Note that apart from “double-degenerate” and “helium-star” channels for the formation of AM CVn-stars, there exists the third, “CV”-channel [778, 779, 522, 596]. In the latter, the donor star fills its Roche lobe at the very end of the main-sequence stage () or just after its completion. For such donors the chemical inhomogeneity inhibits complete mixing at typical for initially non-evolved donors. The mixing is delayed to lower masses, and as a result the donors become helium dwarfs with some traces of hydrogen. After reaching the minimum orbital periods , these binaries start to evolve to longer orbital periods. However, in this scenario the birth rate of systems that can penetrate the range of periods occupied by the observed AM CVn-stars, especially , is much lower than the birth rate in the “double-degenerate” and “helium-star” channels; below we shall neglect it.
The importance of AM CVn stars both for astrophysics and for physics in general stems from the fact that, because of their very short orbital periods and relative brightness due to proximity to the Sun, they are expected to be strong GWR sources and the first Galactic objects discovered by space GW interferometers like LISA (or eLISA), i.e., they will serve as “verification binaries” [744*]. Besides, AM CVn stars can be precursors of SN Ia [784*, 727*, 662] as well as still-hypothetical order-of-magnitude–weaker explosive events called SN .Ia [48, 697, 702, 576].
A comprehensive review paper on AM CVn stars covering their observational properties and some theoretical aspects was published recently by Solheim . Therefore, we shall provide below only basic information about these binaries.
In a binary with stable mass transfer, change of the radius of the donor exactly matches the change of its Roche lobe. This condition combined with the approximate size of the Roche lobe given by Eq. (35*) (which is valid for low ) provides the following relation between the orbital period of the binary and the mass and radius of the donor:7 and most probably, in the merging of components, if a supernova explosion is avoided. Of course, Eq. (71*) oversimplifies the conditions for a stable mass exchange. A rigorous treatment must include tidal effects, angular momentum exchange, and the possible super-Eddington immediately after RLOF by the donor and the associated common envelope formation, as well as the possible ignition of accreted helium on the WD surface [829*, 269, 464*, 246].
Currently, only some of these effects have been studied to a certain extent. In Figure 22* the distribution of birthrates of the possible progenitors of AM CVn stars in the DD channel from [518*] is over-plotted with constraints imposed by the mass exchange stability conditions, the Eddington accretion rate, and the efficiency of tidal interaction (in fact, this is the bottom part of Figure 17* applied to the population of AM CVn stars). It is clear that the substantial number of AM CVn stars formed via the “double-degenerate” scenario may exist only in the case of efficient tidal coupling. As shown in [518*], the difference in the expected numbers of stars between two limiting cases of synchronization time and amounts to two orders of magnitude.
AM CVn systems are so close that in some of them the accretion stream may hit the accretor directly, and the accretion disk does not form [512*]. The candidate “direct impact” systems currently include the shortest orbital period known systems HM Cnc (), V407 Vul (), ES Cet (), see  for discussion and references.
Until quite recently, the evolution of systems with WD donors was calculated using Eq. (70*) and applying different relations for zero-temperature WDs close to , such as given in [882, 506, 809*], ignoring detailed models of dwarfs. For He-star donors, radii were approximated by fits obtained in the evolutionary models of semidetached systems (e.g., from [776*]). Early calculations of this kind may be found in [551, 812, 194, 784*, 626, 748, 809*, 790*, 518*, 519*, 661]. Figure 23* shows examples of the evolution of systems with a helium degenerate donor or a low-mass “semidegenerate” helium star donor and a carbon-oxygen accretor with initial masses that are currently thought to be typical for progenitors of AM CVn systems – and for double degenerates and and for He-star systems, respectively. In Figure 23* the mass–radius relation  is used for WD, while for the low-mass He-star – relation from . The same equations are applied to obtain the model of the population of AM CVn-stars discussed in Sections 10 and 11.
