© ESO, 2017

1. Introduction

For several decades, white dwarfs (WDs) have been considered a key component of the progenitors of a number of astrophysical transient phenomena. These include archetypical Type Ia supernovae (SNe Ia), Type Iax (SNe Iax; see Li et al. 2003; Foley et al. 2013), theoretically predicted subluminous SNe “.Ia” (Bildsten et al. 2007; Kilic et al. 2014), and other “peculiar” transients (Woosley & Kasen 2011). Owing to the key importance of SNe Ia in particular as a tool for high-redshift distance measurements (Perlmutter et al. 1999; Riess et al. 1998; Schmidt et al. 1998), observationally disentangling these kinds of transients and theoretically understanding the differences between their production mechanisms would have potentially far-reaching consequences.

Even in the light of numerous attempts, to date no direct progenitor of a SN Ia has been unambiguously identified, although a number of non-detections (see Kelly et al. 2014; Nelemans et al. 2008; Maoz & Mannucci 2008; Schaefer & Pagnotta 2012; Li et al. 2011) have considerably limited the available parameter space for candidate systems. Of these, Li et al. (2011), who attempted to identify the progenitor of the SN Ia 2011fe, is of particular interest with respect to the study of He-accreting models as it apparently excludes a large part of the available parameter space for He-donor stars but leaves open the possibility of a low-mass helium giant donor.

Interestingly the picture is somewhat clearer where SNe Iax are concerned. McCully et al. (2014) succeeded in identifying a possible progenitor of SN 2012Z as a luminous blue source, possibly a massive main sequence star or blue supergiant. However, a helium giant of ~2.0M_⊙ is not excluded. The ambiguity of the source is manifested in the failed attempt by Foley et al. (2015) to identify the progenitor of SN 2014dt, excluding some of the progenitor parameters found by McCully et al. (2014), thereby hinting at the existence of a variety of SN Iax progenitors.

These findings point to a possible connection between helium accreting WDs as progenitors of SNe Ia and/or Iax. Observed cases have been modeled with some success as deflagrations of near-Chandrasekhar-mass WDs (Magee et al. 2016; Foley et al. 2016; Kromer et al. 2015), while the observed properties of the companion stars imply high mass-transfer rates (Ṁ ≳ 10^-6M_⊙/ yr). This leaves open the question whether systems with lower mass accretion rates (~10^-8M_⊙/ yr) and less-than near-Chandrasekhar-mass WDs could also be viable progenitors for any of these transients.

Recently, a possible connection between the theoretically predicted SNe .Ia and AM Canum-Venaticorum systems (AM CVns; Warner 1995; Nelemans 2005) was proposed (Shen et al. 2010; Bildsten et al. 2007; Shen & Bildsten 2009) and extensively studied (Shen & Moore 2014), suggesting a continuum of He-accretion-induced transients defined by the nature of the donor star, the system’s orbital period and the associated mass-transfer rate.

The idea that the accumulation of material at rates of ~10^-8M_⊙/ years from a helium-rich companion on a sub-Chandrasekhar mass carbon-oxygen (CO) WD could lead to an explosive ignition was first seriously explored more than 30 yr ago (Taam 1980a,b; Nomoto 1980, 1982b,a). While it was noted that some of these detonation left the CO core of the WD intact, thereby producing a faint supernova, there was a possibility that the detonation of the He-envelope lead to a secondary detonation of the CO core, resulting in the complete disruption of the star (double detonation mechanism). Later one- and multidimensional studies (Livne 1990; Livne & Glasner 1990, 1991; Livne & Arnett 1995; Benz 1997; Livne 1997; Fink et al. 2007, 2010; Sim et al. 2010; Kromer et al. 2010; Woosley & Kasen 2011) established He-accretion as a promising avenue in the search for the progenitors of explosive transients like SNe Ia. However, synthetic spectra (Woosley & Kasen 2011; Woosley & Weaver 1994; Woosley et al. 1986; Sim et al. 2010; Kromer et al. 2010) suggest that lower amounts of He generally lead to a better agreement with the spectra of regular SNe Ia. Studies trying to establish a link between naturally occurring binary systems and their ability to produce either SNe Iax or Ia using the double detonation mechanism were undertaken by Wang et al. (2013) and Neunteufel et al. (2016).

The state of the WD at ignition and the subsequent detonation or deflagration constitutes the crucial link between the progenitor and the resulting transient. The accretion behavior, including the amount of mass that a WD is able to steadily accrete before He-ignition occurs, has previously been studied for the non-rotating case (Woosley & Kasen 2011, among others) and the rotating case (Yoon & Langer 2004b,a). These studies took into account the effects of a number of rotationally induced instabilities and the concurrent effects of angular momentum transport in differentially rotating stellar interiors. One of the most notable results of these studies was that, especially considering the effects of the dynamical shear instability, chemical mixing was strong and the amount of helium needed to induce ignition with an accretion rate of ~10^-8M_⊙/ yr was very low when compared to the non-rotating case because of rotational energy dissipation. This indicates that angular momentum dissipation significantly impacts the parameter space for possible progenitor systems.

One mechanism for angular momentum transport, which was not taken into account in the above-mentioned studies of WD accretion, was proposed by Spruit (2002). This mechanism (referred to as the Tayler-Spruit mechanism or dynamo in the following) relies on the Tayler instability (Tayler 1973), and occurs in the radiative parts of differentially rotating stars. Small seed fields may be amplified to a predominantly toroidal magnetic field structure, with the associated magnetic torque leading to angular momentum transport which reduces the shear.

The effects of this mechanism on stellar evolution has been studied for massive stars by Maeder & Meynet (2003) and Heger et al. (2005). The case for low-mass stars was investigated by Maeder & Meynet (2004), and for the Sun by Eggenberger et al. (2005). Apart from having major implications for the evolution of non-degenerate stars, as discussed in these studies, it is concluded that the Tayler-Spruit mechanism tends to induce flat rotational profiles in the radiative zones of stars. It is therefore expected that the Tayler-Spruit mechanism could have a significant impact on the evolution of WDs, which – by their nature – are mostly non-convective. Angular momentum transport as provided by the Tayler-Spruit mechanism indeed appears to be required to understand the slow rotation of isolated neutron stars Heger et al. (2005) and white dwarfs (Suijs et al. 2008), and the spin-down of red giant cores as implied by asteroseismic observations (Mosser et al. 2012).

This present study attempts to further refine our understanding of binary systems in which an accreting WD ignites its He-shell. To do this, we investigate the response of rotating CO WD models to helium accretion, taking into account the effects of the Tayler-Spruit mechanism. Each model is calculated up to the point of ignition of the accumulated helium envelope. We then discuss the impact of variations of initial model parameters on the mass of the helium envelope and densities at ignition and the implications of our results for observations.

This paper is organized as follows. In Sect. 2 we describe the computational framework used, the mathematical description of the physics under consideration (Sect. 2.1), and the chosen initial models and input parameters (Sect. 2.2). In Sect. 3 we discuss the results obtained through our computations, first with an emphasis on the structural response of the WD to accretion (Sect. 3.1), then with regard to the observational implications of its state at the point of helium ignition (Sect. 3.3). We compare rotating models with and without including the Tayler-Spruit mechanism in Sect. 4, and discuss the fates of our simulated model WDs in Sect. 5. In Sect. 6 we discuss our findings, putting them into context with previously published works on the subject.

2. Numerical methods and physical assumptions

2.1. Numerical methods

The binary evolution code (BEC) is a well-established computational framework capable of performing detailed one-dimensional numerical experiments of single or binary star systems (Langer et al. 2000; Yoon & Langer 2004b). The framework is capable of performing detailed hydrodynamical simulations of rotating and degenerate systems, including mass accretion (Heger et al. 2000; Heger & Langer 2000).

2.1.1. Implementation of the Tayler-Spruit mechanism

Angular momentum transport within a stellar model is assumed to follow a diffusive process (as in Heger et al. 2000; Heger & Langer 2000; Endal & Sofia 1978; Yoon & Langer 2005), defined by a diffusion equation of the form (1)where ω is the local angular velocity, t time, M_r the mass coordinate, i the local specific moment of inertia, r the radius, ρ the local density and ν_turb the local shear viscosity as defined by the active diffusive processes.

It is possible to accurately compute models of diffusive processes in non-degenerate stars using an explicit initial value approach (see Heger et al. 2000). However, the sensitive dependence of some rotationally induced instabilities on the exact form of the angular velocity profile tends to lead to significant numerical errors in explicit schemes under certain circumstances. The pertinent problem here is encountered in cases of fast angular momentum redistribution occurring on a dynamical timescale, as encountered in studies of the dynamical shear instability. In order to deal with this problem, a non-linear solving routine was implemented by Yoon & Langer (2004b) in order to study the effects of dynamical shears. Further relevant mixing processes included in the framework are the secular shear instability, Goldreich-Schubert-Fricke instability and Eddington-Sweet circulation.

