He-accreting WDs

He-Accreting WDs: accretion regimes and final outcomes

Abstract

The behaviour of carbon-oxygen white dwarfs (WDs) subject to direct helium accretion is extensively studied. We aim to analyze the thermal response of the accreting WD to mass deposition at different time scales. The analysis has been performed for initial WDs masses and accretion rates in the range (0.60 - 1.02)  and   , respectively. Thermal regimes in the parameters space , leading to formation of red-giant-like structure, steady burning of He, mild, strong and dynamical flashes have been identified and the transition between those regimes has been studied in detail. In particular, the physical properties of WDs experiencing the He-flash accretion regime have been investigated in order to determine the mass retention efficiency as a function of the accretor total mass and accretion rate. We also discuss to what extent the building-up of a He-rich layer via H-burning could be described according to the behaviour of models accreting He-rich matter directly. Polynomial fits to the obtained results are provided for use in binary population synthesis computations. Several applications for close binary systems with He-rich donors and CO WD accretors are considered and the relevance of the results for the interpretation of He-novae is discussed.

keywords:
Binaries: general, Supernovae:general, White Dwarfs, Accretion
12

1 Introduction

Accretion of helium onto carbon-oxygen white dwarfs (CO WDs) plays an important role in several astrophysical processes. Most significantly, it may have relevance to the problem of the Supernovae Ia (SNe Ia) progenitors. Among hypothetical evolutionary paths to SNe Ia are semidetached close binary stars in which CO WDs grow in mass up to the Chandrasekhar mass limit () by accretion of matter directly from non-degenerate or degenerate helium-rich companions (e.g., Tutukov & Yungelson, 1996; Yoon & Langer, 2003; Solheim & Yungelson, 2005; Wang et al., 2009), as well as explosions of sub-Chandrasekhar mass CO WD via “edge-lit” or “double-detonation” mechanism, in which detonation in the He-layer at the surface of an accreting WD triggers detonation of CO-accretor via shock waves that compress the latter (e.g., Nomoto, 1980, 1982a; Livne, 1990; Livne & Glasner, 1991; Limongi & Tornambè, 1991; Woosley & Weaver, 1994; Livne & Arnett, 1995; García-Senz, Bravo & Woosley, 1999; Fink, Hillebrandt & Röpke, 2007; Sim et al., 2010; Fink et al., 2010; Woosley & Kasen, 2011; Schwab et al., 2012; Townsley, Moore & Bildsten, 2012; Moll & Woosley, 2013; Shen & Bildsten, 2014a; Moore, Townsley & Bildsten, 2013).

Recent modification of the classical “double-degenerate” scenario (Webbink, 1984; Iben & Tutukov, 1984) envisions “violent” or “prompt” merger so that detonation is initiated at the interface of two merging WD in C-O or He-C-O mixture and determines the final complete burning of both WDs (e.g., Pakmor et al., 2010, 2011, 2012, 2013; Kromer et al., 2013; Moll et al., 2014). It is worth noting that some authors questioned whether carbon detonation is really prompt (e.g., Raskin et al., 2012). Moreover it has been shown that the location of initial explosion does depend on the numerical resolution as well as on the initial configuration adopted in the computations (Dan et al., 2012, 2013). Double-detonation and violent merger scenarios are currently considered as alternative or complementary to the “classical” single-degenerate and double-degenerate scenarios for SNe Ia. For a review of the SNe Ia progenitors and the observational constraints to theoretical models see, e.g., Hillebrandt et al. (2013); Höflich et al. (2013); Maoz, Mannucci & Nelemans (2013); Postnov & Yungelson (2014), while for the recent results of binary population synthesis (BPS) for SNe Ia from different channels see Ruiter, Belczynski & Fryer (2009); Wang et al. (2009); Mennekens et al. (2010); Ruiter et al. (2011); Toonen, Nelemans & Portegies Zwart (2012); Nelemans, Toonen & Bours (2013); Ruiter et al. (2013); Wang, Justham & Han (2013).

For the double-detonation scenario, the onset of the initial He-detonation depends on the accretion rate and the mass of He retained at the WD surface3. For both scenarios the percentage of the He-rich matter transferred from the donor and effectively retained by the accreting WD and nuclearly processed into C/O rich mixture up to either the He-detonation or the merger may be important since simulations show that physical conditions adequate to reproduce normal SNe Ia are more favourable in massive accretors –  for double detonation (Livne & Glasner, 1991; Fink et al., 2010; Kromer et al., 2010; Piro, Thompson & Kochanek, 2014) and  for violent mergers (Pakmor et al., 2013; Kromer et al., 2013). This means that the transfer of He-rich matter onto new-born WD and its conversion into C and O may be necessary, as CO WDs typically form with lower masses.

In the “classical” single-degenerate scenario for Chandrasekhar mass SNe Ia retention efficiency of He is a crucial parameter, since the most important physical process involved is the nuclear burning of H into He and then into C-O-Ne mixture (see, e.g., Bours, Toonen & Nelemans, 2013, for recent discussion). Retention efficiency of helium and He-burning regimes are, for instance, also important for formation and evolution of AM CVn stars (Nelemans et al., 2001), the origin of faint and fast transients possibly associated with single He-layer detonation at the surface of accreting WDs, like still hypothetical SNe .Ia (see, e.g., Bildsten et al., 2007; Shen et al., 2010; Waldman et al., 2011; Woosley & Kasen, 2011; Sim et al., 2012; Raskin et al., 2012; Shen & Bildsten, 2014a; Kasliwal, 2012) and, possibly, some subluminous SN Ia (Wang, Justham & Han, 2013). Other problems include, e.g., the interpretation of events suggested to be “Helium Novae” (Ashok & Banerjee, 2003; Rosenbush, 2008).

The large majority of studies devoted to the thermal response of CO WDs accreting He-rich matter, focused on defining physical conditions for the onset of either a very strong He-flash or a He-detonation. (Sugimoto & Fujimoto, 1978; Nomoto, Nariai & Sugimoto, 1979; Nariai, Nomoto & Sugimoto, 1980; Taam, 1980a, b; Nomoto, 1982b, a; Fujimoto & Sugimoto, 1982; Woosley, Taam & Weaver, 1986; Nomoto & Hashimoto, 1987; Iben & Tutukov, 1991; Limongi & Tornambè, 1991; Woosley & Weaver, 1994; Piersanti, Cassisi & Tornambé, 2001; Bildsten et al., 2006; Shen & Bildsten, 2007, 2009; Woosley & Kasen, 2011). As a result, various accretion regimes were roughly defined, depending both on the accretion rate and on the CO WD mass. In particular, Nomoto (1982b) suggested that for a high value of the accretion rate , but still below the Eddington limit, an extended He-rich layer is piled-up and WD expands to giant dimensions. The minimum value of  for this is defined by the rate at which He burns into CO-rich matter at the base of the He-envelope. Basing on the computations of massive pure-He stars by Uus (1970), Nomoto derived an analytical formula for such a limiting value:

(1)

where is mass of the CO core in , varying in the range (0.75 – 1.38) . For  lower than  , but larger than  , He-burning is stable and the CO core increases in mass steadily (Iben & Tutukov, 1989). For lower values of the accretion rate, but larger than  , He-burning proceeds through recurrent flashes, whose strength increases with decreasing , for a fixed mass of the CO core. In this case the He-flashes are too weak to develop any dynamical effects, even if the large energy release can trigger the expansion of the accreting structure and, hence, the interaction with its binary companion (Taam, 1980a; Fujimoto & Sugimoto, 1982). Finally, for    a dynamical He-flash occurs when a critical amount of He-rich matter has been accreted. Woosley & Kasen (2011) investigated in great details the physical properties of He-accreting WDs at low accretion rates and define very accurately the transition from novalike He-flashes to He-detonation (see their Figure 19). They demonstrated that the He-layer above the CO core could detonate only if the density at the ignition point is larger than a critical value, around .

Kato & Hachisu (2004) (hereinafter KH04) calculated the retention efficiency  of He-accreting WD for CO cores in the range (0.6 – 1.3)  and   .  is defined as the ratio of the mass effectively accumulated onto the CO core after one full He-flash driven cycle and the amount of matter transferred from the donor during the same cycle. The computations by KH04 were based on the “optically thick wind theory”, which presumably describes the continuum-radiation driven wind operating very deeply inside the photosphere (Kato & Hachisu, 1994). Even if a huge efforts have been applied to investigate the thermal response of CO WDs accreting He-rich matter, an overall picture is still missing. In particular, in the regime characterised by recurrent He-flashes the possibility that accreting objects could overfill their Roche lobe has not been explored so far, despite this issue is of a paramount importance to determine the long-term evolution of binaries harbouring He-donors. The present work is aimed to the systematic analysis of the parameter space , considering fully evolutionary models of CO WDs with initial mass in the range (0.6 – 1.05) 4 and accretion rates  . Basing on our results, we intend to define the limits of different accretion regimes and the possible outcomes of the accretion process. Moreover, at variance with previous studies, we investigate the effects of the previous evolution, namely of the accretion history, on the actual thermal response of accreting WDs.

Cool Models Heated Models LABEL M060 M070 M081 M092 M102 M060 M070 M081 M092 M102 (in ) 0.60000 0.70000 0.80833 0.91962 1.02061 0.59678 0.70185 0.81033 0.91897 1.02048 0.3839 0.4308 0.5396 0.5371 0.4870 0.3839 0.4308 0.5396 0.5371 0.4870 (in ) 0.5158 0.6580 0.7921 0.9114 1.0157 0.5173 0.6580 0.7921 0.9116 1.0158 (in ) 3.37 1.40 0.73 0.39 0.18 1.62 0.67 0.30 0.13 0.06 7.8647 7.8644 7.8759 7.9915 8.0798 7.8318 7.8495 7.8593 7.9012 7.9528 6.4981 6.7518 7.0219 7.2792 7.5403 6.4552 6.7387 7.0187 7.2995 7.5779 0.5358 0.5637 0.4881 1.3760 2.7042 3.3358 3.7567 4.0224 4.2488 4.4‘87 4.7996 4.8424 4.8606 5.0989 5.4154 5.2821 5.4382 5.5655 5.6850 5.7789 -1.8083 -1.8799 -1.9542 -1.9868 -1.9557 -1.3745 -1.4752 -1.5969 -1.7235 -1.8254
Table 1: Physical properties of the initial CO WDs (Cool Models) and after the first mass transfer episode (Heated Models). We list the total mass of the model , the mass fraction abundance ratio of carbon over oxygen at the center C/O, the mass extention of the He-deprived core , the mass extention of the more external layer where the helium abundance by mass fraction is larger than 0.05 , the temperature in K and the density in at the center, the surface luminosity , the effective temperature in K and the surface radius . For more details see text.
Figure 1: Profiles in the plane for the Cool (dashed lines) and Heated Models (solid line). Each panel refers to one CO WD as labelled. In the profiles of Heated Models we mark with a filled circle the He/CO interface and with and open circle the He-burning shell.