Equations (25*), (35*), and (70*) imply that for the mass loss rate scales as . As a result, for all combinations of donor and accretor masses, the – lines form two rather narrow strips within which they converge with decreasing .
The “theoretical” model of evolution from shorter periods to longer ones is supported by observations, which found that the UV luminosity of AM CVn-stars is increasing as the orbital period gets shorter, since shorter periods are associated with higher . The orbital period change at the level was quite recently found for the first discovered eclipsing AM CVn type binary SDSS J0926+3624 (. ). This value of is consistent with the mass-exchange rate close to , expected both in the “double-degenerate” and “helium-star” channels of the evolution of AM CVn stars for the period of SDSS J0926+3624. Sion et al.  estimated accretion rates in several AM CVn binaries by means of analyses of their far and near-UV spectra and found that in their sample; the system ES Ceti with the shortest has the highest , while the longest orbital period system GP Com (46.5 min) has the lowest .
About 1/3 of AM CVn stars demonstrate outbursts [623*]. The properties of an outburst are well described by the thermal-viscous disk instability model, see  and references therein. Crudely, disks are unstable for mass accretion rates .
However, we should note that the time span between the formation of a pair of WDs and their contact may range from several Myr to several Gyr [790*, 518*]. This means that the approximation of zero-temperature white dwarfs is not always valid. of WD at the onset of mass transfer may be as high as almost , as estimated by Deloye et al.  for WD progenitors of donors of AM CVn stars from [518*]. This produces differences in period distributions of stars at the onset of mass transfer: while pairs with zero-temperature WDs have narrow distribution , systems with finite entropy WDs have and for about 80% of the model stars .
Deloye et al. incorporated the pure helium equation of state and the low temperature opacities in their computations of (still unique) series of full evolutionary models of mass-losing donors with different initial mass and electron degeneracy.30 Their main finding is that the evolution of the donor includes three distinct phases: (i) the onset of mass transfer in which the WD contracts, mainly due to the fact that the very outer layers are never degenerate, (ii) the phase of adiabatic expansion, and (iii) the final phase when the thermal time scale of the donor becomes shorter than the mass-loss time scale and the white dwarf cools, contracts, and evolves to a nearly degenerate configuration. Early computations were not able to account for these transitions, since mass-radius relations for zero-temperature WD were used.
In Figure 24* we show, after Deloye et al., evolution for AM CVn systems with initial masses and and the dependence of the relation on the initial binary parameters and the degeneracy parameter for pairs with different initial component masses and electron degeneracy. Note that the initial stage of decreasing is very short and hardly observable (however, we do observe HM Cnc). At the final stage of evolution, the tracks with different initial degeneracy merge. The comparison with Figure 23* shows that the tracks for the “intermediate” adiabatic mass loss stage based on the zero-temperature relation and on finite entropy models do not differ significantly, justifying early calculations, while in the stage of cooling at , the mass loss rate becomes so small that stars will again be hardly observable. The main difference between the approximate and full computations arises from taking into account the initial non-zero temperature and should be reflected in after contact. This may affect the expected GW signal from AM CVn stars, which must be dominated by the shortest period systems (and the closest ones), see Section 10.
Yet another point, which may slightly change the evolution of the double-degenerate family of AM CVn stars based on the “traditional” -relation, is the possible presence of a low-mass () but a geometrically thick (several ) nondegenerate layer of hydrogen supported by - nuclear burning at the surface of the donor at RLOF. Models of mass-transfer in such systems [131, 337] show that (i) they may make contact at and (ii) for the first of evolution radii of the donors decrease and systems evolve to shorter periods, until the H-rich layer is almost shed. This may be the case of AM CVn-type star HM Cancri (= RX J0806.3+1527), which is also the star with the shortest known  and the only member of the AM CVn class of binaries with a suspected presence of hydrogen.