In addition to the dynamical shear instability, the Tayler-Spruit mechanism is capable of acting on dynamical timescales during the secular evolution of an accreting white dwarf. While an explicit treatment of the Tayler-Spruit mechanism, using the prescription by Spruit (2002), was successfully used to study angular momentum transport in massive stars (Heger et al. 2005), the above-mentioned problems regarding numerical stability preclude its use for the purposes of this study. Therefore, a new implicit numerical treatment is required, which was constructed on the basis of the implicit routines for the dynamical shear instability created by Yoon & Langer (2004b). We use essentially the same mathematical prescription as Yoon & Langer (2004b).

Spruit (2002) does not explicitly limit his derivations to non-degenerate matter, which is dealt with in this study through the use of appropriate material descriptions, as explained later in this section. However, the assumption of shellular rotation might not be entirely applicable to white dwarfs (as discussed by Heger et al. 2000; Yoon & Langer 2004b), although the angular momentum distribution of shellular rotation exhibits enough similarities to cylindrical rotation to serve as an adequate approximation.

It is assumed that the ordering of characteristic frequencies is such that (2)where ω_A is the well-known Alfvén frequency, ω is the angular velocity and N is the Brunt-Väisälä frequency. The Brunt-Väisälä frequency can be decomposed into a chemical buoyancy frequency (N_μ) and a thermal buoyancy frequency (N_T) such that(3)The prescription assumes a locally stably stratified stellar interior, which translates into the condition that N²> 0.

In order for the dynamo to work, the local angular velocity gradient has to be large enough to overcome the stabilizing effects of the buoyancy forces. This is accomplished by demanding that the dimensionless differential rotation rate satisfies (4)with (5)the minimal rotation rate for dynamo action.The minimal rotation rates for the stabilizing effects of the chemical and thermal stratification, respectively q₀ and q₁, can be written (6)and (7)where η is the magnetic diffusivity and κ the thermal diffusivity.

We compute the magnetic diffusivity using the electrical conductivity according to Nandkumar & Pethick (1984), which has been found to adequately describe the conditions in the non-degenerate and the degenerate parts of the star. The thermal diffusivity is computed taking into account the conditions in degenerate matter (see Yoon & Langer 2004b).

The “effective viscosity”¹ for a predominantly chemically stable medium is given as (8)and for a predominantly thermally stable medium as (9)These expressions are then connected with a patching formula in the form (10)where ν is assumed to represent the effective viscosity for a thermally and chemically stable medium and f(q) is a term introduced to ensure that ν vanishes smoothly as q approaches q_min, and is defined as (11)We note that as N_T and N_μ approach zero independently, indicating that the medium is becoming less stably stratified, ν_e0 and ν_e1 approach infinity. This means that as ν_e0 approaches infinity ν_e1 dominates Eq. (10) and vice versa. The thus computed effective viscosity is then summed with those resulting from all other active diffusive processes in ν_turb (see Eq. (1)). Since the given Tayler-Spruit viscosity is singular in a number of variables, we introduce an artificial upper limit corresponding to the local maximum of the shear viscosity attainable through the dynamical shear instability. The condition reads thus (12)where h_p is the local pressure scale height and τ_dyn the local dynamical timescale. Since the shortest timescale any diffusive process in a star can act on is the dynamical timescale and the dynamical shear instability acts on this timescale it is a justifiable lower limit for the operating timescale of the Tayler-Spruit mechanism.

The rate of energy dissipation is estimated following Kippenhahn & Thomas (1978) and Mochkovitch & Livio (1989) as (13)where is the differential rotation rate (i.e. q = | r × σ |). The value of ν_turb changes on short timescales. For the purposes of the non-linear diffusion solver, ϵ_diss is calculated as the time average of the energy dissipation during each evolutionary time step of the entire stellar evolution code.

It should be mentioned at this point that tests on massive star models produced good agreement between the upgraded implicit diffusion solver and the explicit solver used by Heger et al. (2005).

2.1.2. Equation of momentum balance

For later reference, we note that BEC implements centrifugal effects via the description provided by Endal & Sofia (1978). The equation of momentum balance is given as (14)where P is pressure, t is time, r_p is radius, and m_p is mass, the last two as contained within an equipotential surface. The effects of rotation are described by the quantity f_p, where (15)with S_p the area of the closed isobaric surface, g the absolute of the local effective gravitational acceleration and ⟨ g^-1 ⟩ the mean of its inverse. The correcting factor f_p is obtained iteratively and is consistent with the stellar structure equations (see Endal & Sofia 1976), but limited such that, locally, f_p does not decrease if v/v_K > 0.6. This causes our methodology to generally underestimate effects relating to centrifugal forces wherever the local rotational velocity v approaches the Keplerian value v_K, which is defined as (16)where all variables are defined in the customary way.

2.2. Initial models and input parameters

In this study, we use CO WD models of masses 0.54 M_⊙, 0.82 M_⊙, 1.0 M_⊙ and 1.1 M_⊙. These models were created by evolving a helium star of appropriate mass up to the end of the helium main sequence and then artificially removing most of the remaining helium envelope. This results in a WD of about 60% carbon and 40% oxygen (though retaining a small, <0.01 M_⊙, helium layer for the 0.82 M_⊙ and 1.0 M_⊙ models), an effective temperature of about 2 × 10⁵ K, and a surface luminosity of log (L/L_⊙) ≈ 2.5. The models are then spun up to a surface velocity between 7 km s^-1 and 20 km s^-1, which is in agreement with measurements of the rotational velocities of isolated white dwarfs (see Heber et al. 1997; Koester et al. 1998; Kawaler 2003; Suijs et al. 2008; Mosser et al. 2012), and then allowed to cool to surface luminosities of about log (L/L_⊙) = 0, −1, −2 (see Table 1). It is not immediately obvious whether these values are adequate. Since any He star+CO WD system will be a product of one common envelope event (or possibly two), the thermal history of the WD will be different from those usually obtained for isolated WDs (e.g., Renedo et al. 2010). However, since the details of common envelope evolution are currently far from being completely understood (see Ivanova et al. 2013), the exact disposition of a WD at the onset of mass accretion is subject to some uncertainty. Enlarging the parameter space to include the most likely initial luminosities is one way to address this problem. We note that the discrepancies in initial rotational velocities are expected to have negligible impact on the final result. Since the accreted material efficiently spins up the WD, surface velocities will reach values up to three magnitudes higher than the initial one long before any helium burning is expected to set in. For the purposes of this study, rotational velocities of the magnitude presented in Table 1 can be thought of as negligible. This is clarified in Sect. 3. In addition, non-rotating models are computed for comparison.

White dwarfs in binary systems are expected to accrete mass through a Keplerian accretion disk. The material falling onto the WD is therefore expected to move with a velocity close to the Keplerian value at the point of impact (Paczynski 1991; Popham & Narayan 1991), carrying a specific angular momentum of j_kepler. Within the context of the BEC framework, the amount of specific angular momentum (j_acc) carried by accreted matter is manually controlled by the choice of f_acc in the expression j_acc = f_acc·j_kepler. The choice of f_acc = 1.0 is the most natural one. In order to investigate conditions deviating from this assumption, we also include systems calculated with f_acc = 0.6 and f_acc = 0.3.

If the star is rotating critically, we avoid supercritical rotation by assuming that no further angular momentum is accumulated, the excess momentum being dissipated in the accretion disk. This is accomplished by setting f_acc = 0 if further accretion would lead to supercritical rotation.

The accretion rates chosen in this study range between 1 × 10^-8M_⊙/ yr and 1 × 10^-7M_⊙/ yr, which are commensurate with average mass-transfer rates achievable in a short-period He star+CO WD system with the He star (M ≤ 1.0 M_⊙) undergoing He main sequence evolution. Consequently, the accreting material is assumed to consist of helium with an admixture of hydrogen-burning products corresponding to solar metallicity. At accretion rates higher than 1 × 10^-7M_⊙/ yr the WD is expected to undergo a series of nova outbursts with individual ejecta masses that are comparatively low (Kato & Hachisu 2004; Piersanti et al. 2015). This mechanism is expected to have a profound impact on the outcome of the evolution of He star+CO WD systems (Neunteufel et al. 2016), but the nova cycle and the evolution of complete systems are both beyond the intended scope of this work. For studies of the regime of accretion rates Ṁ ≥ 3 × 10^-6M_⊙/ yr the reader is referred to Yoon & Langer (2003) and more recently Brooks et al. (2016). In this regime, a period of steady burning and/or shell flashes potentially leads to a helium shell detonation, with the expectation of a detonation of the CO core of the WD.