In § 2 we describe our evolutionary code and the input physics. We also present the main properties of the initial CO WD models. In § 3 we present our results and define various accretion regimes. In § 4 we discuss under what conditions central C-ignition could occur in a Chandrasekhar-mass WD, thus triggering an explosion. § 5 is devoted to the accumulation efficiency in models accreting directly He-rich matter and as a by-product of H-burning in H-accreting WDs. In § 6 we discuss the formation of He-rich donors in close binaries and we analyze the applications of our results to several types of systems in which stable accretion of pure He onto a CO WD may occur. Our final considerations are reported in § 7.

2 Input physics and numerical methods

All the models presented in this study have been computed with an updated version of the F.RA.N.E.C code, the original version being described in Chieffi & Straniero (1989). The setup of the code is the same as in Piersanti, Straniero & Cristallo (2007); in particular, we consider solar chemical composition models adopting the solar mixture GS98 derived in Piersanti, Straniero & Cristallo (2007). Tables of radiative opacities for temperature lower than K have been derived through the web facility provided by the OPAL group (http://opalopacity.llnl.gov/new.html see Iglesias & Rogers, 1996), while at higher temperatures we adopt the tables derived from the Los Alamos Opacity library (Huebner et al., 1977). The contribution of electron conduction to the total opacity has been included according to the prescription by Potekhin et al. (1999). We adopt the equation of state computed by Straniero (1988) and successive upgrades (Prada Moroni & Straniero, 2002). Matter is accreted by assuming that it has the same specific entropy as the external layers of the CO WD; this is equivalent to the assumption that the energy excess is radiated away (Piersanti et al., 2000, and references therein). We fix the chemical composition of the accreted matter to X=0, Y=0.98, Z=0.02, where X, Y and Z are the mass fraction abundance of hydrogen, helium and metals, respectively. Since the He-donor has formed via nuclear burning of H-rich matter, we assume that all the initial CNO elements have been converted into (i.e., where is the abundance by number of the -isotope and the subscripts and refer to the initial MS star and the final He-donor star, respectively). For all the elements heavier than oxygen we assume a scaled-solar chemical composition.

The adopted nuclear network includes elements from H to Fe, linked by -, - and -capture reactions as well as decays. We do not consider the NCO chain () because it has been demonstrated that its contribution to the energy budget does not have a sizable effects on the physical properties of accreting WDs (Piersanti, Cassisi & Tornambé, 2001).

The convective mixing is modelled by means of the time dependent algorithm introduced by Sparks & Endal (1980) and successively modified by Straniero, Gallino & Cristallo (2006) (for a recent review see also Straniero, Cristallo & Piersanti, 2014). In this case the degree of convective mixing between two points inside the convective zone depends on the corresponding turnover timescale.

In order to obtain the initial CO WD models we evolved intermediate mass stars from the pre-main sequence to the cooling sequence, by assuming that they are components of interacting binary systems. For models M060 and M070 we assumed that their progenitors experienced a common envelope episode during the red giant branch phase, while for progenitors of models M081, M092 and M102 we assumed a Roche lobe overflow (RLOF) in the hydrogen-shell burning stage while they possessed radiative envelopes. Some physical properties of the initial WD models are summarised in Table 1 (Cool Models).

The deposition onto the Cool Models of He-rich matter at a relatively high rate (let say,    ) determines the heating of the physical base of the He-rich layer, because the adopted initial models are compact and cool (their luminosity is definitively lower than that of a post-AGB star of the same mass). Hence, the compressional heating timescale at the surface of the accretor is definitively lower than the inward thermal diffusion timescale. As a consequence the temperature at the base of the He-rich layer increases and He-burning is ignited when the local temperature attains  K. However, due to the partial degeneracy at the ignition point5, a thermonuclear runaway occurs, even if the resulting flash does not become dynamical. Accreting WD reacts to the injection of a huge amount of nuclear energy by expanding to giant dimensions (). When considering that these objects are components of binary systems, it turns out that they have to experience a RLOF, losing part of the matter previously accreted.

We simulated this first flash episode by adopting   . Moreover, for all the considered models we fix the radius of the Roche lobe to . It is worth noticing that, even if such an assumption is completely arbitrary, the physical properties of the final structure do not depend on the exact value of , since the expansion to giant dimensions occurs on a very short timescale, smaller that the nuclear timescale of the He-burning shell6 and of the inward thermal diffusion timescale.

Figure 2: Possible accretion regimes as a function of the WD initial mass and accretion rate. Solid lines represent the transition between different accretion regime as obtained by linearly interpolating the results of our computations (open and filled symbols). Interpolation formulae are presented in Appendix A.1.

We find that during this first mass transfer episode, the amount of material effectively deposited onto the WDs is practically zero (in some cases the pre-existing He-rich zone is also eroded). The only effect of the first He-flash is just to modify the physical properties of the He-rich layer (mainly temperature and density profiles), so that the He-shell attains the conditions for steady burning. This is clearly depicted in Fig. 1 where we show, for each considered model, the profile in the plane for WDs at the beginning of the first mass transfer episode (dashed lines) and after the RLOF episode, when the WDs has attained its maximum effective temperature at the beginning of the cooling sequence (solid lines). Henceforth, we address these models with modified thermal structure of the envelope as Heated Models. In Table 1 we report also some relevant physical quantities referring to each model after the first He-flash episode. The CO core is defined as the portion of the star where the helium abundance is lower than by mass fraction, while the He-burning shell is defined as the mass coordinate where He-burning is at maximum. The physical and chemical conditions for is a maximum are far from the He/CO interface, but close to the mass coordinate where He abundance is 0.05 by mass fraction. The He-burning shell does not correspond to the maximum temperature because, after the RLOF episode, it moves outward, where temperature is lower, while the new-synthesized CO layer contracts and heats up.

3 Accretion Regimes

We accrete He-rich matter directly onto heated CO WDs (see previous section) by adopting values for the accretion rate in the range . The starting point for all the computations is defined along the high luminosity branch, during the blueward evolution of the post-first-flash steady-state structure.

Our results are summarised in Fig. 2 where we show the possible accretion regimes as a function of the WD initial mass and accretion rate. The lines marking the transition from one accretion regime to another have been obtained by considering the thermal response of Heated Models to the mass transfer. For example, for the model M081 we fix the transition from the Steady Accretion to the Mild Flashes at = , since for =  the model is in a steady state while for =  it experiences recurrent mild flashes (see below for the definitions of different regimes).

3.1 Dynamical He-Flashes regime

At low accretion rates, the physical base of the He-rich mantle cools down and becomes degenerate. Later on, due to the continuous deposition of matter, it heats-up and He-burning is ignited. Owing to the degeneracy at the ignition point, the nuclear energy delivered by the -reactions is stored locally producing the increase of temperature and causing a thermonuclear runaway (He-flash). However, the degeneracy level at the ignition point is only a necessary condition to trigger a dynamical burning event. Indeed, the He-flash triggers the formation of a convective shell which extends outward as the temperature increases at the base of the He-rich zone. Convective shell injects fresh He in the burning zone fuelling the thermonuclear runaway and, hence, the continuous increase of the local temperature. As a matter of fact, if the mass of the He-rich layer (and, hence, the total He abundance by mass) is too small, the propelling effect of convective mixing rapidly exhausts, the flash quenches and the accreting WD expands (for a more detailed discussion see § 3.2).

For accretion rate lower than   the whole structure becomes isothermal very rapidly, independently of the initial temperature profile, since the inward thermal diffusion timescale is very small as compared to the accretion timescale ( yr, while yr). This also implies that the energy delivered by the mass deposited on the surface of the WD can not produce a local increase of temperature. As a consequence, the nature of the final He-flash (explosion or not) depends only on the interplay between the neutrino cooling of the physical base of the He-shell and the heating driven by the compression of the whole structure. Since the latter quantity depends on the mass growth timescale, it turns out that, for a fixed total mass of the initial WD, the nature of the final He-flash depends only on the accretion rate: if it is smaller than a critical value , the He-shell becomes strongly degenerate and the resulting flash turns into an explosion.

M060
1 1.255 0.658 658.4 0.587 5.246 72.061 0.737
5 1.092 0.495 99.1 0.601 7.579 13.452 0.571
15 0.971 0.374 24.9 0.608 8.619 4.995 0.448
M070
1 1.277 0.575 698.0 0.698 5.124 75.664 0.616
5 1.123 0.421 84.2 0.715 7.648 12.802 0.457
20 0.913 0.211 10.6 0.735 9.442 1.839 0.252
M081
1.5 1.263 0.453 301.7 0.818 6.071 45.067 0.461
6 1.144 0.333 55.6 0.825 7.851 10.913 0.342
25 0.912 0.102 4.1 0.826 9.947 0.929 0.111
M092
1.5 1.301 0.383 255.2 0.918 5,799 59.903 0.392
7 1.180 0.262 37.4 0.930 7,844 10.624 0.271
30 0.963 0.045 1.491 0.922 10.534 0,590 0.053
M102
1.5 1.318 0.299 199.7 1.019 5.883 92.234 0.305
8 1.204 0.185 23.1 1.028 8.075 8.791 0.190
50 1.039 0.020 0.4 1.021 11.148 0.427 0.025
Table 2: Models experiencing dynamical He-flashes. For each model we list as a function of the accretion rate (in  ), the final mass in   the accreted mass in , the accretion time in yr, the mass coordinate where He-burning is ignited in , the temperature in K and density in when He-burning is ignited. The last column gives the mass of the zone where helium abundance by mass fraction is larger than 0.01 in . A more detailed table with small increments in is available online.

In our computations, following Bildsten et al. (2007) and Shen & Bildsten (2009), we check the onset of a dynamical flash by comparing , the heating timescale due to release of nuclear energy, and , the dynamical timescale, at the He-burning shell. If becomes smaller than , the He-rich zone above the He-burning shell can not readjust to a new equilibrium configuration and, hence, their evolution decouples, driving to the formation at the interface of an overpressure which triggers the explosion. These two characteristic timescales have been computed according to the following relations:

(2)

where is the specific heat at constant pressure, T — the temperature,  — the rate of energy production via nuclear burning,  — the pressure scale height and  — the local value of the sound velocity.

