WDs.   A flowchart schematically presenting the typical scenario for formation of low-mass compact binaries with white-dwarf components and some endpoints of evolution is shown in Figure 5View Image. Of course, not all possible scenarios are plotted, but only the most probable routes to SNe Ia and 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 overflow their Roche lobes when they have He- or CO-cores. In stars with a mass below (2.0– 2.5)M ⊙, helium cores are degenerate and if these stars overflow the Roche lobe prior to He-ignition, they produce helium white dwarfs. Binaries with non-degenerate He-core donors (M ≳ (2.0 –2.5)M ⊙) first form a He-star + MS-star pair that may be observed as a subdwarf (sdB or sdO) star with MS companion. When the He-star completes its evolution, a pair harbouring a CO white dwarf and a MS-star appears.

If after the first common-envelope stage the orbital separation of the binary a ≃ several R ⊙ and the WD has a low-mass (≲ 1.5M ⊙) MS companion the pair may evolve into contact during the hydrogen-burning stage or shortly after because of loss of angular momentum by a magnetically coupled stellar wind and/or GW radiation. If, additionally, the mass-ratio of components is favourable for stable mass transfer a cataclysmic variable may form. If the WD belongs to the CO-variety and accreted hydrogen burns at the surface of the WD stably, the WD may accumulate enough mass to explode as a type Ia supernova; the same may happen if in the recurrent outbursts less mass is ejected than accreted (the so-called SD scenario for SNe Ia originally suggested by Whelan and Iben [451]; see, e.g., [219470Jump To The Next Citation Point128134Jump To The Next Citation Point103Jump To The Next Citation Point135217] and references therein for later studies).

Some CV systems that burn hydrogen stably or are in the stage of residual hydrogen burning after an outburst may be also observed as supersoft X-ray sources (see, e.g., [426340177471179178]).

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.

The outcome of the evolution of a CV is not completely clear. It was hypothesized that the donor may be disrupted when its mass decreases below several hundredth of M ⊙ [359]. Note, that for q ≲ 0.02, that may be attained in Hubble time, and the conventional picture of mass exchange may become non-valid, since the circularization radius becomes greater than the outer radius of the disk. Matter flowing in from the companion circularises onto unstable orbits. At q ≈ 0.02, matter is added at Rcirc onto orbits that can become eccentric due to the 3:1 resonance. At q ≈ 0.005 the circularization radius approaches the 2:1 Lindblad resonance. This can efficiently prevent mass being transferred onto the compact object. These endpoints of the evolution of binaries with low-mass donors, were, in fact, never studied.

The second common envelope may form when the companion to the WD overfills its Roche lobe. If the system avoids merger and the donor had a degenerate core, a close binary WD (or double-degenerate, DD) is formed. The fate of the DD is solely defined by GWR. The closest of them may be brought into contact by AML via GWR. The outcome of the contact depends on the chemical composition of the stars and their masses. The lighter of the two stars fills the Roche lobe first (by virtue of the mass–radius relation R ∝ M − 1∕3). For a zero-temperature WD the condition of stable mass transfer is q < 2∕3 (but see the more detailed discussion in Section 9). The merger of the CO-WD pair with a total mass exceeding MCh may result in a SN Ia leaving no remnant (“double-degenerate SN Ia scenario” first suggested by Webbink [443Jump To The Next Citation Point] and independently by Tutukov and Yungelson [418Jump To The Next Citation Point]) or in an AIC with formation of a single neutron star [261Jump To The Next Citation Point]. The issue of the merger outcome for Mtot > MCh still remains an unsolved issue and a topic of fierce discussion, see below. For total masses lower than MCh the formation of a single WD is expected.

If in a CO + He WD dwarfs pair the conditions for stable mass exchange are fulfilled, an AM CVn system forms (see for details [417Jump To The Next Citation Point422Jump To The Next Citation Point281Jump To The Next Citation Point251Jump To The Next Citation Point122Jump To The Next Citation Point]).

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Figure 6: The age of merging pairs of helium WDs. Two components of the distribution correspond to the systems that experienced in the course of formation two or one common envelope episodes, respectively.