2.3. Ignition conditions

All models are computed up to the point of unstable He-ignition, which we define as the point where the helium burning timescale τ_nuc,He approaches the dynamical timescale of the star (i.e. τ_nuc,He ≈ τ_dyn). Owing to our methodology, we are unable to follow the ignition through to the final outcome of the process (i.e. detonation or deflagration). It is therefore necessary to apply some quantitative criteria in order to distinguish one outcome from the other. Possible criteria to distinguish detonations from deflagrations have been previously proposed by Woosley & Weaver (1994) and Woosley & Kasen (2011). Both of these criteria use the density at the point of ignition as a measure to determine whether the ignition would subsequently develop into a (supersonic) detonation or a (subsonic) deflagration. The defining factor in both cases is whether the density at the point of ignition is sufficient to preclude the premature quenching of the thermonuclear runaway through thermally induced expansion. Woosley & Kasen (2011) introduce a critical density dependent on the ignition temperature. Woosley & Weaver (1994) set the critical density to a constant value of ρ_crit = 1 × 10⁶ g/cm³ while Woosley & Kasen (2011) give (17)However, as stated by Woosley & Kasen (2011), the given function is an approximation which fails to properly identify detonating systems of the same study. The lowest predicted ignition density of all of the detonating systems in the above-mentioned study is ρ_ign ≃ 6.8 × 10⁵g/cm³ with most ignition densities exceeding ρ_crit = 1 × 10⁶ g/cm³. We therefore assume that the choice of a constant critical ignition density of ρ_crit = 1 × 10⁶ g/cm³ as put forward by Woosley & Weaver (1994) is justified for the purposes of this study.

3. Results

In this section, we discuss the results obtained from our simulations. A selection of defining parameters of the systems under consideration (though limited to systems with f_acc = 1.0 for the sake of brevity) at the point of He-ignition are presented in Table A.1. For the sake of clarity, it should be mentioned that in the following section, “rotating” always refers to models in the context of this study, i.e. rotating models with an active Tayler-Spruit dynamo, dynamical shear instability, Eddington-Sweet circulation and secular shear instability and “non-rotating” refers to non-rotating models with no angular momentum diffusion. Rotating models as defined here are explicitly compared with non-magnetic rotating models (i.e. rotating models without an active Tayler-Spruit dynamo) in Sect. 4.

3.1. Rotational profile

Fig. 1 Rotational velocity profile of model sequence Lm20510-5 (Minit = 0.54 M⊙) (A) and Lm21010-5 (Minit = 1.0 M⊙) (B) at three different times during its evolution (facc = 1.0, Ṁ = 5 × 10-8M⊙/ yr and log (Linit/L⊙) = −2 in both cases). Line i corresponds to t = 0 yr, the start of mass accretion, m to the midpoint of the accretion phase and f to just after the point of He-ignition. In subfigure A) this corresponds to m at t ~ 1.55 Myr and f at t = 3.11 Myr. In subfigure B) the corresponding times are m at t ~ 0.3 Myr and f at t = 0.64 Myr. The y-axis is normalized to unity with the maximum value of each graph indicated in ωmax.

White dwarfs tend to be convectively stable throughout their interiors. Accordingly, the Tayler-Spruit mechanism will work on the entirety of their structures. When compared to dynamical shears, the strength of their associated shear viscosities is of the same magnitude for a given condition, but the critical rotational velocity gradient required to initiate the Tayler-Spruit mechanism (Eq. (5)) is usually much smaller. Without angular momentum accretion, the dynamo will remain active until the velocity gradient is decreased to that of the threshold gradient.

Fig. 2 Evolution of the surface velocities of model sequences L00510-5, Lm20510-5 (A) and L01010-5, Lm21010-5 (B). All model sequences are computed up to He-ignition. The y-axis is normalized to the critical rotational velocity. Menv is the mass of the helium envelope and corresponds to the amount of helium accreted. All models were computed with Ṁ = 5 × 10-8M⊙/ yr and facc = 1.0.

Figure 1A shows the radial velocity profile of model sequence Lm20510-5². As can be seen, any residual differential rotation present at the beginning of mass accretion (line i in Fig. 1A) is damped by the Tayler-Spruit mechanism until a situation close to solid-body rotation is attained (line m). A very small radial velocity gradient remains, but is barely noticeable in the plot. Line f in Fig. 1A shows the profile just after (≪1 s) He-ignition. With the onset of rapid expansion, conservation of angular momentum in the expanding layers leads to the reemergence of an angular velocity gradient. The situation is similar for more massive WDs, as shown in Fig. 1B.

We note that there is no significant change in the rotational profile even as it passes areas with significant gradient in the chemical composition. Such a change might be expected due to the dependence of the effective viscosity on the chemical stratification (see Eq. (6)).

Figure 2 shows the surface velocity of four representative model sequences at two different initial luminosities of log (L_init/L_⊙) = 0 and log (L_init/L_⊙) = −2 (L00510-5, Lm20510-5A and L01010-5, Lm21010-5B) in units of ω_surf/ω_crit (ω_crit being the critical surface velocity) with respect to time. Less massive WDs spin up faster than more massive ones (see Fig. 2) due to their relatively smaller rotational inertia. The fact that the models with M_init = 0.54 M_⊙ never quite reach ω_crit is related to the treatment of supercritical accretion as explained in Sect. 2. The prescription calls for f_acc = 0 if further accretion leads to supercritical rotation, which means that critical rotation itself can never actually be reached.

Figure 2 also shows that the fraction of the critical rotational velocity achievable by a WD strongly depends on the initial mass of each model. The reasons for this, while somewhat obvious, as well as the implications of this behavior are discussed in greater detail in Sect. 3.3. It should be noted, though, that none of our model sequences of M_init> 0.82 M_⊙ reach critical rotation before He-ignition, which affects the mass-dependency of final He-shell masses, as is discussed in Sect. 3.3.

3.2. Chemical profile

Fig. 3 Chemical profile of model sequence Lm20510-5 (Minit = 0.54 M⊙) (A) and Lm21010-5 (Minit = 1.0 M⊙) (B). The remaining initial parameters are the same for both model sequences: facc = 1.0, Ṁ = 5 × 10-8M⊙/ yr and log (Linit/L⊙) = −2. Included are the three most abundant isotopes He4, C12 and O16 in units of relative mass per shell over the shell’s mass coordinate. Solid lines represent the profile at the time of the start of mass accretion at t = 0 yr, dashed lines at the midpoint of the accretion phase at t ~ 1.55 Myr(A) and t ~ 0.3 Myr(B), dotted lines just after the point of He-ignition at t = 3.11 Myr(A) and t = 0.64 Myr(B). The inset is a zoom of the area of the initial envelope-core interface (the interface is shown as a dashed gray line), where the initial profile has been omitted for the sake of clarity.

Figure 3 shows the chemical development of two representative model sequences, Lm20510-5 and Lm21010-5. As can be seen, there is noticeable chemical mixing at the boundary layer of the less massive system (Fig. 3A), while the more massive system (Fig. 3B) shows barely any mixing at the boundary layer. Rotationally induced mixing has been found to be much more pronounced in non-magnetic rotating models (Yoon & Langer 2004a). With the Tayler-Spruit mechanism suppressing the overwhelming majority of any initially present angular velocity gradient, the only region where appreciable chemical mixing can plausibly take place is the outermost layer of the envelope where an angular velocity gradient is maintained by the continuous accumulation of rotating matter. With growing envelope mass, the area exhibiting the largest angular velocity gradient moves away from the initial surface mass coordinate. This means that with growing envelope mass, chemical mixing at the core-envelope interface gradually diminishes. The differences in the mixing behavior between the two presented model sequences are due to a difference in initial chemical profiles.

We therefore conclude that chemical mixing in these systems is fundamentally weak, but highly dependent on the initial chemical profile. The presence of a preexisting He-envelope considerably weakens the magnitude of expected chemical diffusion.

The expectation of an adequately well-defined interface between the original core and the accumulated material throughout the evolution of our WD models leads us to refer to this interface as the “core-envelope interface” from this point on.