Figure 3: Accreted mass as a function of the accretion rate for models experiencing a dynamical He-flash. Each set of symbols refers to a different initial model, as labelled in the figure. Solid lines represent polynomial fits to the data (see Appendix A.2).

Our results are summarised in Table 2, where we report, as a function of the accretion rate, the total final mass (), the accreted mass () and the corresponding accretion time (), the mass coordinate of the point where He-flash is ignited () and the value of temperature () and density () at the ignition point for some representative computed models. In the last column we report , defined as the total mass of the layer where He abundance is larger than 0.01 by mass fraction. Note that the latter is slightly larger than the accreted mass since the initial WD models are capped by a He-rich layer, determined by the previous evolution, which is much smaller than the mass to be accreted for igniting a dynamical He-flash (see § 2). Polynomial fits of as a function of the accretion rate for all the models displayed in Fig. 3 are provided in Appendix A.2. Where comparable, for M=(0.6 – 1.0) , minimum accreted masses necessary for a dynamical He-flash are in very good agreement, within factor of less than 2, with the estimates obtained by Shen et al. (2010). As well, in agreement with the latter study we find that the ignition mass depends on . In Fig. 3 we present the accreted mass as a function of the accretion rate for each initial WD model. At variance with Nomoto (1982b) and in agreement with Shen et al. (2010), we find that the maximum value of the accretion rate still producing a dynamical He-flash slightly depends on the WD mass (see Fig. 2). By interpolating the data reported in Table 2 we obtained (in  )7

(3)
4 1.300 0.382 181.239 0.918 5.823 58.871 0.393
6 1.298 0.380 143.235 0.918 5.870 56.824 0.391
8 1.290 0.372 101.557 0.918 6.040 49.957 0.383
9 1.259 0.341 62.223 0.918 6.606 30.342 0.351
10 1.140 0.222 13.316 0.928 8.259 6.808 0.233
15 0.995 0.077 2.903 0.928 9.912 1.053 0.085
20 0.984 0.066 2.403 0.926 0.096 0.882 0.074
Table 3: The same as in Table 2, but for model M092 accreting He-rich matter at  , with different values of the characteristic timescale , as listed in the first column in yr. See text for more details.

In order to investigate the dependence of our results on the history of the accretion rate we computed an additional set of models, by adopting as initial CO WD the M092 model and a time-dependent accretion rate, given by:

(4)

where is a characteristic timescale varying in the range (4 - 20) Myr. The results are summarised in Table 3, where we list the same quantities as in Table 2.

Figure 4: The same as in Fig. 3, but for the M092 CO WD accreting He-rich matter with an exponentially decaying accretion rate, as reported in the Figure. Different models have different decay timescale, as labelled. Heavy dots represent total accreted mass for models with constant accretion rate.

In Fig. 4 we plot the amount of accreted mass as a function of the accretion rate for the models listed in Table 3 and, for comparison, we also show the total accreted mass of models with constant accretion rate (filled dots). As it can be noticed, the mass accreted before the onset of the dynamical He-flash depends both on the accretion rate and on the accretion history. In particular Fig. 4 reveals that the leading parameter determining is the actual value of  at the onset of the He-flash, while the previous mass transfer process has a minor role, just reducing by less than 10% the value of the accreted mass. In any case, if  decreases very rapidly, the accreting WD loses memory of the previous evolution and the evolution to the explosion occurs exactly as it would if  was kept from the very beginning constant and equal to its final value.

3.2 Strong Flashes Regime

For slightly larger values of the accretion rate the He-flash does not become dynamical, as the nuclear timescale at the base of the He-rich layer never approaches . However, the released energy is huge, so that an expansion to giant dimensions occurs. In order to illustrate the evolution of models experiencing this accretion regime we show in Fig. 5 the evolution in the HR diagram of the model M102 accreting He-rich matter at  . We mark by filled dots and letters some crucial moments during the evolution. In our analysis we follow the approach used by Piersanti et al. (2000) to discuss the evolution of H-accreting models (see also the references therein). The evolution is counterclockwise along the track and we start our discussion from point A in Fig. 5, the bluest point of the whole track. This point represents a bifurcation in the evolution of He-accreting models. At this stage the He-burning shell is very close to the surface and cool and, in addition, the mass of the He-rich zone has been reduced below a critical value at which release of nuclear energy exceeds release of the gravothermal one. Hence, the energy production via He-burning rapidly decreases and the gravothermal energy becomes the main source of energy8. During the following evolution the model approaches the WD radius appropriate to its mass, while the He-burning luminosity continues to decrease and after yr it attains its minimum () at point B.

During the following evolution which lasts for yr, due to the continuous deposition of matter, the physical base of the He-shell begins to heat up and nuclear burning via 3-reactions gradually resumes. Note that, even if the degeneracy of the He-burning shell is not high, the local nuclear timescale is shorter than the timescale for the thermal response of the star to a structural change. As a consequence, He-burning turns into a flash. Very soon, the flash triggers the formation of a convective shell (point C in Fig. 5) which rapidly increases in mass, attaining very soon the surface (point D). It is important to remark that also the inner border of the convective zone moves inward and this causes that the He-burning shell becomes more internal. In the model considered here the convective unstable zone attains its maximum mass extention after yr. The onset of convection has two main effects: on one hand the nuclear energy delivered locally by He-burning is transferred outward, thus limiting the thermonuclear runaway; on the other hand, convection dredges down fresh helium into the burning zone so that the thermonuclear runaway speeds up. When the evolutionary timescale becomes very short the feeding of the He-burning shell by convection becomes not efficient and the structure reacts by evolving toward high luminosity. After day it attains point E along the HR track, where the He-burning luminosity is at a maximum: . The continuous expansion of the whole He-rich zone occurs at the expense of its thermal energy, so that He-burning occurs at a progressively lower rate; hence, the flash-driven convection starts to recede very soon after day (point F in Fig. 5) and it definitively disappears at point G ( yr).

Figure 5: Evolution in the HR diagram of the M102 model accreting He-rich matter at  = . The points along the track represent A: the bluest point; B: minimum He-burning luminosity; C: start of flash-driven convection; D: Maximum extension of the flash-driven convective shell; E: maximum He-burning luminosity; F: beginning of the flash-driven convective shell backtrack; G: die down of flash-driven convection; H: appearance of surface convection. For more details see text.

The He-flash drives the transition of the accreting model from a low state, corresponding to the cooling of the structure, to a high state, corresponding to the high luminosity branch. In fact, the He-flash provides the thermal energy needed for a quiescent He-burning to set in, modifying the temperature and the density at the base of the He-rich layer. For the Strong Flashes regime the amount of nuclear energy delivered during the flash largely exceeds the energy required for such a transition. This implies that the large thermal energy produced by He-burning is initially locally stored and, later on, redistributed along the whole zone above the He-burning shell. Under this condition the He-rich layer has a too large thermal content which has to be dissipated before quiescent He-burning could set in. Hence, the expansion toward the red part of the HR diagram is the natural consequence of the strong flash. Obviously, the lower the accretion rate, the stronger the resulting He-flash and, hence, the larger the final radius attained by the accreting model. During the expansion along the high luminosity branch, surface convection sets in (point H) and it penetrates inward as the effective temperature decreases. The expansion from point G to H occurs in about yr. We forcibly halt the computation when the surface temperature becomes smaller than 11300 K, since the adopted opacity tables are inadequate to describe models with lower . Indeed, due to the overlap of the flash-driven convective shell and convective envelope, the surface heavy elements abundance (mainly ) increases well above the total metal content usually adopted in the computation of low temperature opacity tables.

When considering that the accreting WD is a component of an interacting binary system, it turns out that during the expansion phase a fraction of the matter previously accreted can be lost by the WD, thus limiting the growth in mass of the CO core. This problem has been investigated in detail by KH04, who derived the amount of mass effectively retained by the accreting WD in the framework of the optically thick wind theory (Kato & Hachisu, 1994). According to such an approach, after the He-flash, during the expansion to giant dimensions, wind mass loss starts when the photospheric temperature decreases to a critical value (). During the redward evolution the wind mass loss rate increases as the photospheric temperature decreases until the WD attains thermal equilibrium and stops to expand, coming back blueward. The continuous loss of the matter reduces the layer above the He-burning shell and, when the photospheric temperature of the star becomes larger than the critical value, it becomes negligible. The following evolution is driven by He-burning which reduces the mass of the He-rich layer, up to the bluest point in the HR diagram, when the burning dies and the structure becomes again supported by gravothermal energy. Kato & Hachisu (1994) claimed that the strong optically thick wind prevents the onset of a RLOF. Moreover, they suggested that, even if a RLOF could occur, a common envelope phase is avoided in any case, because the wind velocity is much larger than the orbital velocity of the two components in the binary system, so that the heating and the consequent acceleration of the lost matter is practically inefficient. However, as discussed in KH04, for CO WDs less massive than 0.8  the optically thick wind does not occur and all the matter is effectively deposited onto the WD, ”if the binary separation is large enough for the expanded envelope to reside in the Roche lobe”. Moreover, KH04 analyse the behaviour of CO WDs only for very large values of the accretion rate, not exploring the whole parameter space reported in our Fig. 2.