The current Galactic merger rate of close binary WDs is about 50% of their current birth rate [286Jump To The Next Citation Point423Jump To The Next Citation Point]. It is not yet clear how the merger proceeds; it is possible that for He + He or CO + He pairs a helium star is an intermediate stage (see, e.g., [126]). It is important in this respect that binary white dwarfs at birth have a wide range of separations and merger of them may occur gigayears after formation. Formation of helium stars via merger may be at least partially responsible for the ultraviolet flux from the giant elliptical galaxies, where all hot stars finished their evolution long ago. This is illustrated by Figure 6View Image which shows the occurrence rate of mergers of pairs of He-WDs vs. age.

If the donor has a nondegenerate He-core (M ≳ (2.0– 2.5) M ⊙) and the system does not merge, after the second CE-stage a helium subdwarf + WD system may emerge. If the separation of components is sufficiently small, AML via GWR may bring the He-star into contact while He is still burning in its core. If MHe ∕Mwd ≲ 1.2, stable mass exchange is possible with a typical M˙ ∼ 10− 8M ⊙ yr−1 [370Jump To The Next Citation Point]. Mass loss quenches nuclear burning and the helium star becomes “semidegenerate”. An AM CVn-type system may be formed in this way (the “nondegenerate He-core” branch of evolution in Figure 5View Image). 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 [383]11. If the He-star completes core He-burning before RLOF, it becomes a CO-WD. In Figure 5View Image it “jumps” into the “double degenerate” branch of evolution.

Table 5: Occurrence rates of SNe Ia in the candidate progenitor systems (in yr–1), after [468]. SG stands for sub-giant, RG for red giant, and XRS for X-ray source.

Donor CO-WD MS/SG He-star He-WD RG

Counterpart Close binary WD Supersoft XRS Blue sd AM CVn Symbiotic star
Mass transfer mode Merger RLOF RLOF RLOF Wind
Young population 10–3 10–4 10–4 10–5 10–6
Old population 10–3 10–5 10–6

Type Ia supernovae.   Table 5 summarizes order of magnitude model estimates of the occurrence rate of SNe Ia produced via different channels. For comparison, the rate of SNe Ia from wide binaries (symbiotic stars) is also given. The estimates are obtained by a population synthesis code used before in, e.g., [421470Jump To The Next Citation Point423Jump To The Next Citation Point] for the value of common envelope parameter αce = 1. The differences in the assumptions with other population synthesis codes or in the assumed parameters of the models result in numbers that vary by a factor of several; this is the reason for giving only order of magnitude estimates. The estimates are shown for T = 10 Gyr after the beginning of star formation.

A “young” population had a constant star formation rate for 10 Gyr; in the “old” one the same amount of gas was converted into stars in 1 Gyr. Both populations have a mass comparable to the mass of the Galactic disk. We also list in the table the types of observed systems associated with a certain channel and the mode of mass transfer. These numbers have to be compared to the inferred Galactic occurrence rate of SNe Ia: (4 ± 2) × 10–3 yr–1 [54]. Table 5 shows that, say, for elliptical galaxies where star formation occurred in a burst, the DD scenario is the only one able to respond to the occurrence of SNe Ia, while in giant disk galaxies with continuing star formation other scenarios may contribute as well.

For about two decades since the prediction of the possibility of the merger of pairs of white dwarfs with total mass ≥ MCh, the apparent absence of observed DDs with proper mass and merger times shorter than Hubble time was considered as the major “observational” difficulty for the DD scenario. Theoretical models predicted that it may be necessary to investigate for binarity up to 1,000 field WDs with V ≲ 16 –17 for finding a proper candidate [288Jump To The Next Citation Point]. Currently, it is likely that this problem is resolved (see Section 8).