3.3. Accretion behavior and ignition

Fig. 4 Evolution of the temperature-density profile of a rotating (A) and non-rotating (B) WD with 0.54 M⊙ accretion rate is 5 × 10-8M⊙/ yr, facc = 1.0 and log (Linit/L⊙) = −1. Blue dots indicate the core-envelope interface. Initial (at the beginning of mass accretion) and final (at the point of helium ignition) profiles are indicated. The given masses (left to right) correspond to the presented profiles (initial to final). We note that the profiles marked as “initial” are not the first models in their evolutionary sequence, having been evolved for about one kyr each.

Fig. 5 Evolution of the temperature-density profile of a rotating (A) and non-rotating (B) WD with 1.0 M⊙. All other parameters and labels as in Fig. 4.

Figures 4 and 5 show the evolution of representative model sequences in the ρ-T-plane. The model sequence in Fig. 4A depicts model sequence Lm10510-5, which means that M_init = 0.54 M_⊙, log (L_init/L_⊙) = −1 and f_acc = 1.0. Figure 4B is the same but non-rotating. The initial profile is as at the beginning of the mass accretion phase, and the final profile is just before (≤1 s) He-ignition as defined in Sect. 2. It should be noted, however, that the rotating model accumulated a significantly larger amount (0.05 M_⊙) of helium at the point of detonation. This is a consequence of solid-body rotation. Helium ignition requires both sufficiently high temperatures and sufficiently high densities, the threshold temperature being a function of local density. In the rotating non-magnetic case, the WD would, at the point of He-ignition, show significant differential rotation, introducing a much larger amount of energy dissipation (Eq. (13)) into the uppermost layers of the He-envelope, thereby providing the necessary temperature to facilitate He-ignition after a much smaller amount of helium has been accumulated (see Yoon & Langer 2004b). The Tayler-Spruit mechanism is active at much smaller rotational velocity gradients than, for example, the dynamical shear instability under the conditions encountered in the present model sequences. This means that, as described in Sect. 3.1, a small rotational velocity gradient is induced which extends throughout the stellar envelope (i.e., a small, non-localized gradient). The energy dissipation caused by the Tayler-Spruit mechanism is correspondingly small (locally, practically ϵ_diss ≈ 0). This means that a temperature increase resulting from energy dissipation due to the Tayler-Spruit mechanism will be smaller, but will affect a wider area when compared to that expected from, for example, the dynamical shear instability. Therefore, in order to increase the temperature of the stellar material at any given point sufficiently to induce He-ignition, a larger amount of angular momentum, and thus helium, has to be accumulated than in the rotating non-magnetic case. Furthermore, since the WD is rotating critically (see Fig. 2A, this model sequence falls between the two depicted ones), the centrifugal force serves to decrease the local density of the stellar matter at any given coordinate compared to the non-rotating case, in turn increasing the temperature required for He-ignition. We therefore expect a significant increase in the required amount of material to be accreted before ignition. It should be noted at this point that the evolution of the rotational profile is the dominant factor in this prediction. This becomes evident when comparing the depicted model sequences with models computed with a variation of values of the f_acc-parameter. We find that when M_init is held constant at 0.54 M_⊙, at high accretion rates, the final accumulated amount of helium decreases with increasing f_acc (see Fig. 7). Exceptions to this exist at low accretion rates. These exceptions are discussed in detail in Sect. 3.4.5.

Figure 5 is the same as Fig. 4, but the model sequence’s initial mass is M_init = 1.00 M_⊙. Contrary to the less massive example, the effects of rotation lead to a decrease in the required mass to be accreted in order to initiate He-ignition. Partially this is a consequence of the physical differences in WDs of different mass at the onset of mass accretion (see Table 1). Smaller size at a higher mass makes newly deposited material move at much higher angular velocity for a given value of f_acc. This leads to larger amounts of dissipational energy being introduced in the system. This combined with the compressional heating incurred by the accumulating material falling onto the WD with a significantly higher surface gravity than in less massive cases, leads to a much faster increase in temperature in the outer layers of the WD (compare Figs. 4A and 5A).

Rotational inertia of white dwarfs increases with increasing mass (see, e.g., Piersanti et al. 2003). The higher the rotational inertia of a solidly or quasi-solidly rotating object, the smaller the increase in rotational velocity incurred through accretion of a given amount of rotational momentum. This leads to He-ignition occurring before the WD has a chance to achieve critical or near-critical rotation (see Fig. 2), in turn leading to much diminished centrifugal effects compared to the case of lower masses.

In both of the cases presented above, it is useful to comment on the NCO reaction. The effects of the NCO reaction on accretion scenarios have been investigated before (e.g., Woosley & Weaver 1994; Hashimoto et al. 1986). The consensus is that the reaction is largely, but not completely (Hashimoto et al. 1986), independent of temperature, the dominant factors in a CO WD being a sufficiently high local abundance of nitrogen and a local density of more than 10⁶ g/cm³. In the present scenario, in the absence of strong chemical mixing (Sect. 3.2), the effect of increased energy generation due to the NCO reaction would be most important at the core-envelope interface. In both the rotating and non-rotating model sequences with M_init = 0.54 M_⊙, the interface never reaches densities high enough to facilitate the NCO reaction. In the heavier model sequences, only the non-rotating WD exhibits high enough densities at the interface, leading to an acceleration of the temperature increase due to the NCO reaction in the overlying layers and therefore earlier He-ignition than if the NCO reaction were neglected. In our model sequences the NCO-reaction plays a limited role insofar as it provides some additional heat in models with low initial luminosities. It only leads to a slightly reduced final helium envelope mass and to the igniting layer being located somewhat lower in the star. The majority of the heat is still introduced through the accretion process itself. In models with higher initial luminosities, the role of the NCO-reaction is eclipsed by the residual heat of the WD.

The answer to the question whether rotating magnetic WDs accumulate more massive He-envelopes at He-ignition than non-rotating ones therefore depends strongly on whether the WD can achieve near-critical rotation before He-ignition, and therefore on the initial and on the efficiency of angular momentum accumulation.

Figure 6 shows the time evolution of the location of the maximum temperature in the star (T_max) of several representative model sequences at different initial masses (0.54 M_⊙, 0.82 M_⊙ and 1.0 M_⊙) with two initial luminosities of log (L_init/L_⊙) = 0 and log (L_init/L_⊙) = −2, calculated with f_acc = 1.0.

Fig. 6 Mass coordinates of the point of the temperature maximum (Tmax) over time (solid colored lines) of several representative systems with initial masses of 0.54 M⊙(A), 0.82 M⊙(B) and 1.0 M⊙(C). The solid black line in each plot indicates the corresponding total mass of the WD (MWD), red lines correspond to an initial luminosity of log (Linit/L⊙) = 0 and blue lines to log (Linit/L⊙) = −2. The dashed black line indicates the initial mass of the WD. The green dots indicate the He-ignition points. All depicted sequences were calculated with facc = 1.0 and Ṁ = 5 × 10-8M⊙/ yr. The gray arrows indicate the point of ignition in relation to MWD. Mr,ign is the mass coordinate of the igniting shell.

Since the initial temperature of WDs of the same mass is, across the entire interior, significantly higher at higher initial luminosities, the tendency of M(T_max) to stay at lower values in models with higher initial luminosities is expected. If the core temperature is higher, a larger amount of thermal energy has to be introduced into the star in order to move M(T_max) towards the surface, albeit aided by extant heat in the core.

It is easily seen that the mass coordinate of He-ignition strongly depends on the initial luminosity of the WD. Higher initial luminosities cause the point of ignition to lie closer to the core-envelope interface (the presented model sequences at log (L_init/L_⊙) = 0 feature ignition coordinates not significantly above the core-envelope interface). As additional heat is introduced close to the WD’s surface, M(T_max) moves towards the surface, but higher initial luminosities imply a higher core temperature, causing the M(T_max) to remain closer to the center of the WD. The visible downturn of M(T_max) in each of the depicted model sequences is the result of the burning point moving towards areas of higher density after ignition.

Woosley & Kasen (2011) investigated more closely two possible ways to induce SNe in sub-Chandrasekhar WDs via what has become known as the double-detonation scenario: in the first scenario, ignition of the He-envelope at the core-envelope interface induces a supersonic shock front that compresses the CO-core, inducing a secondary detonation at the center, resulting in the explosive destruction of the WD. In the second scenario, ignition occurs at some distance from the core-envelope interface inducing an inward and outward shock. The inward propagating shock then proceeds from the helium envelope into the CO core, resulting in a SN in what is known as the “edge-lit” scenario. In the light of this study, if an ignition is able to induce a detonation, the former case would be more likely in an initially hotter and more luminous WD, while the latter outcome would be more likely in an initially colder, lower luminosity WD.

3.4. Impact of parameter variations

In this section we describe the impact of variations of the most relevant input parameters on the entire set of simulated systems.