M060
2.5 0.59743 44.780 0.328 37.45 0.170 0.58686 55.354 0.58757 17.193 0.803 0.170 0.003
3.0 0.59757 34.180 0.407 26.61 0.231 0.58691 44.840 0.58762 17.525 0.801 0.172 0.004
4.0 0.59787 25.970 0.590 17.67 0.335 0.58726 36.577 0.58797 18.195 0.802 0.171 0.004
5.0 0.59818 21.570 0.778 12.78 0.428 0.58755 32.188 0.58832 18.645 0.803 0.170 0.004
6.0 0.59852 18.340 1.003 9.10 0.530 0.58798 28.864 0.58863 19.128 0.803 0.171 0.005
7.0 0.59888 15.750 1.248 6.04 0.645 0.58844 26.166 0.58910 19.491 0.808 0.166 0.005
8.0 0.59928 13.480 1.507 3.47 0.769 0.58891 23.830 0.58958 19.706 0.810 0.164 0.005
9.0 0.59969 11.430 1.811 0.93 0.930 0.58950 21.602 0.59015 20.039 0.822 0.152 0.004
M070
3.0 0.70203 31.220 0.089 26.07 0.167 0.69779 35.457 0.69869 8.495 0.758 0.212 0.003
4.0 0.70209 22.090 0.125 16.77 0.245 0.69784 26.341 0.69873 8.686 0.752 0.219 0.004
5.0 0.70216 18.460 0.163 12.96 0.304 0.69791 22.703 0.69876 8.892 0.752 0.220 0.004
6.0 0.70222 16.110 0.202 10.48 0.358 0.69798 20.353 0.69883 9.026 0.753 0.219 0.004
7.0 0.70229 14.320 0.240 8.65 0.406 0.69803 18.578 0.69892 9.037 0.755 0.217 0.005
8.0 0.70236 12.880 0.282 7.10 0.461 0.69810 17.123 0.69892 9.215 0.756 0.216 0.005
9.0 0.70243 11.580 0.323 5.75 0.517 0.69818 15.899 0.69900 9.253 0.765 0.208 0.004
10.0 0.70250 10.610 0.366 4.68 0.573 0.69829 14.801 0.69908 9.336 0.763 0.209 0.005
M081
4.0 0.81039 24.420 0.067 20.02 0.182 0.80984 24.976 0.80954 5.255 0.703 0.260 0.005
5.0 0.81041 18.700 0.084 14.70 0.217 0.80974 19.380 0.80936 5.054 0.707 0.257 0.005
6.0 0.81043 15.730 0.098 11.75 0.258 0.80958 16.576 0.80928 5.134 0.708 0.257 0.005
7.0 0.81044 13.780 0.115 9.78 0.296 0.80959 14.626 0.80930 5.142 0.710 0.255 0.005
8.0 0.81046 12.340 0.132 8.45 0.322 0.80954 13.254 0.80925 5.095 0.714 0.252 0.005
9.0 0.81047 11.130 0.148 7.24 0.358 0.80954 12.064 0.80926 5.109 0.714 0.251 0.005
10.0 0.81049 10.130 0.163 6.33 0.385 0.80950 11.115 0.80923 5.064 0.716 0.250 0.005
20.0 0.81066 4.720 0.247 1.01 0.797 0.80975 5.619 0.80957 4.814 0.731 0.236 0.005
M092
5.0 0.91899 15.590 0.015 15.27 0.022 0.91828 16.304 0.91742 1.885 0.586 0.352 0.019
6.0 0.91899 12.680 0.017 12.22 0.038 0.91827 13.402 0.91772 1.727 0.603 0.344 0.012
7.0 0.91899 10.910 0.020 10.48 0.041 0.91827 11.633 0.91758 1.836 0.606 0.343 0.013
8.0 0.91900 9.620 0.023 9.13 0.053 0.91828 10.335 0.91764 1.845 0.607 0.342 0.013
9.0 0.91900 8.730 0.023 7.97 0.089 0.91828 9.449 0.91805 1.703 0.609 0.342 0.013
10.0 0.91900 7.980 0.027 7.10 0.113 0.91829 8.689 0.91807 1.810 0.611 0.340 0.013
20.0 0.91903 4.220 0.052 2.73 0.359 0.91832 4.915 0.91873 1.781 0.621 0.334 0.013
30.0 0.91906 2.640 0.088 1.10 0.596 0.91838 3.311 0.91862 1.980 0.643 0.315 0.011
40.0 0.91910 1.920 0.104 0.45 0.776 0.91844 2.559 0.91878 1.791 0.676 0.286 0.008
50.0 0.91914 1.450 0.114 0.03 0.983 0.91856 1.996 0.91893 1.630 0.809 0.161 0.002
M102
8.0 1.02047 7.250 0.010 7.43 -0.023 1.02019 7.540 1.01955 0.740 0.545 0.355 0.020
9.0 1.02047 6.460 0.000 6.49 -0.005 1.02021 6.730 1.01970 0.740 0.553 0.361 0.018
10.0 1.02047 5.850 0.010 5.75 0.019 1.02020 6.110 1.01982 0.750 0.555 0.365 0.018
20.0 1.02048 3.120 0.010 2.22 0.291 1.02017 3.430 1.02060 0.780 0.569 0.370 0.021
30.0 1.02049 2.120 0.020 1.18 0.449 1.02018 2.420 1.02051 0.920 0.596 0.351 0.019
40.0 1.02049 1.620 0.030 0.67 0.594 1.02019 1.910 1.02053 0.910 0.623 0.330 0.015
50.0 1.02050 1.300 0.030 0.38 0.714 1.02023 1.550 1.02056 0.860 0.652 0.306 0.012
60.0 1.02051 1.060 0.030 0.17 0.844 1.02025 1.300 1.02060 0.800 0.693 0.270 0.008
Table 4: For each model listed in Table 1 experiencing Strong Flashes, we report as a function of the accretion rate  in  , the value of the total mass at the bluest point along the HR-diagram loop in , the mass transferred up to the onset and after the RLOF episode, and , respectively, in , the mass lost during the RLOF episode in , the accumulation efficiency , the mass coordinate of the He-burning shell at the epoch of the maximum extention of the flash-driven convective shell in , the maximum extention of the flash-driven convective shell in , the mass coordinate of the He-burning shell at the end of the RLOF in , and the average chemical composition by mass fraction of the matter ejected during the RLOF.
Figure 6: Accumulation efficiency   as a function of the accretion rate for models of different initial mass as labelled inside the figure. Solid lines display polynomial fits to the data (see Appendix A.3). The dotted and dashed lines represent the accumulation efficiency determined by KH04 for CO WDs with initial mass 0.9 and 1.0 , respectively (their Eqs. 4 and 5).

At variance with KH04, in the present work we do not assume that the optically thick wind operates and we compute the post-flash evolution of CO WDs experiencing strong flashes by assuming that a RLOF occurs. When the surface radius of the accreting WD becomes larger than the corresponding we subtract mass by requiring that the star remains confined inside its lobe. When the WD definitively starts to contract, we allow the accretion to resume and we follow the evolution up to the bluest point of the loop in the HR diagram. Hence, the accumulation efficiency along the cycle is computed as:

(5)

where is the mass lost during the RLOF, while and are the mass accreted onto the WD before and after the RLOF episode, respectively. In real binaries, the value of is determined by the parameters of the system itself, i.e. masses of primary and secondary components and separation. For this reason, we computed some tests by fixing the initial WD mass (namely the M102 model) and varying the value of in the range 1 – 45 9. Our results show that the uncertainty in the estimated  is smaller than 7-8 %. For this reason, we set in the computations of all the models experiencing Strong Flashes. The values of  computed according to Eq. (5) are displayed in Fig. 6 as a function of the accretion rate for each CO WD model. Note that they refer to the first Strong He-flash experienced by all the Heated Models listed in Table 1. As it can be noticed, decreasing the accretion rate, the accumulation efficiency reduces very rapidly. Indeed, lowering , the resulting He-flash is stronger and, hence, a larger amount of mass has to be removed in order to dissipate the extra-energy content of the whole He-rich layer. For the M102 model, the accumulation efficiency for  lower than   , becomes negative; this means that during the RLOF episode the thin He-rich layer already existing at the bluest point in the HR diagram has been partially eroded. Polynomial fits of  as a function of the accretion rate for all the models displayed in Fig. 6 are provided in Appendix A.3.

In Table 4 we list, for each initial CO WD model, the accretion rate, the total mass at the bluest point along the loop in the HR diagram, the amount of mass transferred from the donor before and after the RLOF, the mass lost during the RLOF episode and the corresponding accumulation efficiency, as well as some other relevant physical quantities. By inspecting Table 4 it is evident that for each initial CO WD the amount of matter accreted after the RLOF is negligible, even if it increases as the accretion rate increases. Moreover, Table 4 reveals that, for the computed models, the flash-driven convective shell extends all over the accreted layer and the most external zone of the He-rich mantle already existing at the epoch of bluest point in the HR diagram loop. This implies that at the onset of the RLOF episode the zone above the He-burning shell has been completely homogenized and, hence, it is depleted in helium and enriched in carbon, the oxygen production resulting marginally efficient. In the last three columns of Table 4 we report the average chemical composition of the ejected matter. As it can be noticed, for a fixed initial CO WD, the lower the accretion rate, the stronger the He-flash and, thus, the more efficient the He-consumption and the larger the carbon abundance. Moreover, the larger the initial CO WD, the lower the matter accreted before the RLOF and, hence, the mass of the convective shell, so that, on average, the dilution of the He-burning ashes is lower and the resulting abundance in the ejecta becomes larger.

As shown in Table 4 for a fixed initial CO WD, , the post-RLOF mass of the He-rich envelope above the He-burning shell , is largely independent of . This reflects the property that the location along the high luminosity branch, i.e. the effective temperature and, hence, the surface radius, depend on the mass of the He-layer above the burning shell. Notwithstanding, Table 4 reveals that still slightly depends on the accretion rate: in low mass initial CO WDs models it is larger for higher , while in more massive ones it exhibits a maximum for intermediate values of . A further inspection of Table 4 reveals that at the end of the RLOF the location of He-burning shell becomes closer to the surface as  increases. Both these circumstances suggest that the strength of the He-flash, which depends inversely on the accretion rate, plays a role in determining the actual value of the retention efficiency.

Encouraged by an anonymous referee to investigate in more detail such an issue, we compute several toy models, by fixing the initial CO WD (M102 model) and accretion rate (= ), and by activating, after the ignition of He-burning, an extra energy source in the layer where helium abundance is larger than 0.01 by mass fraction. This allows us to vary the strength of the flash, while as well as the ignition point remain unaltered. For the sake of simplicity we parametrize the energy delivered by this fake source as , where is the nuclear energy produced at the mass coordinate and a free parameter. Negative value of means that energy is subtracted from the He-rich layer. When the He-flash quenches and the luminosity of the He-burning shell becomes lower than 100 times the surface luminosity, we deactivated the extra energy source. Our results are summarized in Table 5 (lines 1—4): the larger the energy injected into the He-rich layer, the larger the flash-driven convective shell, the larger the mass loss during the RLOF, the more internal the position of the He-burning shell after the RLOF episode and, hence, the lower the accumulation efficiency.