The merger of pairs of WDs occurs via an intermediate stage in which the lighter of the two dwarfs transforms into a disc [417Jump To The Next Citation Point29261Jump To The Next Citation Point237Jump To The Next Citation Point] from which the matter accretes onto the central object. It was shown for one-dimensional non-rotating models that the central C-ignition and SN Ia explosion are possible only for M˙a ≲ (0.1 –0.2)M˙Edd [292402]. But it was expected that in the merger products of binary dwarfs M˙a is close to M˙Edd ∼ 10−5 M ⊙ yr−1 [261] because of high viscosity in the transition layer between the core and the disk. For such an M˙a, the nuclear burning will start at the core edge, propagate inward and convert the dwarf into an ONeMg one. The latter will collapse without a SN Ia [169]. However, an analysis of the role of deposition of angular momentum into a central object by Piersanti and coauthors [316Jump To The Next Citation Point317Jump To The Next Citation Point] led them to conclusion that, as a result of the spin-up of rotation of the WD, instabilities associated with rotation, deformation of the WD, and AML by a distorted configuration via GWR, an Ma˙ that is initially ∼ 10−5M ⊙ yr−1 decreases to −7 −1 ≃ 4 × 10 M ⊙ yr. For this ˙ Ma close-to-center ignition of carbon becomes possible. The efficiency of the mechanism suggested in [316317] is disputed, for instance, by Saio and Nomoto [365] who found that an off-center carbon ignition occurs even when the effect of stellar rotation is included, if M˙a > 3 × 10− 6M ⊙ yr− 1. The latter authors find that the critical accretion rate for the off-center ignition is hardly changed by the effect of rotation. The problem has to be considered as unsettled until a better understanding of redistribution of angular momentum during the merger process will become available.

Because of a long absence of apparent candidates for the DD scenario and its theoretical problems, the SD scenario is often considered as the most promising one. However, it also encounters severe problems. Even stably burning white dwarfs must have radiatively driven winds. At M˙accr ≲ 10− 8M ⊙ yr−1 all accumulated mass is lost in nova explosions [332461]. Even if M˙accr allows accumulation of a He-layer, most of the latter is lost after the He-flash [16555315], dynamically or via the frictional interaction of the binary components with the giant-size common envelope. As a result, mass accumulation efficiency is always < 1 and may be even negative. On the other hand, it was noted that the flashes become less violent and more effective accumulation of matter may occur if mass is transferred on a rate close to the thermal one or the dwarf is rapidly rotating [162Jump To The Next Citation Point470173134Jump To The Next Citation Point103465464]. Thus, crucial for this SN Ia scenario are the range of donor masses that may support mass-loss rates “efficient” for the growth of WD, mechanisms for stabilizing mass loss in the necessary range, convection and angular momentum transfer in the accreted layer that define the amount of mass loss in the outbursts, and the amount of matter that escape in the wind12. If the diversity of SNe Ia is associated with the spread of mass of the exploding objects, it would be more easily explained in a SD scenario, since the latter allows white dwarfs to grow efficiently in mass by shell burning, which is stabilized by accretion-induced spin-up. This inference may be supported by the discovery of the “super-Chandrasekhar” mass SN Ia SN 2003fg (mass estimate ∼ 2M ⊙ [154174]). Even under assumption of the most favourable conditions for a SN Ia in the SD scenario, the estimates of the current Galactic occurrence rate for this channel do not exceed 1 × 10–3 yr–1 [134], i.e. they may contribute up to 50% of the lowest estimate of the inferred Galactic SN Ia occurrence rate.

On the observational side, the major objection to the SD scenario comes from the fact that no hydrogen is observed in the spectra of SNe Ia, while it is expected that ∼ 0.15 M ⊙ of H-rich matter may be stripped from the companion by the SN shell [248Jump To The Next Citation Point]13. Hydrogen may be discovered both in very early and late optical spectra of SNe and in radio- and X-ray ranges [89248Jump To The Next Citation Point216]. Panagia et al. [305] find a firm upper limit to a steady mass-loss rate for individual SN systems of ∼ 3 × 10−8 M ⊙ yr−1. As well, no expected [24853] high luminosity and/or high velocity former companions to exploding WD were discovered as yet14. The SD scenario also predicts the existence of many more supersoft X-ray sources than are expected from observations, even considering severe problems in estimating incompleteness of the samples of the latter (see for instance [83]).

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 a SN Ia and in computations of the explosions themselves, do not allow to exclude any type of progenitors. The existence of at least two families of progenitors is suggested by observations (see, e.g., [247]). A high proportion of “peculiar” SN Ia of (36 ± 9)% [218] suggests a large spread in the ignition conditions in the exploding objects that also may be attributed to the diversity of progenitors. Note that a high proportion of “peculiar” SNe Ia casts a certain doubt to their use as standard candles for cosmology.