Precursors of He-star components of AM CVn stars with initial mass are red giants with non-degenerate helium cores (Figure 9*). Helium stars with almost do not expand in the course of evolution (by less than 30% ). Their lifetimes are comparable to the main-sequence lifetimes of their precursors (Eq. 68*) and the orbital angular momentum loss via gravitational waves radiation may bring them to RLOF before the exhaustion of He in their cores [674, 302, 790]. Mass transfer proceeds steadily if the initial mass ratio of the He star and the WD is less than approximately 3 . The evolution of low-mass semidetached systems with He-donors was explored in detail by Yungelson . Referring the reader for details to this paper, we show in Figure 25* as an example a set of tracks for the system with initial masses of components , which have at the formation (immediately after completion of common envelope evolution) orbital periods from 20 to 130 min. The initial abundances of He and heavy elements at “He-ZAMS” are assumed to be 0.98 and 0.02, respectively. RLOF in these systems happens at orbital periods of about 20 to 30 min when He abundance in their cores ranges from 0.98 to 0.066, from the least evolved models at RLOF to the most evolved ones. The initial stage of increasing Ṁ lasts for 10 Myr only. The stars reach minimum periods of 10 min. Their mass at this time is close to . As shown in , the relation very weakly depends on the total mass of the system. Mass loss leads to the cooling of stellar interiors and quenching of nuclear burning by the time when the minimum of is reached, see Figure 26*. After the minimum of , the chemical composition of the matter transferred to the companion is “frozen” and does not change during the subsequent evolution. In the course of further evolution the matter of donors becomes increasingly degenerate, and their mass-ratio relation approaches that of WD.
As mentioned in Section 7, until recently it was deemed that, at accretion rates of He onto a WD below several units of , of He accumulates at the WD surface and then detonates; converging detonation waves propagate to the center of the accretor and ignite carbon detonation close to the center. Prior to the minimum He-donors provide . Correspondingly, Nelemans et al. [518*, 519*] considered “optimistic” and “pessimistic” scenarios for the formation of AM CVn stars in the He channel, depending on the amount of He that may accrete, detonate and destroy the AM CVn star precursor: or . The difference in the predicted number of observed systems reached factor 2 in favor of the first scenario. However, a comparison of the accretion and mass loss rates in Figures 25* and 14* suggests that the donors instead experience strong flashes of He-burning than detonations, and the model space density of AM CVn stars in [518*, 519*] is, most probably, an underestimate. This exacerbates the known problem of a possibly-almost factor 50 deficit of the observed AM CVn star spatial density compared to theoretical predictions [651, 90*]. However, as noted in , the “observational” estimate of the number of AM CVn stars depends on the assumed intrinsic distribution of their magnitudes, and uncertainty in this distribution may also contribute to the mismatch of theoretical expectations and observations.
An accurate knowledge of the formation channels and the space density of interacting double-degenerates is necessary for a better understanding of the stability of mass exchange, processes in common envelopes and proper modeling of the gravitational wave foreground (Section 10). The same is true for ultra-compact X-ray sources.
The chemical composition of IDD may reveal their origins [515, 521]. In the double degenerate channel, the matter transferred is a product of low-temperature CNO-burning, and the abundances of He, C, N, and O depend on the WD progenitor’s mass. In the He-star channel, the abundances depend on the initial mass of the donor and the extent of He-depletion by the time of RLOF (Figure 26*). In the “evolved CV-channel”, the matter may also contain the traces of hydrogen. As concerns UCXB, it was shown in [877*] that the initial masses of their donors are limited from above by . This means that some of the donors may descend from “hybrid” WDs. However, the latter lose their He-rich envelopes at very early phases of mass exchange, and, since they are descendants of helium stars, the matter transferred by them should not contain He. Nelemans et al.  constructed a set of diagnostic diagrams with abundance ratios (in mass) , , , , as functions of for different groups of donors. As an example, we show in Figure 27* such a diagram for the ratio .