We perform also another test, by putting and preventing the mixing in the flash driven convective shell. In this way the He-burning shell is not re-fuelled so that the resulting He-flash is definitively less strong ( is almost an order of magnitude lower than in the standard case) and the corresponding accumulation efficiency approaches almost unity (line 5 in Table 5). In order to further test the sensitivity of  on the amount of helium dredged down by convective mixing, we computed an additional model. In this case, during the phase when the outer border of the flash-driven convective shell coincides with the stellar surface, we artificially alter the chemical composition in the outermost  zone of the WD after each time step by restoring the local abundances of all elements as in the accreted matter (see §2). In this way, the reservoir of helium-rich matter which could feed the He-flash is increased by about 50%. As shown in Table 5 (line 6), the resulting He-flash is stronger and, correspondingly, the retention almost halves.

The results of all the toy models listed in Table 5 clearly suggest that the strength of the He-flash plays a pivotal role in determining the accumulation efficiency. In fact, it determines the energy content of the He-rich layer in accreting WDs at the onset of the RLOF and, hence, the amount of mass that has to be lost in order to attain the physical condition suitable for the accreting WD to start its blueward evolution. Moreover, the strength of the He-flash determines the maximum inward shift of the He-burning shell during the flash-driven convective episode as well as the duration of the expansion phase up to the RLOF, affecting the exact value of . Note that the energy delivered by the He-flash is determined mainly by the value of  for a fixed initial CO WD, even if our results also demonstrate that the convective mixing acts as a propelling mechanism of the He-flash itself, thus affecting the final value of the accumulation efficiency.

0.0 4.223 0.117 0.453 0.242 1.02053 0.91
-0.5 2.559 0.075 0.648 0.237 1.02091 0.96
0.2 4.609 0.124 0.423 0.244 1.02037 1.00
0.5 5.022 0.130 0.393 0.245 1.02029 1.02
No Convective Mixing
0.0 0.628 0.019 0.913 0.253 1.02160 0.83
Altered Chem. Composition
- 9.282 0.158 0.251 0.242 1.02027 0.76
Table 5: Selected physical properties of the test models computed by artificially varying the thermal content of the He-rich layer (see text for more details). For comparison we report also the standard case . represents the maximum luminosity of the He-burning shell and is expressed in unit. The other quantities are the same as in Table 4 and have the same unit.

For the sake of comparison, in Fig. 6 we also show the values of  obtained by KH04 in the framework of the optically thick wind scenario for a 0.9 (dotted line) and 1.0  (dashed line) CO WDs. The differences in the estimated retention efficiency reflect the different assumptions concerning the mass loss episode. To make more clear this issue, let us recall that the huge energy released during the He-flash is stored in the He-shell as thermal energy, since the nuclear timescale is shorter than the radiative diffusion timescale due to partial degeneracy of the matter. Hence, the flash-driven convective episode redistributes the thermal energy excess, so that the thermal content of the layer above the He-burning shell increases while its physical dimensions (both in radius and mass) remain practically unaltered. In this way the thermal energy of the He-rich layer becomes too large for its very compact configuration and, hence, it has to be dissipated. Such a situation is similar to what occurs to an ideal gas evolving at constant volume (and, hence, constant density): if the temperature is increased then the pressure has to increase. In He-flashing structure the overpressure determined by the increase of the thermal content triggers the expansion of the whole He-rich layer, making work against gravity. In this way the volume increases, the density decreases as well as the specific heat; thus, the thermal content of the He-rich zone decreases. The Roche geometry defines a finite volume in the space so that matter residing inside the corresponding lobe represents the expanding WD while the mass passed through it is lost from the binary system. Hence, due to the continuos expansion, the portion of the star remaining inside the lobe has a lower density and, hence, its specific heat decreases.

0.1 1.426 0.086 0.60 0.594 0.91863 1.335
1.0 1.444 0.112 0.16 0.895 0.91886 1.566
10.0 1.450 0.114 0.03 0.983 0.91893 1.630
Table 6: The same as in Table 4, but for model M092 accreting He-rich matter at    and with different values of the Roche lobe radius, as listed (in solar units).
Figure 7: Accumulation efficiency  as a function of the WD total mass at the bluest point along the loop in the HR diagram, for all the models listed in Table 1. Different curves refer to different accretion rates, as labelled (in  ).

In the RLOF model expansion determines the mass loss, while in the KH04 model wind mass-loss represents an additional mechanism favouring the reduction of the thermal content of the expanding He-rich layer. As a consequence, for low values of the accretion rate, i.e. for very powerful non-dynamical He-flashes, in the RLOF case a larger amount of mass has to be lost and, consequently, the accumulation efficiency is lower than in the case of the KH04 models. For high values of , i.e. for less energetic non-dynamical He-flashes, in the KH04 computation mass loss starts very soon, when the accretor is still very compact (the surface radius is smaller than 0.1 ) so that in this case the expansion does not play a significant role in reducing the thermal energy excess. On the other hand, in the Roche lobe scenario, the WD continues to expand to larger radii, so that the thermal energy excess in the He-envelope is already largely reduced when the mass loss episode induced by the presence of the Roche lobe occurs. As a consequence, in the latter case  is larger than in the KH04 models. Such a conclusion is confirmed by the values of  we obtain for an additional set of models, where we compute accretion of He-rich matter at   onto the M092 “Heated Model” but varying the size of the Roche lobe. As it is shown in Table 6, the lower , the larger the amount of matter lost from the accretor to remove the thermal energy excess determined by the He-flash. It is worth noticing that, if it is assumed that the strong wind by KH04 is at work, the retention efficiency for all these models should be  =0.858. Table 6 also reveals that decreases as the Roche lobe radius is reduced. This because, after the RLOF episode, during the blueward excursion along the high luminosity branch of the loop in the HR diagram the evolutionary timescale becomes longer and a sizable amount of matter can be accreted (see also Table 4). As a consequence, the lower the Roche lobe radius the shorter the time spent in steady burning condition, the smaller the value of . Moreover we found that also depends mildly on the exact value of , as the redward evolution after the end of the flash-driven convective episode is slower with respect to the time-span of the He-flash itself (see the discussion at the beginning of this section). Our results suggest that in relatively wide systems the accumulation efficiency weakly depends on the radius of the critical lobe, but it may reduce by up to a factor 2 in extremely tight systems.

In order to investigate the asymptotic behaviour of CO WDs experiencing Strong Flashes, we computed a sequence of flashes for each WD mass and  combination. For low values of  the computation of each flash episode is very time consuming, due to the large number of models in the sequence because of very small time steps and certain problems in determining the physical structure of the accretor (the models are close to becoming dynamical). For this reason, in some cases we computed just two flash episodes. For each sequence we determine the retention efficiency according to Eq. (5) and in Fig. 7 we plot the results. A table listing the accumulation efficiency plotted in Fig. 7 is available online. In the M060 and M070 cases we plot also the values of  for models accreting at  =  and  = , respectively, which start their evolution in the Mild flash Regime and, then, experience strong He-flashes (for more detail see §3.3).

The thermal content of accreting models, as determined by the accretion history before entering the Strong Flashes regime, has an important role in determining the actual values of , as, for example, in AM CVn type systems in which  in  yr after beginning of mass-transfer declines from    to    (see Fig. 13). In fact, during the previous evolution, occurring in the mild flashes regime, an extended hot layer is piled-up on the initial CO WD. This represents a “boundary condition” completely different to the small helium layer present as a consequence of the pre-heating procedure. Hence, for fixed , in the models which did not have stages of steady burning or “mild flashes” accretion, He-flash will be stronger, leading to a larger mass loss and, hence, to a lower . For example, for   , model M060 has =1 when its total mass at the bluest point is equal to 0.7251 , while, for the same accretion rate, model M070 has =0.429 when 0.7265 . The same is also true at     for the model M070, having =0.905 at 0.8118 , and M081, having =0.783 at 0.8107 . However, pulse by pulse, the thermal content of the He-rich zone is modified, as thermal energy is diffused inward on a timescale depending mainly on the CO core mass. Therefore, the mean temperature level at the base of the accreted He-rich layer becomes a function only of the actual mass of the CO core and of the mass of the helium layer (which is determined by  for each value of ). Therefore, the differences in the accreting models with different accretion history are smeared off, so that their accumulation efficiency attains an asymptotic value. This is evident when comparing M060 and M070 models accreting at  = , for which we obtain =0.389 at 0.7649 and =0.337 at 0.7645, respectively. For all these models the general trend is that, for fixed ,  decreases increasing the total mass of the accreting WDs.

These considerations are still valid for more massive initial CO WDs, as it can be derived by an inspection of Fig. 7. Indeed, for high values of the accretion rate, the accumulation efficiency increases in the M092 and M102 models, since the energy delivered by both the deposition of matter and the He-flash is employed to heat up the most external layers of the He-deprived core, thus modifying the degeneracy level of the physical base of the He-shell. Therefore, pulse by pulse He-flashes become less strong so that the fraction of mass effectively retained during the episode increases. On the other hand, for low , the compressional heating timescale becomes longer than the inward thermal diffusion timescale. Therefore, He-flashes are ignited in more degenerate matter and the corresponding  decreases as increases.

Model
M060 0.736278 5.47712 3.85658 0.73113 3.6606 8.3296 0.81164
M070 0.737064 5.48664 3.85790 0.73411 3.4683 8.2744 0.40375
M060 0.920639 5.65280 4.21510 0.91970 3.3369 8.3352 0.96985
M070 0.919911 5.66284 4.23922 0.91864 3.5007 8.3761 0.95435
M081 0.919113 5.65428 4.21838 0.91816 3.3541 8.3391 0.94949
M092 0.919064 5.68452 4.24309 0.91832 3.3638 8.3507 0.59358
Table 7: Selected properties of models with the same accretion rate and different initial masses. From left to right we list the initial model, the total WD mass at the bluest point in the HR diagram, the corresponding effective temperature and luminosity, the mass coordinate of the He-burning shell, and the corresponding values of density and temperature. In the last column we report the accumulation efficiency. See text for more details.

3.3 Mild Flashes Regime

For slightly larger values of the accretion rate, CO WDs accreting He-rich matter experience Mild Flashes. In this case the evolution is similar to models experiencing Strong Flashes: a He-rich layer is piled-up via accretion, determining the compressional heating of the He-shell up to the moment when the physical conditions suitable for He-ignition are attained. However, as  is larger than in the previously discussed case, energy losses can not counterbalance compressional heating, so that the He-shell does not become degenerate at all. As a consequence, the resulting He-flash is very mild and it delivers just the amount of energy to determine the transition of the model from the low to the high state. As a result, the radii of CO WDs experiencing this accretion regime typically remain smaller than , so, excluding the most compact AM CVn stars, no interaction with the companion could occur and all the matter transferred during each cycle is effectively deposited onto the accretors. For AM CVn stars with separations formation of a short-living common envelope may be envisioned and then  should decrease.