As shown in the flowchart in Figure 5View Image, there are configurations for which it is expected that stable accretion of He onto a CO-WD occurs: in AM CVn systems in the double-degenerate formation channel and in precursors of AM CVn systems in the helium-star channel. In the latter systems the mass exchange rate is close to −8 −1 (1– 3) × 10 M ⊙ yr, practically irrespective of the combination of donor and accretor mass. It was suggested that in such systems the accumulation of a ∼ 0.1M ⊙ degenerate He-layer onto a (0.6– 0.9)M ⊙ accretor is possible prior to He-detonation and that the latter initiates a compressional wave that results in the central detonation of carbon [232234459233]; even if central carbon ignition does not occur, the scale of the event is comparable to weak SNe [220164414]. For a certain time these events involving sub-Chandrasekhar mass accretors (nicknamed “edge-lit detonations”, ELD) that may occur at the rates of ∼ 10–3 yr–1 were considered as one of the alternative mechanisms for SNe Ia, although it was shown by Höfflich and Khokhlov [150] that the behaviour of light-curves produced by them does not resemble any of the known SNe Ia. Thus, until recently, the real identification of these events remained a problem. However, it was shown recently by Yoon and Langer [463], who considered angular-momentum accretion effects, that the helium envelope is heated efficiently by friction in the differentially rotating spun-up layers. As a result, helium ignites much earlier and under much less degenerate conditions compared to the corresponding non-rotating case. If the efficiency of energy dissipation is high enough, detonation may be avoided and, instead of a SN, recurrent helium novae may occur. The outburst, typically, happens after accumulation of 0.02M ⊙. Currently, there is one object known, identified as He-nova – V445 Pup [439440910]. If He-novae are really associated with mass-transfer from low-mass helium stars to CO white dwarfs, then the estimates of the birth rate of the latter systems (≈ 0.6 × 10–3 yr–1[473] and of the amount of matter available for transfer (≈ 0.2M ⊙) give an occurrence rate of He-novae of ∼ 0.1 yr–1, i.e. one He-nova per several 100 “ordinary” hydrogen-rich novae.

As we mentioned above, intermediate mass donors, before becoming white dwarfs, pass through the stage of a helium star. If the mass of the latter is above ≃ 0.8 M ⊙, it expands to giant dimensions after exhaustion of He in the core and may overflow the Roche lobe and, under proper conditions, transfer mass stably [163]. 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 MCh and a SN Ia become possible, as it was shown explicitly by Yoon and Langer [462]. However, the probability of such a SN Ia is only ∼ 10–5 yr–1.

Ultra-compact X-ray binaries.   The suggested channels for formation of UCXBs in the field are, in fact, “hybrids” of scenarios presented in Figures 4View Image and 5View Image. In progenitors of these systems, the primary becomes a neutron star, while the secondary is not massive enough. Then, several scenarios similar to the scenarios for the systems with the first-formed white dwarf are open. A white dwarf may overflow the Roche lobe due to systemic AML via GWR. A low-mass companion to a neutron star may overflow the Roche lobe at the end of the main sequence and become a low-mass He-rich donor. A core helium-burning star may be brought in contact by AML due to GWR; mass loss quenches nuclear burning and the donor becomes a helium “semidegenerate” object. An additional scenario is provided by the formation of a neutron-star component by AIC of an accreting white dwarf. We refer the reader to the pioneering papers [411Jump To The Next Citation Point289394370412Jump To The Next Citation Point102434] and to more recent studies [286Jump To The Next Citation Point472Jump To The Next Citation Point32331226168Jump To The Next Citation Point429430193183]. An analysis of the chemical composition of donors in these systems seems to be a promising way for discrimination between systems of different origin [279280168448]: Helium dwarf donors should display products of H-burning, while He-star descendants should display products of He-burning products. Actually, both carbon/oxygen and helium/nitrogen discs in UCXBs are discovered [278Jump To The Next Citation Point].

In globular clusters, UCXBs are formed most probably by dynamical interactions, as first suggested by Fabian et al. [94] (see, e.g., [34317243223527] and references therein for the latest studies on the topic).

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