Figure 7 shows the final masses of all computed model sequences with respect to the mass accretion rate. We emphasize that the coordinate in the graphs does not correspond to the mass of the accumulated envelope but to the total mass of the star. The mass of the envelope can be obtained by subtracting the mass of the corresponding initial model (black lines) from the mass of the model sequence at He-ignition. We identify two distinct cases for accretion rates with Ṁ ≥ 3 × 10^-8M_⊙/ yr (effects at accretion rates lower than this, applicable to initial masses M ≥ 0.82 M_⊙ will be discussed separately), differentiated by their ignition mass when compared to the non-rotating case: The high initial mass case in which the WD ignites its helium envelope after accumulation of a smaller amount of helium than the non-rotating case and the low initial mass case in which the WD ignites its envelope with a larger amount of helium than in the non-rotating case. There is also the intermediate case where the behavior of the WD is principally governed by the accretion rate. Systems with accretion rates lower than Ṁ< 3 × 10^-8M_⊙/ yr form a separate case, exhibiting vastly increased He-shell masses at detonation when compared to the non-rotating case as well as higher accretion rates. These systems will be collectively referred to as the low accretion rate case.

3.4.1. Low initial mass case

The behavior of the low initial mass case is illustrated by the model sequences with M_init = 0.54 M_⊙. The effects of close-to-critical rotation, discussed in Sect. 3.1 and Sect. 3.3, driving up ignition masses become obvious here. At all initial luminosities the model sequences with M_init = 0.54 M_⊙ exhibit significantly higher envelope masses at He-ignition than in the non-rotating case (by up to ~ 0.14 M_⊙). The observation that this behavior is related to the ability of the accumulated angular momentum to spin up the WD is strongly supported by the evident correlation between the increase of the required accretion and f_acc, since a variation in f_acc immediately and artificially impacts the amount of angular momentum imparted on the star for a given time period and therefore directly controls its spin-up timescale. The switch (with f_acc = 0.6 showing higher ignition masses than f_acc = 1.0) at Ṁ ≃ 4−5 × 10^-8M_⊙/ yr in this behavior seen in Fig. 7 should be noted for further reference (see explanation in Sect. 3.4.5).

Systems with f_acc = 0.3 generally do not spin up quickly enough for the WD to reach surface velocities close to v_crit, usually undergoing He-ignition at v_surf ~ 0.25 v_crit. As expected, these systems therefore most closely resemble non-rotating ones, with exceptions at the lowest considered mass accretion rates if log (L_init/L_⊙) ≤ −1.

Fig. 7 Total masses (M) of all computed model sequences of this study with respect to constant mass accretion rate (Ṁ). The presented graphs differ in the chosen initial luminosity of the white dwarf model with log (Linit/L⊙) = −2 in graph A), log (Linit/L⊙) = −1 in B) and log (Linit/L⊙) = 0 in C). Black solid lines correspond to the initial mass of each model sequence with the initial mass being indicated. Colored lines indicate the final mass of each model sequence with green corresponding to Minit = 0.54 M⊙, red to Minit = 0.82 M⊙, blue to Minit = 1.00 M⊙ and yellow to Minit = 1.10 M⊙. Line types indicate chosen rotational parameters. Solid colored lines for non-rotating models, dashed lines for facc = 1.0, dotted lines for facc = 0.6 and dash-dotted lines for facc = 0.3. Dots indicate the masses of individual models.

At log (L_init/L_⊙) = −2, the initial temperatures are low enough, ensuring that the amount of required material is high enough to allow for a significant increase in rotational velocity, to ensure that the process described above is active over the entire range of considered mass accretion rates.

We conclude that the rotating magnetic case leads to higher envelope masses in relatively low-mass CO WDs of about 0.54 M_⊙ compared to the non-rotating case.

3.4.2. High initial mass case

The behavior of the high initial mass case is illustrated by the model sequences with initial masses of M_init = 1.00 M_⊙ and M_init = 1.10 M_⊙. Unlike the low initial mass case, WDs of this kind undergo He-ignition after a smaller amount of material has been accumulated than in the non-rotating case. As described in Sects. 3.1 and 3.3, the reason for this lies in the expectation, due to their higher rotational inertia, that higher mass WDs are spun up more slowly than lower mass objects. This means that higher mass WDs will not be spun up appreciably before He-ignition. The difference in ignition masses with respect to the value of f_acc in this case stems from the more efficient heating caused by rotational energy dissipation because of the larger rotational velocity gradient induced through accretion at larger values of f_acc. This is reflected in the ordering of the ignition masses with the divergence from the non-rotating case being smallest for f_acc = 0.3 and largest for f_acc = 1.0. It should be noted that the decrease in ignition masses in the high initial mass case in the rotating magnetic case mirrors the behavior of rotating non-magnetic ones (cf. Yoon & Langer 2004a). The non-rotating case and the rotating non-magnetic case can be thought of as upper and lower limits for the He-envelope mass at ignition for the high initial mass case (M ≥ 1.0 M_⊙).

3.4.3. Intermediate case

The intermediate case is represented in this study by model sequences with M_init = 0.82 M_⊙. These systems exhibit the behavior of high initial mass case systems at high mass-transfer rates and low values of the f_acc-parameter and the behavior of low initial mass case systems at low mass-transfer rates and high values of the f_acc-parameter. Lower initial luminosity increases the propensity of the system to show an increased He-envelope mass at ignition when compared to the non-rotating case. The reason for this behavior again lies in the interplay between the spin-up of the WD and the heating effects of accretion: higher mass accretion rates decrease the amount of helium needed to result in an ignition, allowing less time to spin up the WD. The intermediate case can therefore be viewed as a transitory case between the high and low initial mass cases. We again emphasize the expectation of the largest helium envelope masses to be reached to pass from model sequences with f_acc = 1.0 to model sequences with f_acc = 0.6 with decreasing Ṁ.

3.4.4. Low accretion rate case

Fig. 8 Temperature profiles over time of individual representative model sequences. Mr is the mass coordinate, t the time since the start of the accretion phase. The color bar indicates temperature and the black line corresponds to the current total mass of the WD. All model sequences were computed with initial parameters of Minit = 0.54 M⊙, log (Linit/L⊙) = 0 and facc = 1.0. Model sequence A) (L00510-1) uses a mass accretion rate of Ṁ = 1 × 10-8M⊙/ yr, model B) (L00510-2) Ṁ = 2 × 10-8M⊙/ yr and model C) (L00510-4) Ṁ = 4 × 10-8M⊙/ yr. It should be noted that the temperature color scale was restricted to the indicated range. Temperatures higher and lower than the limits do occur. The gap visible in plot B) is due to a subsequence of models computed with a larger time step than the rest of the sequence, resulting in a physically irrelevant discontinuity of the available data.

Fig. 9 Maximum temperature over time for a number of representative model sequences. All model depicted model sequences were calculated with Minit = 0.54 M⊙ and facc = 0.3. Blue lines indicate a mass accretion rate of Ṁ = 3 × 10-8M⊙/ yr, red lines Ṁ = 2 × 10-8M⊙/ yr and green lines Ṁ = 1 × 10-8M⊙/ yr. Solid lines indicate an initial luminosity of log (Linit/L⊙) = −2, log (Linit/L⊙) = −1 and dotted lines log (Linit/L⊙) = 0.

As can be seen in Fig. 7, systems with Ṁ < 3 × 10^-8M_⊙/ yr exhibit two features absent in the cases with Ṁ ≥ 3 × 10^-8M_⊙/ yr. First, the helium shell mass at detonation is much larger than in the corresponding non-rotating case and in the cases with higher mass accretion rates. Second, the helium shell mass at detonation is practically independent of the initial luminosity of the WD.

Apart from the already discussed diminished effect of the NCO-reaction, both of these features are explained by the diminished effects of compressional heating at lower mass accretion rates. The compressional heating timescale (τ_c) is proportional to the mass accretion timescale (τ_acc), heating being further decreased by rotation (e.g., Piersanti et al. 2003). Therefore, if the initial temperature profile of the WD is higher than can be maintained through the given rate of mass accretion, the WD will cool until an equilibrium between radiative cooling and compressional heating is achieved. If the WD’s initial temperature profile is lower than the equilibrium profile, the WD will be heated. Figure 8 shows that this leads to a period of low temperatures and flat temperature profiles in systems with mass accretion rates Ṁ < 3 × 10^-8M_⊙/ yr, while systems with higher mass accretion rates maintain a temperature maximum in the upper parts of their envelope, which in turn aids ignition once an adequate amount of material has been accumulated. This accounts for the significantly increased envelope masses for Ṁ< 3 × 10^-8M_⊙/ yr.