The above-described diagnostic of abundances in the spectra of known IDD and other means of analysis suggest that HM Cnc, CP Eri, GP Com, V396 Hya, SDSS J124058.03–015919.2, SDSS J080449.49+161624.8 almost definitely belong to the WD-family, while AM CVn, HP Lib, CR Boo, V803 Cen, SDSS J173047.59+554518.5 may have helium-star donors [521, 725, 88]. Significantly-evolved He-star donors are ruled out for SDSS J113732.32+405458.3 and SDSS J150551.58+065948.7 .
Among the UCXBs there are two systems (4U 1626-67 and 4U 0614+09) in which detection of C and O lines but no He lines suggests hybrid white dwarf or very evolved helium star donors. For 4U 1916-05, the detected He and N lines suggest a He white dwarf or an unevolved helium star donor . For the rest, data are not sufficient for classification.
By means of population synthesis modeling, Solheim and Yungelson  found that accumulation of by WD in the double-degenerate channel of formation and evolution of IDD is possible at a rate of , while Ruiter et al.  provided an estimate that is an order of magnitude higher. Note that in both studies SN Ia require massive WD accretors, which are not typical for observed IDD.
As described above, in the WD-family of AM CVn stars, the mass transfer rate initially is as high as and decreases at later times. While during the initial phase of accretion the nuclear burning in the accreted He envelope of the WD is thermally stable, it becomes thermally unstable as drops (Figure 14*). In the He-star family, the mass transfer rate before reaching the orbital period minimum is several , i.e., accretors are in the strong He-flashes regime ab ovo. The “ignition” mass for He-outbursts increases with decreasing . This yields 10 He-flashes in the before the amount of accreted mass required to achieve He-ignition becomes larger than the mass of the donor (Figure 28*), while de facto the binaries are already in the “detonation” regime of He-burning (Figure 14*). Thus, there must be the “last” most powerful outburst with an ignition mass of which can occur in the hydrodynamic regime. It can lead to He-detonation, which can be potentially observable as a faint thermonuclear supernova . Since the mass transfer rate drops very rapidly after RLOF, it is plausible that the observed AM CVn stars already experienced their last strong He-flash (which could be of the SN .Ia scale) and are now slowly transferring He without any expected nuclear-powered phenomena. Supernovae .Ia probably do not initiate detonations of low-mass accretors of AM CVn stars since the masses of the exploding shells are insufficient for this [713, 698].
As estimated in , SN .Ia may happen every 5000 to 15 000 yr in an E/S0 galaxy. Their brightness and duration are about 1/10 of the ordinary SN Ia event. The rise times of the light curves in all bands are very rapid ( 10 days). The explosion nucleosynthesis in these events primarily yields heavy -chain elements (40Ca through 56Ni) and unburned He (up to 80%) [702, 493]. It is interesting that with the 1-day dynamic-cadence experiment on the Palomar Transient Factory,31 it is expected to annually discover a few such events.
SN 2010X (PTF 10bhp) with extremely rapid decay ( =5 day) was, for instance, suggested as a candidate SN .Ia . Another candidate may be a faint type Ib supernova, SN 2005E, in the halo of the nearby isolated galaxy, NGC 1032. Spectroscopic observations and analysis reveal high velocities of the ejecta, dominated by helium-burning products including a large amount of radioactive 44Ti, just as expected for SN .Ia. However, as noticed by Drout et al. , e.g., severely stripped-envelope SN Ic explosions may mimic SNe .Ia.
It is possible that strong flashes of He-burning can be identified with He-novae, like H-outbursts are identified with classical novae. The sole known object that may be a prototype of He-novae – V445 Pup erupted in 2000, and its optical and IR spectra are hydrogen-deficient [15, 853]. Provisional of V445 Pup is 0.65 day . Several other candidate He-novae have been suggested, but remain still unexplored . The estimated pre-outburst luminosity of V445 Pup is . Apparently, it is associated with a more massive system than the typical AM CVn star, but one cannot exclude nova-scale events among AM CVn’s.