Figure 8: Profiles in the for models with the same total mass at the bluest point, but with different initial mass and equal accretion rate ( =  and  =  in the upper and lower panel, respectively). Filled dots mark the position of the He-burning shell. In the upper panel the dotted lines represent the M060 Heated Model.

It is well known, and we find it as a result of our computations too, that for a given value of the accretion rate, He-flashes become stronger, as the total mass of the CO WD increases. As a consequence the models may transit from mild flashes regime into the regime of the strong ones. However, the possibility of such a transition depends not only on the actual mass of the accreting WD, but also on the previous accretion history which, in turn, fixes the thermal content of the CO core. Such an occurrence appears evident when comparing the long term evolution of models with the same  and different initial mass (see Table 7). For example, model M070 accreting at   experiences strong He-flashes from the very beginning (0.70185), while model M060 accreting at the same  enters the Strong Flashes regime when its total mass has increased to 0.72514 . In the latter model the deposition of matter has deeply modified the temperature profile in the CO core underlying the He-rich layer with respect to the M070 model, as it is clearly seen in the upper panel of Fig. 8, where we plot the temperature profile as a function of density for the two models when their total mass is . The same considerations are still valid when considering the evolution of a model accreting He-rich matter at  , as it is shown in the lower panel of the same Figure. Note that in this case the temperature profiles in the inner zones of the CO cores of models M060 (dotted line) and M070 (dashed line) became practically coincident. It is worth noting that the difference in the temperature profiles of the models with the same total CO core mass and accretion rate, but with different accretion histories, leads to different accumulation efficiency, as it comes out by an inspection of the last column in Table 7. Some differences between models M060 and M070 are due to the fact that the nuclear energy delivered during the last  Myr of evolution corresponding to the mild flashes regime has been transferred inward very efficiently, so that the whole structure of the former model became hotter and less dense with respect to the M070. As well, the structures do not have exactly the same CO-core mass.

3.4 Steady Accretion Regime

In the Steady Accretion regime, by definition, the rate at which He is converted via nuclear burning into a CO-rich mixture is very close to the rate at which He-rich matter is transferred from the donor. As a consequence, models experiencing this regime evolve in the HR diagram along the high luminosity branch of the typical loop. The long-term evolution of these models is determined by the interplay of two different factors:
(i) With increase of the mass of the CO core the luminosity level of the models becomes larger. In order to counterbalance larger radiative energy losses, the shell has to burn helium at a higher rate. This determines a progressive reduction of the mass of the He-rich mantle;
(ii) The external layers of the CO core are hot and expanded, since they have been piled-up via nuclear burning. Contraction of this zone delivers thermal energy which represents an additional source to balance the radiative loses from the surface. Hence, He-rich matter has to be burnt at a lower rate. In any case, for a model with a constant or a decreasing accretion rate, the mass of the He-rich envelope progressively reduces and when it becomes smaller than a critical value the accretor enters the Mild Flashes regime.

In Fig. 9 we show accretion rate at which the transition from the Steady to the Mild Flashes regime occurs as a function of the WD total mass. For comparison we also plot the transition line as derived for Fig. 2 (open circles with thick solid line). This figure reveals that the thermal content of the most external zones of the CO core just below the He-burning shell affects the value of the WD total mass at which the transition occurs. Such a conclusion is reinforced when considering that different initial models converge to the same value when about (0.05–0.1)  of He-rich matter has been accreted.

Figure 9: Evolution of the accretion rate at which the transition from the Steady Accretion to the Mild Flashes regimes occurs as a function of the WD total mass . Different lines refer to different initial models. Heavy solid line and the open dots represent the lower limit of the Steady Accretion zone shown in Fig. 2.

3.5 RG Regime

If the accretion rate is definitively larger than the rate at which helium is nuclearly processed into a CO-rich mixture, a massive He-rich mantle is piled-up. As a consequence, accreting WD resembles a post-AGB star: the more massive the helium envelope, the lower the effective temperature and, hence, more expanded the structure. Models in the RG accretion regime evolve redward in the HR diagram, developing very soon a surface convective layer which penetrates inward as the structure expands. When considering that accreting WDs are components of interacting binary systems, it turns out that, depending on the geometry of the systems, they could overfill their Roche lobe, so that a part (if not all) of the matter transferred from the donor has to be ejected from the binary system.

(  ) () () () (K) () () () (K)
He-accreting CO-accreting
M102
2 1.3746 0.0000 9.3230 8.4321 1.3733 0.0000 9.3230 8.4321
3 1.3347 1.3204 6.3845 8.7761 1.3559 1.3445 6.4328 8.7743
M092
2 1.3747 0.0000 9.3230 8.4321 1.3740 0.0000 9.3195 8.4358
Table 8: Physical properties of some selected models experiencing C-ignition. For more details see the text.

In order to define the lower limit in the plane for the RG regime we adopt the same procedure as for models in the Strong Flashes regime. In particular, for each model listed in Table 1, we determine the values of the  for which the accreting WD expands, thus attaining an effective temperature of 11300 K. Hence, we assume that the Roche lobe radius of accretor is equal to 10 and we force that by subtracting mass from the WD. We compute the minimum rate for the RG regime as:

(6)

where is the rate at which matter is transferred from the donor, is the amount of mass lost via RLOF and – the time step. Our results suggest that for a given initial model the larger is the accretion rate, the more rapid is the expansion, so that the transition to the RG regime occurs at smaller WD total mass (see Fig. 10). However, after a short transition phase, all the models with the same initial mass converge to the same limiting value, clearly indicating that depends on the thermal content of the CO core as determined both by the compression and the thermal energy flowing inward from the He-burning shell. When comparing models with different initial masses the same conclusion is still valid.

Figure 10: Evolution of as a function of the WD total mass. Solid lines refer to models with different initial mass and the same accretion rate   . Dotted, dashed, long-dashed and dot-dashed lines refer to the M060 model accreting He-rich matter at  , respectively. The heavy solid line represents the upper limit of the Steady Accretion zone, as in Fig. 2.

If an accreting WD is in the RG regime, it is quite reasonable to assume that the mass excess with respect to the maximum value defined by Eq. (6) is lost by the system. However, the further evolution of this kind of binary systems strongly depends on the amount of angular momentum carried away by the lost matter as it determines the evolution of the separation and, hence, the rate at which the donor continues to transfer mass. As a matter of fact, if the mass transfer occurs at a rate typical of the RG regime both the components of the binary system should become immersed in a common envelope.

Figure 11: Profiles in the plane for some selected models attaining the physical conditions suitable for C-ignition. Solid and dashed lines refer to He-accreting and CO-accreting WDs, respectively. The location of the He-burning shell in He-accreting models is marked by a filled dot. The dotted line represents the ignition line.

4 Evolution up to the C-ignition

In principle, if the He-donor is massive enough, accreting WDs could attain the physical conditions suitable for C-ignition. In this regard, it is important to remark that in He-accreting WDs a part of the nuclear energy released via nuclear burning is transferred inward though it does not provide a significant heating of the underlying CO core. In fact, the thermal evolution of the CO core is driven by the deposition of CO-rich material, the ashes of the He-burning shell. As a consequence, it can be argued that for values of the accretion rate lower than  C-burning will be never ignited in He-accreting CO WDs. If CO-rich matter is directly accreted onto CO WDs, for such values of  the structure can grow in mass up to the Chandrasekhar mass limit, when the strong homologous compression determines the physical conditions suitable for C-ignition at the center (for example, see Piersanti et al. 2003 and references therein). However, according to the results discussed in § 3.2, for   , He-accreting WDs enter the Strong Flashes regime, independently of their initial mass and the amount of matter effectively accreted decreases very rapidly to zero as  increases. For this reason, in the considered range of accretion rates, the final outcome is a very massive CO WD.

For larger , the extant results for WD accreting CO-rich matter suggest that central C-ignition occurs for   , while for larger values C-burning is ignited off-center (see Piersanti et al. 2003 and references therein). In order to define the maximum for which a C-deflagration Supernova could occur, we computed the long-term evolution of He-accreting WDs with accretion rates (1 – 3) . The results are summarised in Table 8, where we report the mass coordinate where C-burning is ignited, the values of density and temperature at that point and the total mass at the ignition time10. In Fig. 11 we plot the profile of the last computed structure in the plane for the model M102 accreting at    and    (upper and lower panels, respectively). Solid lines refer to He-accreting WDs, while dashed lines to CO-accreting WDs. As it can be noticed, models igniting Carbon at the center have the same physical properties, independently of the chemical composition of the accreted matter. In this case, the evolution up to the ignition is driven by the contraction of the whole accreting WD as it approaches . At variance, the temperature and density profiles in models igniting Carbon off center depend on the presence or not of the He-burning shell. In fact, since C-burning occurs very close to the surface, thermal energy flowing from the He-burning shell keeps hotter the underlying CO layer.

According to the previous considerations, central C-burning in highly degenerate physical conditions can occur only if  . All the He-rich matter accreted in the Steady and Mild Flashes regimes is retained by the WD and determines the growth in mass of the CO core. The matter accreted in the Strong Flashes regime is only partially retained (when  is very low it is not retained at all). When the WD enters the Dynamical Flashes regime, it could explode if the necessary amount of He-rich matter appropriate to its current mass is accreted.

5 Retention efficiency of H-accreting WDs.

In SD systems, where the donor has a H-rich envelope, a He-rich layer can be piled up onto the CO WD if the H-accretion rate is larger than the upper limiting rate for the strong nova-like H-flashes (e.g. see Cassisi, Iben & Tornambe, 1998, and references therein). If H-accretion occurs steadily, the rate of H-burning and, hence, of He-accumulation are by definition equal to the accretion rate. On the other hand, if the H-accreting WD experiences mild H-flashes11, the matter accreted before the H-flash is nuclearly processed during the high luminosity phase after the H-flash so that averaging along the loop the rate of He-deposition is almost equal to the rate of H-accretion.