Independence from initial luminosity is explained by the same process. Figure 9 shows the maximum envelope temperature of representative models with respect to mass accretion rates and initial luminosity. As explained above, models with mass accretion rates of Ṁ < 3 × 10^-8M_⊙/ yr evolve towards a state of equilibrium, defined by an equilibrium temperature which depends on the mass accretion rate. Once equilibrium is reached, further evolution becomes independent of initial parameters. Systems with Ṁ ≥ 3 × 10^-8M_⊙/ yr never reach this equilibrium, their evolution being influenced by their initial disposition up to the point of He-ignition.

As is shown in Sect. 3.5, systems conforming to the low accretion rate case are the most likely candidates for He-detonations (as opposed to deflagrations), but their high mass requirements make their natural occurrence highly unlikely. mass-transfer rates of Ṁ < 3 × 10^-8M_⊙/ yr are expected to require a mass donor of M ≤ 0.9 M_⊙. Such a low-mass donor does not have a mass budget large enough to provide the required amount of helium (see Neunteufel et al. 2016).

3.4.5. f_acc-dependence

As noted in the previous sections, in the low initial mass and intermediate cases, the ordering with respect to final He-envelope mass of models calculated with different values of the f_acc-parameter depends on the mass accretion rate. In this section we will explain why this switch occurs and why it is related to the mass accretion rate.

It is important to note that this behavior is unexpected insofar as the effects of rotation should be stronger for more rapid rotation (see Eq. (14)). Since f_acc controls the amount of angular momentum accumulated per unit mass, a model calculated with a larger value of the f_acc-parameter should be rotating faster than in the case with a smaller f_acc after the same amount of material has been accreted. This criterion is satisfied in our models. Since the conditions for the onset of helium burning primarily depend on the local density and temperature, a fast, quasi-solidly, rotating star will have to accumulate a larger amount of helium than a slower rotating one in order to facilitate He-ignition. Therefore, naively, one would expect the ignition mass of a given WD to increase with the f_acc-parameter.

Fig. 10 Densities at the point of highest temperature (ρ(Tmax)) with respect to the total current mass of the accumulated He-envelope (MHe). The colored dots indicate the current surface velocity of the model. Depicted are model sequences lm10810-3, lm10806-3 and lm10803-3. The model sequences differ in the chosen facc-parameter.

Figure 10 shows the evolution of the density at the point of maximum temperature, ρ(T_max), with respect to the mass of the accumulated He-envelope, M_He. Color indicates the surface rotational velocity of the WD (owing to quasi-solid-body rotation, the angular velocity does not differ significantly at the surface or anywhere else in the star). Depicted are three model sequences, all with initial mass M_init = 0.82 M_⊙, initial luminosity log (L_init/L_⊙) = −1 and constant mass accretion rate Ṁ = 3 × 10^-8M_⊙/ yr for three different values of f_acc. The initial decrease in density is explained by the fact that, at the start of the accretion phase, the WDs have the highest temperature in the central region. With proceeding accretion of matter, thermal energy is introduced in the outermost layers of the WD while heat is radiated away from the center. This leads to a flattening of the temperature gradient with a peak somewhere in the upper envelope (cf. Figs. 4 and 5). Once high enough local temperature and density, are reached, the helium will ignite and a flame will propagate inwards, leading to the increase of ρ(T_max) immediately before the end of the model sequence. Keeping in mind that Eq. (14) leads us to expect that slower rotational velocities lead to earlier ignition, the small helium shell mass of the f_acc = 0.3 model sequence is easily explained by the observation that this model does not reach rotational velocities close to critical. Both the f_acc = 1.0 and the f_acc = 0.6 model sequences reach critical rotational velocities, the former sooner than the latter. After the initial decrease, the f_acc = 1.0 model sequence exhibits lower values of ρ(T_max) than the f_acc = 0.6 model sequence. Owing to the relatively higher rotational velocity of the f_acc = 1.0 model sequence, this is expected.

Fig. 11 Local rotational velocity (v) divided by the local Keplerian velocity (vK) with respect to mass coordinate (M). Depicted are model sequences lm10810-3 and lm10806-3. Solid lines represent the facc = 1.0 case (lm10810-3), dotted lines the facc = 0.6 case (lm10806-3). The color bar indicates the time-coordinate of the profile. The black lines indicate the mass coordinate of the point of ignition for the respective model. The solid black line is valid for facc = 1.0 and the dotted line for facc = 0.6.

Fig. 12 Density at the point of ignition (ρign) plotted against mass accretion rate. The black dotted line indicates the minimum density for a supersonic runaway according to Woosley & Weaver (1994). Only model sequences with facc = 1.0 included.

The subsequent unexpected increase in ρ(T_max) in the f_acc = 1.0 model sequence at an earlier time than in the f_acc = 0.6, leading to helium ignition, occurs because of a combination of the following factors: as the WD accumulates mass, the radial coordinate of any given mass coordinate decreases, increasing the local Keplerian velocity (Eq. (16)). At the same time, as long as the WD has not reached critical rotation, accumulation of angular momentum through mass accretion increases the local rotational velocity relative to the Keplerian velocity. As discussed in Sect. 2, once a model sequence reaches critical rotation, f_acc is assumed to be 0. This means that angular momentum is only accumulated at the rate allowed by the increase in mass of the WD instead of the rate dictated by the f_acc-parameter. This in turn means that, at lower mass coordinates, the local value of v/v_K decreases due to the increase in value of v_K. This behavior is evident in Fig. 11, which shows the evolution of the local value of v/v_K at the point of ignition with respect to the total mass of the WD for the same model sequences with different values chosen for the f_acc-parameter. The earlier ignition of the model sequence with f_acc = 1.0 is therefore precipitated by the fact that it reaches critical rotation earlier than models with a smaller value of f_acc.

3.5. Ignition densities and detonations

Figure 12 shows the density at the point of ignition at the time of ignition (ρ_ign) in relation to the chosen mass accretion rate (Ṁ). Most importantly, the expected ignition densities in rotating models are significantly lower than in the non-rotating case. There is a correlation of higher ignition densities with higher initial masses, lower mass accretion rates and lower initial luminosities.

Apart from our baseline non-rotating models, only low accretion rate case systems reach high enough ignition densities for detonation. However, taking into account the fact that the conditions of the mass accretion phase must be present in a physical CO-WD He-Star system in order to produce an observable event, the low accretion rate case presents us with two mutually exclusive conditions for detonation: 1) low accretion rates and 2) large amounts (~0.45M_⊙) of transferred mass. As shown by Neunteufel et al. (2016), mass-transfer rates of Ṁ ≤ 3 × 10^-8M_⊙/ yr are only to be expected in close binary systems for He-donors of M ≤ 0.9 M_⊙, varying with the evolution of the donor. Such a donor star would need to lose a majority of its mass in order to produce a detonation. The same criterion rules out the possibility that the required low mass accretion rates could be provided through wind accretion from a more massive donor. Previous studies (e.g., Theuns et al. 1996) suggest that a companion star might be able to accrete a few percent (~1−2%) of a primary’s wind mass loss rate. A He-star exhibiting sufficiently high wind mass loss to lead to an accretion rate of ~1 × 10^-8M_⊙/ yr on an accompanying WD would be unlikely to be able to sustain it for long enough to allow for the accumulation of ~0.5M_⊙ of He.

We note that our non-rotating models, including the predictions of whether the WD is capable of producing a detonation, are in reasonably good agreement with previous results such as those obtained by Woosley & Kasen (2011), generally showing a discrepancy of less than 5% in final helium shell mass. Larger discrepancies of up to 30% in final helium shell mass in the comparable model sequences of initial masses around 0.8 M_⊙ at mass accretion rates of 2−3 × 10^-8M/ yr can be attributed to the presence of more massive He-layer in our 0.82 M_⊙-model. Comparisons with comparable models obtained by Piersanti et al. (2014) yield a discrepancy of 5−20%, which can be attributed to the higher initial luminosities used in that source. We also note that the chosen methodology does not take into account changes in the mass accretion rate, which would occur in any natural interacting binary. We nevertheless conclude that, while it may be theoretically possible to induce a He-detonation on a rotating WD with an active Tayler-Spruit dynamo, the conditions necessary to do so are unlikely to exist in a naturally occurring binary system. Any naturally occurring accretor will far more likely undergo a helium deflagration rather than a detonation. However, we stress again the limitations of using the prescription introduced in Woosley & Weaver (1994) and point to Woosley & Kasen (2011) where detonations have been found to occur at lower (but not much lower) ignition densities than the ρ_crit = 10⁶ g/cm³ used here. More detailed calculations using a more specialized simulation framework would have to be used in order to resolve this problem.

As far as resulting transients are concerned, we find it unlikely that any of the rotating models under consideration here might be able to produce classical Type Ia SNe. Returning to Woosley & Kasen (2011) for reference (although only non-rotating WDs are discussed, and so this comparison must be considered approximate), Type Ia-like spectra and light curves depend on low He-shell masses ≲0.05 M_⊙ at ignition and on the ignition turning into a supersonic shock. As discussed above, comparable He-shell masses only occur in our model sequences with M_init ≥ 1.0 M_⊙ and Ṁ ≥ 5 × 10^-8M_⊙/ yr, which are incapable of producing a detonation when rotation is included. On the other hand, the only systems in our sample which are able to produce a detonation are systems with M_init ≤ 0.82 M_⊙ at mass accretion rates Ṁ ≤ 3 × 10^-8M_⊙/ yr. But the He-shell masses at detonation of these systems, which – as explained above – we do not expect to occur naturally, are large enough to render a SN Ia-like spectrum very unlikely.

We emphasize at this point that our parameter space excludes accretion rates of Ṁ < 4 × 10^-8M_⊙/ yr for WDs with M_init ≥ 1.0 M_⊙. Although we cannot specifically exclude ignitions developing into detonations in this part of the parameter space, the resulting He-shell masses would nevertheless suggest resulting spectra unlike those of classical SNe Ia.

However, the comparatively high ignition masses and ignition densities just below 10⁶g/cm³, in systems with accretion rates of Ṁ ≃ 2−3 × 10^-8M_⊙/ yr, could correspond acceptably well with the properties of the fast, faint calcium-rich SNe, as discussed by Waldman et al. (2011).

4. Comparison with the non-magnetic rotating case

As mentioned before, the impact of rotation with non-magnetic angular momentum diffusion and energy dissipation, the non-magnetic rotating case, has been studied by Yoon & Langer (2004a); it differs from the systems in this study, the magnetic rotating case, in its neglect of the Tayler-Spruit mechanism.

Table 2 allows a comparison between systems the taken from Yoon & Langer (2004a) and this study. While there are some differences in the initial mass of the model sequences, these are sufficiently small to make this comparison meaningful.

Comparing models with f_acc = 1.0, it is evident that the amount of helium required to induce ignition is significantly smaller in the rotating non-magnetic case than in the rotating magnetic case. The impact of the inclusion of the Tayler-Spruit mechanism is drastic; the state of initially very similar systems is very different at the point of helium ignition. For example, in the rotating non-magnetic case sufficiently high ignition densities are precluded by a strong localized input of thermal energy through energy dissipation near the core-shell interface (see Sect. 3.3); instead, given a sufficiently large mass budget, a rotating, magnetic system would actually result in a helium detonation.

Another important difference is that in the rotating, non-magnetic case, the amount of accreted helium until ignition is smaller when compared to the non-rotating case, while it is larger in the rotating, magnetic case.

However, in both the rotating, non-magnetic case and the rotating, magnetic case, we expect He-detonations to be an unlikely outcome of the WD’s evolution. In the rotating, non-magnetic case this is a consequence of the already mentioned localized input of thermal energy through viscous dissipation. In the rotating, magnetic case, the reason lies in the flatter density stratification brought about by quasi-solid-body rotation in cases with Ṁ ≥ 3 × 10^-8M_⊙/ yr and the expected insufficient mass budget of prospective mass donors in cases with Ṁ ≤ 3 × 10^-8M_⊙/ yr. Owing to the significant differences in the He-shell mass at ignition, the transients resulting from helium ignition in rotating non-magnetic and rotating magnetic systems would differ significantly from each other.

5. Evolutionary fates

As described in Sect. 3.5, our magnetic, rotating WDs are not promising progenitor models for classical Type Ia supernovae or for other potential transients requiring the ignition of the underlying CO core. The question naturally arises: what do they result in?

To discuss this question, we may divide the parameter space of our models into three parts: the low accretion rate regime, i.e., Ṁ ≤ 2 × 10^-8M_⊙/ yr; the high accretion rate regime, i.e., Ṁ ≥ 4 × 10^-8M_⊙/ yr; and the transition regime between the two.

In the low accretion rate regime, ignition densities would be sufficiently high to produce detonations. However, as argued above, since a high mass of helium needs to be accreted for the ignition to occur, this may not happen in nature. Instead, the accretion may stop before the helium can ignite (Neunteufel et al. 2016), leading to a close binary consisting of a CO white dwarf covered by a degenerate helium envelope and a He white dwarf with some amount of CO in its core. Gravitational wave radiation may lead to a merger of such binaries, which could either produce a cataclysmic event (e.g., van Kerkwijk et al. 2010; Dan et al. 2014) or an R CrB star (e.g., Staff et al. 2012; Menon et al. 2013).

In the high accretion rate regime, our rotating models accrete only 0.01−0.2 M_⊙ until helium ignites (Fig. 7), which occurs at densities below ~3 × 10⁵g/cm³ (Fig. 12) and renders helium detonations unlikely. Such events might be comparable to the faint He-flashes discussed by Bildsten et al. (2007) in the context of AM CVn systems.

In the transition regime, however, we find some rotating models that only ignite helium after accreting slightly more than 0.2 M_⊙, at a density above ~5 × 10⁵g/cm³ (e.g., Models Lm20810-3, Lm10810-3, L00810-3). When our density criterion for He detonation is applied to these models, a helium shell deflagration is predicted to occur. While our density criterion is uncertain (Sect. 2.3), we cannot exclude that a detonation would occur in this case. However, the ignition conditions in these models are similar to those investigated by Waldman et al. (2011), who argue that even if a helium detonation does occur, the CO core might remain unaffected in this situation.

It was shown by Waldman et al. (2011) that the helium shell explosion with parameters corresponding to our transition regime models, even if it occurs as a detonation, leads to an incomplete burning of the accumulated helium and a rich production of intermediate mass elements (see also Hashimoto et al. 1983). Waldman et al. (2011) suggest that the resulting explosions may correspond to the observationally identified class of Ca-rich supernovae, which are a fast and faint subtype of Type Ib supernovae (Perets et al. 2010), although arguments favoring a core-collapse origin have been put forward as well (Kawabata et al. 2010).

Overall, it appears unlikely in the light of our models that helium accreting white dwarfs provide the conditions for classical Type Ia supernovae. Instead, they may be good progenitors for fast and faint hydrogen-free transient events, with helium novae occurring for the lowest accumulated helium masses (≤ 0.01 M_⊙) at the faint end, up to subluminous Type I supernovae like the Ca-rich supernovae at the bright end.

It has been shown observationally by Perets et al. (2011) that Ca-rich supernovae tend to be offset from their host galaxies by a considerable distance and located in regions where low-mass progenitors in old stellar populations would be expected. The corresponding progenitor age is subject to an ongoing debate. Yuan et al. (2013) argue for delay times of 10 Gyr or more, while Suh et al. (2011) give characteristic ages as low as 300 Myr. Calculating delay times for the systems under consideration here is beyond the scope of this paper; however Wang et al. (2013) used similar masses in their study, but neglected rotation. According to their estimates, a delay time for sub-Chandrasekhar mass detonations in He+WD systems close to 1 Gyr is most probable. Questions concerning the delay times of these transients are equally applicable to white dwarf as to core-collapse progenitors (Yuan et al. 2013; Zapartas et al. 2017) and will have to be addressed in future research.

6. Conclusions

We have carried out detailed simulations of CO-WDs accreting helium-rich matter at constant accretion rates of 10^-8−10^-7M_⊙/ yr, allowing for the accumulation of angular momentum and rotational energy dissipation and considering the effects of the Tayler-Spruit mechanism, dynamical shear instability, secular shear instability and the Goldreich-Schubert-Fricke instability up to the point of He-ignition.

We find that inclusion of the Tayler-Spruit mechanism profoundly impacts the expected rotational profile of the WD, inducing quasi-solid-body rotation on the object. We also find that, due to the apparent rotational profile, the role of the NCO-effect in facilitating ignition of the helium envelope is diminished compared to the non-rotating case. Our calculations show a stark difference in the relation of the amount of helium accreted at the point of ignition between rotating and non-rotating model sequences. The magnitude of the difference depends on the initial mass of the WD. Low-mass WDs generally accrete up to 0.1 M_⊙ more (0.3 M_⊙ in total) at accretion rates of Ṁ ≥ 3 × 10^-8M_⊙/ yr and up to 0.2 M_⊙ more (0.7 M_⊙ in total) at accretion rates of Ṁ < 3 × 10^-8M_⊙/ yr. High-mass WDs accrete on the order of 0.01 M_⊙ less than in the non-rotating case.