M060 M070 M081 M092
He-Acc. H-acc. He-Acc. H-acc. He-Acc. H-acc. He-Acc. H-acc.
0.5871 0.5866 0.6983 0.6981 0.8092 0.8090 0.9188 0.9186
4.9218 3.8365 4.6535 3.2002 4.2944 2.8316 5.6805 3.4903
1.3902 1.7216 1.4458 1.8366 1.5054 1.9540 1.4653 1.9311
1.3985 1.4413 0.9268 0.8033 0.5281 0.4460 0.4156 0.3076
9.2495 9.2780 4.9761 4.5877 1.4159 1.3212 1.1457 1.0369
3.1191 3.0973 7.3273 4.1301 7.8490 5.2372 32.5150 7.4220
2.2862 0.9673 0.5615 0.1640
0.659 0.649 0.5209 0.5797 0.6561 0.5928 0.3236 0.4518
Table 9: Selected physical properties of models M060, M070, M081 and M092 accreting H- and He-rich matter at a rate given by Eq. (7). The listed values of the He-shell position (), its density () and temperature (), the mass of the He-rich () and H-rich () layers refer to the epoch of He-flash ignition. represents the maximum luminosity produced via He-burning during the flash episode. In the last row we report the retention efficiency for the computed models.

Usually the long term evolution of H-accreting CO WDs is determined by assuming that the behaviour of the He-rich layer is equal to that in a model accreting directly He-rich matter at the same rate. According to the range of  values derived by Cassisi, Iben & Tornambe (1998) and Piersanti et al. (1999) for the steady H-burning and mild H-flashes in H-accreting WDs, the helium layer piled-up via nuclear burning can undergo either a dynamical or a strong non-dynamical flash. However, on a general grounds, the H-burning shell can modify the thermal content of the He-rich zone, so that H-accreting WDs could have an evolution different from those accreting He-rich matter at the same . Piersanti et al. (2000) demonstrated that, for low values of , corresponding to the Strong He-Flashes regime, the mass of the He-rich layer becomes so large that the physical base of the He-shell, where the He-flash will be ignited, is insulated from the overlaying H-burning shell. In this case the mass of the He-rich zone as well as the temperature and the density at the He-ignition are practically the same in models accreting H- or He-rich matter at the same rate. On the other hand, Cassisi, Iben & Tornambe (1998) showed that for WDs accreting H-rich matter in the Steady regime thermal energy flows from the H-burning shell inward, keeping the whole He-rich zone hotter with respect to models accreting directly He-rich matter at the same rate. As a consequence, in H-accreting models He-burning is ignited when the mass of the He-rich zone is smaller. This might have important consequences in determining the total matter retention efficiency of H-accreting WDs.

Figure 12: The parameter space suitable for steady accretion of H-rich matter is displayed (grey area) overimposed to the accretion regimes derived in the present work for He-accreting WDs. Heavy dashed line corresponds to Eq. (7).

To better illustrate this issue we computed some additional models, by accreting H-rich matter onto the M060, M070, M081 and M092 Heated Models. The chemical composition of the accreted matter has been set to X=0.7, Y=0.28, Z=0.02, elements heavier than helium having a scaled-solar distribution. Moreover, we assumed that the CO WD is accreting in the Steady H-burning regime at the fixed accretion rate

(7)

In Fig. 12 we show the steady accretion regime for H-accreting models (grey area) as derived by Piersanti et al. (1999), overimposed to the possible accretion regimes of He-accreting WDs as derived in § 3. The long dashed line represents the value of provided by Eq. (7).

For comparison, we compute also models with the same accretion rate, but with chemical composition of the accreted matter X=0, Y=0.98, Z=0.02 as in § 3. In Table 9 we report some relevant quantities of the computed models.

In the M070, M081 and M092 models the total amount of mass transferred to the WD as well as the mass of the He-rich layer at the onset of the He-flash are lower in the H-accreting case. Moreover, due to the flow of thermal energy from the H-burning shell inward, the bottom of the He-shell, where the He-flash is ignited, is hotter and less dense with respect to the same model accreting directly He-rich matter. As a consequence, in the H-accreting case the He-flash is less strong, as suggested by the lower value of . Similar considerations are valid also for the M060 case, even if the differences between the H- and He-accreting cases are less marked, because the accreted matter represents a small fraction ( 15%) of the He-rich zone at the onset of the He-flash. Due to the He-flash, all the models expand to giant dimension, so we assume that the accreting WD fills its Roche lobe when its surface radius becomes larger than 10 . At the onset of the RLOF we stop the accretion and compute the following evolution up to when the WD definitively contracts. The obtained retention efficiency is listed in the last row of Table 9.

Our results are in good agreement with the findings by Cassisi, Iben & Tornambe (1998), while they contradict the recent work by Newsham, Starrfield & Timmes (2013). A direct comparison of the computations in the latetr work and our results is possible only for the M070 model accreting at  corresponding to the model with =0.7 , accreting H-rich matter at  =1.6  . In this case we find that the strong He-flash is ignited when a mass of  has been accreted onto the WD, while Newsham, Starrfield & Timmes (2013) halted their computation after the total accreted mass is , i.e. well before the physical conditions for igniting He-burning in the He-rich layer could be attained.

Recently, Idan, Shaviv & Shaviv (2013) (hereinafter ISS13) analyzed the very-long term evolution of H-accreting massive WDs. They considered high value of , corresponding to steady H-burning in a ”red-giant-like-star” and they find that all the computed models experience a very powerful He-flash driving to the expulsion of the whole accreted He-layer. Their models can not be compared directly with the results of our computations because we consider initial WDs definitely less massive. Moreover, ISS13 included in their computation the effects of thick wind which reduces the average value of the accretion rate. In the case of their model with =1.00  accreting H-rich matter at  =  the strong He-flash occurs when the mass of the He-rich layers attains , after about 33000 yr (4153 H-flashes with a period of 8 yr). This corresponds to an average growth rate of the He-rich layer of about  . This model can be compared with our M102 model accreting He-rich matter directly at the same accretion rate, even if some caveats has to be borne in mind. The initial model in ISS13 is a bare CO core with a temperature of K at the center and of K at the border of the He-deprived core, while in our M102 “Heated Model” the CO core is capped by a He-rich mantle and the temperature in the zone below the He-burning shell is larger (see Table 1). Moreover, model M102 has already experienced one very powerfull He-flash (pre-heating procedure) while ISS13 focus their attention on the very first He-flash. Note that in the latter model the recurrent H-flashes experienced by the accreting WD deliver an amount of nuclearly energy definitively lower than the pre-heating He-flash in our M102 model. This implies that the thermal content of the layer below the He-burning shell in the initial model adopted by ISS13 is definitively lower than in our M102 “Heated Model”. At the end, as discussed at the beginning of this section, the mass of the layer above the He-burning shell at the epoch of the He-flash could be lower in WD accreting helium as a by product of H-burning because thermal energy is diffused inward from the burning shell, thus allowing He to be ignited when a smaller amount of He-rich matter has been piled-up. In our model M102 accreting He-rich matter directly at a strong He-flash is ignited after the deposition of  of matter; the He-flash is ignited at the mass coordinate 1.02050 and the mass coordinate of the He-burning shell moves inward during the flash-driven convective episode down to the mass coordinate 1.02018. Hence, in our model the mass of the He-rich layer above the burning shell is at maximum  (see Table 4), a factor 4 smaller than in the ISS13 computation. In our computation the maximum temperature attained during the He-flash is  K, while the maximum luminosity delivered via He-burning is as high as (see Table 5).

Note that the M102 “Cool Model” accreting He-rich matter at  has negative retention efficiency (see the discussion in §2). At variance, the M102 “Heated Model” accreting He-rich matter at the same rate has a small but positive accumulation efficiency (%). Such an occurence shows that the strength of the He-flash depends on the physical conditions (mainly the temperature) below the He-burning shell. In order to verify if the discrepancy between our results and those in ISS13 depends on the thermal content of the adopted initial CO WD, we evolve our M102 “Cool Model” up to the instant when its temperature at the center decreases to K and its central density increases to . At this epoch the mass coordinate where the He-abundance is larger than 0.01 by mass fraction has a temperature of K This structure is more similar to the one adopted as starting model by ISS13, even if the temperature at the physical base of the He-rich layer is a factor of 3 larger. Hence, we accrete directly He-rich matter at and we find that a very strong He-flash is ignited at the mass coordinate 1.02230 when the accreted mass is . During the flash-driven convective episode the He-shell moves inward to the mass coordinate 1.02017, so that the mass of the He-rich layer above the He-burning shell is . In this case the maximum temperature attained during the He-flash and the maximum luminosity related to He-burning are K and , respectively. By comparing this last model with the previous one, it comes out that, for a fixed value of , the amount of matter to be accreted to trigger a He-flash and, hence, the strenght of the He-flash itself do depend on the thermal content of core underlying the point where He-burning is ignited. We do not follow the RLOF episode of this additional model, but, according to the discussion in §3.2, we can assume that the corresponding value of the accumulation efficiency is similar to the one obtained for the M102 “Heated Model” accreting He-rich matter at  , i.e. . According to the data in Table 4, this latter model accretes  during the evolution prior to the RLOF episode and the maximum mass of the He-rich layer above the He-burning shell is  . Moreover, during the He-flash the maximum luminosity related to He-burning is as high as . This result confirms that the discrepancy between our findings and those by ISS13 depends only on the thermal content of the initial model. In particular, our “Heated” M102 model has already experienced a strong He-flash, which modified the thermal content below the He-burning shell, while the initial CO core adopted by ISS13 is very cold, as they do not adopt any pre-heating procedure.

6 Some Applications

We discuss below several types of binaries with accretion of helium onto CO WDs. Helium WDs and helium stars in close binaries (CB) form in the so-called “case B” of mass-exchange, when the stars overflow Roche lobes in the hydrogen-shell burning stage (Kippenhahn & Weigert, 1967). During the subsequent evolution, some of them can stably transfer mass onto companion. For the purpose of this paper we are interested only in helium WDs and helium-stars with the lowest rates of mass-transfer, which do not result in formation of extended envelopes of WD ().

6.1 CO WDs in semidetached systems with helium WD companions

Helium WD companions to CO WDs have precursors with main-sequence mass . If the time scale of angular momentum loss by a detached pair of He and CO WDs via radiation of gravitational waves is shorter than the Hubble time, He WD which has larger radius than its companion may overfill its Roche lobe, forming an “interacting double-degenerate” system. Observationally, these systems are identified with AM CVn stars (Paczyński, 1967). Nelemans et al. (2001) nicknamed this variety of AM CVn’s “WD-family”. Their precursors may be hidden, e.g., among so-called “extremely low mass” (ELM) detached binary white dwarfs with the mass of visible component , a significant fraction of which is expected to lose mass stably after the contact (see Brown et al., 2013, and references therein). Evolutionary considerations and conditions for stable mass exchange limit initial masses of the donors and accretors in the WD family of AM CVn stars by (0.1 – 0.3)  and (0.5 – 1.0) , respectively (Nelemans et al., 2001; Marsh, Nelemans & Steeghs, 2004; Solheim, 2010). As it was shown by Tutukov & Yungelson (1996) and Nelemans et al. (2001), time-delay between formation of a detached pair of He- and CO-WD and the onset of mass-transfer may last from several Myr to several Gyr. Therefore, finite entropy of Roche-lobe filling WD should be taken into account in evolutionary computations (see Deloye et al., 2007, and references therein). Non-zero entropy of the donors is reflected in the degree of their degeneracy, which may be characterized by central “degeneracy parameter” , where , , are the central density, temperature, and electron Fermi energy, respectively. “Mass of the donor – mass loss rate” relations for two systems: ( and (0.3, 1.025, 3.0) are plotted in Fig. 13. The former system represents an example of a binary with a “hot” low-mass donor in which RLOF occured very soon after formation, while the latter system is an extreme example of the system with a “cold” donor and massive accretor. Typical evolutionary tracks for AM CVn stars should be located between these two curves. The tracks, computed by full-scale evolutionary code with realistic EOS and opacities under the assumption of completely conservative mass exchange were kindly provided by C. Deloye (see Deloye et al., 2007). Actually, if completely nonconservative evolution is assumed or mass of the donor is varied to also typical value of accretor mass 0.2 , the tracks in the -  plane practically do not differ. Heavy dots overplotted on the lower line in Fig. 13 represent the lower limits of   for steady burning, mild flashes and strong flashes regimes, respectively, as derived in the present study. In the strong flashes regime WD experiences about 10 flashes. White Dwarf accumulates  before entering the Dynamical Flashes regime and, since, for  WD it is necessary to accrete at least about 0.2 for a dynamical flash (see Fig. 3), the latter never happens. Thus, our results confirm that the strongest “last” flash really should exist, possibly producing a “faint thermonuclear supernova” (SN .Ia, Bildsten et al. (2007). But note, none of the observed explosive events suggested to be SN .Ia was confirmed so far (Drout et al., 2013).

Figure 13: Mass loss rate by finite-entropy He WDs vs. their mass in the systems with initial masses of donor and accretor ( (solid line) and  (dotted line). is the degeneracy parameter (see text). In the former system , while in the latter system . Heavy dots at the upper line mark the  limits for RG, steady burning, mild and strong flashes regimes (left to right). At the lower line, the same limits for the latter three regimes are marked, since  in this case never is high enough for the formation of an extended envelope. Ticks on both lines mark time elapsed from the beginning of RLOF – 0.1, 1 and 10 Myr, respectively.

At the beginning of mass transfer in the system with more degenerate donor and more massive accretor, the donor loses mass for  yr at a rate exceeding the upper limit for the steady burning of He by the WD. The mass lost by the donor in this regime is , while the accretor may burn only . It is reasonably to assume that the resulting small amount of ejected matter cannot lead to the formation of a common envelope. In the steady accretion, mild and strong flashes regimes the CO WD may accrete additionally . Extrapolation of the data presented in Table 2 suggests that the accretor will experience a dynamical flash. Even if it will evolve into a detonation, it is still under debate, whether detonation of the He-shell may result in a double detonation and destruction of the binary (see, e.g., Moll & Woosley, 2013; Shen & Bildsten, 2014b). Note, in this case the mass of the accumulated helium may be too high to allow the existing theoretical models to reproduce correctly observations of SNe (Kromer et al., 2010). But such events may be hidden among other transients. As well, they hardly contribute significantly to the total rate of SNe Ia, since AM CVn stars are rare themselves (birthrate  yr, Nelemans, Yungelson & Portegies Zwart 2004) and typically have low-mass accretors.

It is interesting to note that all the possible explosive events in AM CVn stars happen during the first several Myr of their lifetime. This means that currently in the Galaxy exist only several of these binaries which may still be “nuclearly active”, while the rest of them may show only accretion-related activity.

However, it is worth to note that, since the delay-time between the formation of CO WD + He WD pair and the beginning of RLOF may be up to several Gyr and the evolution to a dynamical flash also proceeds in Gyr-long time scale, explosions may occur in galaxies of any morphological type.

Figure 14: Mass loss rate by nondegenerate He-donors in binaries vs. mass of the He-star. Upper panel — (0.35, 0.6)  pair; lines a, b, c show evolution of the systems with (initial period , He-abundance in the core) = (20 min., 0.98), (100 min., 0.64), and (144 min, 0.118), respectively. Middle panel — the same for (0.4, 0.8) pair, with (20 min. 0.98), (100 min, 0.51), and (140 min, 0.09). Lower panel — the same for (0.65, 0.8) pair, with (35 min. 0.98), (80 min, 0.4), and (85 min, 0.29). In the shaded regions of the plots He burns in the Strong Flashes regime (the lower limits of them correspond to a completely conservative evolution, while dotted lines mark the same limit for completely nonconservative evolution).

6.2 CO WD in semidetached systems with low-mass helium-star companions

For solar chemical composition, the precursors of nondegenerate He-star components in close binaries (with minimum mass ) have main-sequence mass . The He-star mass is related to the mass of its MS precursor as (in solar units). If , the stars do not expand during core helium burning stage which lasts up to (500 – 700) Myr. If during this time-span angular momentum loss via gravitational waves radiation brings the two components into contact, helium star may fill its Roche lobe and, if the conditions for stable mass loss are fulfilled, mass transfer onto the companion starts. In systems with a CO WD these stars first evolve to shorter orbital periods  min. At the epoch of the period minimum, the He-stars masses decrease to (0.20 – 0.25)  and they start to lose matter that was nuclearly processed in their convective cores prior to contact (RLOF quenches very fast the nuclear burning — Savonije, de Kool & van den Heuvel 1986).

Semidetached low-mass He star + CO WD binaries also are suggested to be a variety of AM CVn type stars (Savonije, de Kool & van den Heuvel, 1986; Iben & Tutukov, 1991; Tutukov & Yungelson, 1996), nicknamed “He-star family” (Nelemans et al., 2001). Typical mass transfer rates in these systems before the period minimum are    (Yungelson, 2008, see also Fig. 14).

In Fig. 14 we show three examples of evolutionary tracks of He-stars in semidetached systems with WD companions from Yungelson (2008) in “mass of the donor - mass-loss rate” plane. The tracks shown in Fig. 14 were computed in conservative approximation, but this does not affect their behaviour compared to the completely nonconservative case (Yoon & Langer, 2004; Yungelson, 2008). Initial combination of masses =(0.35, 0.6) is deemed to be typical for precursors of He-star AM CVn systems, while (0.4, 0.8)  and (0.65, 0.8)  may be more rare (see Fig. 3 in Nelemans et al., 2001). For each combinations of He-stars and CO WDs mass in Fig. 14 we show three characteristic tracks corresponding to the RLOF by a completely unevolved star (line a), a star with He in the core consumed by about 40% to 60% (line b) and a star with significantly He-depleted core (, line c); for lower overall contraction begins and the stars can not fill their critical lobes. Shaded regions in the plots show the domain of strong He-flashes (after Fig. 2 above). The lower border of this region is drawn under the assumption that He-accretion occurs conservatively, i.e., the mass of WD is the sum of its initial mass and mass lost by He-star. For stars experiencing strong flashes the border between dynamical and strong flashes regime is between shaded regions and dotted lines, representing the transition for completely non-conservative evolution.

In the (0.35+0.6)  set of systems, the accreted He never experiences a dynamical flash since such systems predominantly evolve into the Strong Flashes regime of He-burning immediately after the RLOF. Moreover, the donor is not massive enough to provide enough mass for a dynamical flash after entering this accretion regime (see Table B2 in the Additional material in the online version).

In the (0.4+0.8)  set of systems, the accreting WDs stay in the range of mass-exchange rates corresponding to the Dynamical Flashes regime for a part of the pre-period-minimum time, but hardly accumulate enough He to give rise to a dynamical event (Fig. 3). In the Strong Flashes regime they experience outbursts and, even if the corresponding retention efficiency is small, the WD mass should increase. For this case we may safely assume that by reentering the Dynamical Flashes regime the total mass of WDs is close to 1.0 . Then, WDs of this set should experience a dynamical flash shortly after the period minimum, when several 0.01 has been accreted. According to Fink et al. (2010), the detonation of He-shell in this case may initiate the detonation of carbon close to the center of the CO WD.

Finally, in the most extreme case of (0.65+0.8)  systems no strong flashes happen, but dynamical flashes of He onto WD may occur and double detonations may be expected.

Like semidetached WD+WD systems, low-mass He-stars evolve to the period minimum in  yr only. This may mean that most of the AM CVn stars of He-star family existing in the Galaxy already experienced their “last flash” (SN .Ia in the case of detonation) and continue to evolve without any expected thermonuclear events. It is possible that some would-be AM CVn’s ceased their existence due to double-detonations shortly after their birth. The estimation of the rate of the latter events should be addressed by means of population synthesis calculations taking into account relations between critical masses for explosive events, retention efficiency and , which was never made before. The same, in fact, is true for the total population of AM CVn’s, since in the existing studies its formation rate was restricted by rather ad hoc assumptions, while the effects of unstable He-burning were not taken into account. The relevance of such a new study is also emphasized by the fact that the existing “theoretical” models predict significantly larger Galactic population of AM CVn stars than observed and it has been suggested that the “theoretical” models overestimate their number (see, e.g., Carter et al., 2014, and references therein).

6.3 Helium-star channels to SN Ia

Above, we discussed the evolution of CO WDs accreting He from low-mass stellar companions () which do not expand after exhausting He in their cores. More massive He-stars may overflow Roche lobe both in the core He-burning and in the He-shell burning phases (Paczyński, 1971). In the latter case the expansion (up to several 100 ), which occurs in the thermal time scale is limited only by the existence of companion. It was shown, e.g., by Iben & Tutukov (1985), Yoon & Langer (2003), Wang et al. (2009) that upon RLOF both core and shell He-burning stars may lose several 0.1  at a rate   , which corresponds for the accreting CO WDs to burn helium in steady or flashes regimes, depending on the orbital period and combination of masses of the components. If RG formation is avoided, initially sufficiently massive CO WD components may accumulate