Rotating magnetic systems are also found to exhibit significantly increased ignition masses when compared to previous non-magnetic rotating models (Yoon & Langer 2004a).

We find that rotating magnetic WDs, accreting helium at rates Ṁ ≥ 3 × 10^-8M_⊙/ yr, will not reach ignition densities above 10⁶ g/cm³, which indicates that helium ignitions in these systems are unlikely to develop into a detonation. Only systems with Ṁ ≤ 2 × 10^-8M_⊙/ yr exhibit ignition densities larger than 10⁶ g/cm³. However, naturally occurring systems of this type are unlikely to eventually produce a He-detonation since the expected large amount (≳ 0.5 M_⊙) of helium required to induce helium ignition would call for a suitably massive donor star. Such a donor would in turn be unable to provide sufficiently low mass-transfer rates (Neunteufel et al. 2016).

This means that identifying any of these models as the progenitors of classical SNe Ia via the double detonation mechanism is not seen as viable. However, the obtained ignition masses and densities for rotating magnetic WDs with accretion rates of Ṁ ≃ 3 × 10^-8M_⊙/ yr resemble the conditions which appear promising for reproducing the faint and fast Ca-rich supernovae. More research, especially concerning explosion physics, is needed in order to strengthen this indication.

In any case, the stark difference between the expected states of the accreting WD at the time of ignition between the rotating magnetic case, as discussed in this study, the rotating non-magnetic case (e.g., Yoon & Langer 2004a) and the non-rotating case (e.g., Woosley & Kasen 2011) are a clear indication that rotation and magnetic effects should not be neglected in the future study of accretion scenarios in the lead-up to explosive stellar transients.

Acknowledgments

This research was supported by the Korea Astronomy and Space Science Institute under the R&D program (Project No. 3348-20160002) supervised by the Ministry of Science, ICT and Future Planning. Support by the Deutsche Forschungsgemeinschaft (DFG), Grant No. Yo 194/1-1, is gratefully acknowledged. The authors wish to thank the anonymous referee, whose comments added to the clarity and scope of the article. P.N. would like to thank Phillip Podsiadlowski for useful discussions.

References

Appendix A: Additional table

All Tables

All Figures

Fig. 1 Rotational velocity profile of model sequence Lm20510-5 (Minit = 0.54 M⊙) (A) and Lm21010-5 (Minit = 1.0 M⊙) (B) at three different times during its evolution (facc = 1.0, Ṁ = 5 × 10-8M⊙/ yr and log (Linit/L⊙) = −2 in both cases). Line i corresponds to t = 0 yr, the start of mass accretion, m to the midpoint of the accretion phase and f to just after the point of He-ignition. In subfigure A) this corresponds to m at t ~ 1.55 Myr and f at t = 3.11 Myr. In subfigure B) the corresponding times are m at t ~ 0.3 Myr and f at t = 0.64 Myr. The y-axis is normalized to unity with the maximum value of each graph indicated in ωmax.

In the text

	Fig. 2 Evolution of the surface velocities of model sequences L00510-5, Lm20510-5 (A) and L01010-5, Lm21010-5 (B). All model sequences are computed up to He-ignition. The y-axis is normalized to the critical rotational velocity. Menv is the mass of the helium envelope and corresponds to the amount of helium accreted. All models were computed with Ṁ = 5 × 10-8M⊙/ yr and facc = 1.0.
In the text

Fig. 3 Chemical profile of model sequence Lm20510-5 (Minit = 0.54 M⊙) (A) and Lm21010-5 (Minit = 1.0 M⊙) (B). The remaining initial parameters are the same for both model sequences: facc = 1.0, Ṁ = 5 × 10-8M⊙/ yr and log (Linit/L⊙) = −2. Included are the three most abundant isotopes He4, C12 and O16 in units of relative mass per shell over the shell’s mass coordinate. Solid lines represent the profile at the time of the start of mass accretion at t = 0 yr, dashed lines at the midpoint of the accretion phase at t ~ 1.55 Myr(A) and t ~ 0.3 Myr(B), dotted lines just after the point of He-ignition at t = 3.11 Myr(A) and t = 0.64 Myr(B). The inset is a zoom of the area of the initial envelope-core interface (the interface is shown as a dashed gray line), where the initial profile has been omitted for the sake of clarity.

In the text

Fig. 4 Evolution of the temperature-density profile of a rotating (A) and non-rotating (B) WD with 0.54 M⊙ accretion rate is 5 × 10-8M⊙/ yr, facc = 1.0 and log (Linit/L⊙) = −1. Blue dots indicate the core-envelope interface. Initial (at the beginning of mass accretion) and final (at the point of helium ignition) profiles are indicated. The given masses (left to right) correspond to the presented profiles (initial to final). We note that the profiles marked as “initial” are not the first models in their evolutionary sequence, having been evolved for about one kyr each.

In the text

	Fig. 5 Evolution of the temperature-density profile of a rotating (A) and non-rotating (B) WD with 1.0 M⊙. All other parameters and labels as in Fig. 4.
In the text

Fig. 6 Mass coordinates of the point of the temperature maximum (Tmax) over time (solid colored lines) of several representative systems with initial masses of 0.54 M⊙(A), 0.82 M⊙(B) and 1.0 M⊙(C). The solid black line in each plot indicates the corresponding total mass of the WD (MWD), red lines correspond to an initial luminosity of log (Linit/L⊙) = 0 and blue lines to log (Linit/L⊙) = −2. The dashed black line indicates the initial mass of the WD. The green dots indicate the He-ignition points. All depicted sequences were calculated with facc = 1.0 and Ṁ = 5 × 10-8M⊙/ yr. The gray arrows indicate the point of ignition in relation to MWD. Mr,ign is the mass coordinate of the igniting shell.

In the text

Fig. 7 Total masses (M) of all computed model sequences of this study with respect to constant mass accretion rate (Ṁ). The presented graphs differ in the chosen initial luminosity of the white dwarf model with log (Linit/L⊙) = −2 in graph A), log (Linit/L⊙) = −1 in B) and log (Linit/L⊙) = 0 in C). Black solid lines correspond to the initial mass of each model sequence with the initial mass being indicated. Colored lines indicate the final mass of each model sequence with green corresponding to Minit = 0.54 M⊙, red to Minit = 0.82 M⊙, blue to Minit = 1.00 M⊙ and yellow to Minit = 1.10 M⊙. Line types indicate chosen rotational parameters. Solid colored lines for non-rotating models, dashed lines for facc = 1.0, dotted lines for facc = 0.6 and dash-dotted lines for facc = 0.3. Dots indicate the masses of individual models.

In the text

Fig. 8 Temperature profiles over time of individual representative model sequences. Mr is the mass coordinate, t the time since the start of the accretion phase. The color bar indicates temperature and the black line corresponds to the current total mass of the WD. All model sequences were computed with initial parameters of Minit = 0.54 M⊙, log (Linit/L⊙) = 0 and facc = 1.0. Model sequence A) (L00510-1) uses a mass accretion rate of Ṁ = 1 × 10-8M⊙/ yr, model B) (L00510-2) Ṁ = 2 × 10-8M⊙/ yr and model C) (L00510-4) Ṁ = 4 × 10-8M⊙/ yr. It should be noted that the temperature color scale was restricted to the indicated range. Temperatures higher and lower than the limits do occur. The gap visible in plot B) is due to a subsequence of models computed with a larger time step than the rest of the sequence, resulting in a physically irrelevant discontinuity of the available data.

In the text

Fig. 9 Maximum temperature over time for a number of representative model sequences. All model depicted model sequences were calculated with Minit = 0.54 M⊙ and facc = 0.3. Blue lines indicate a mass accretion rate of Ṁ = 3 × 10-8M⊙/ yr, red lines Ṁ = 2 × 10-8M⊙/ yr and green lines Ṁ = 1 × 10-8M⊙/ yr. Solid lines indicate an initial luminosity of log (Linit/L⊙) = −2, log (Linit/L⊙) = −1 and dotted lines log (Linit/L⊙) = 0.

In the text

	Fig. 10 Densities at the point of highest temperature (ρ(Tmax)) with respect to the total current mass of the accumulated He-envelope (MHe). The colored dots indicate the current surface velocity of the model. Depicted are model sequences lm10810-3, lm10806-3 and lm10803-3. The model sequences differ in the chosen facc-parameter.
In the text

Fig. 11 Local rotational velocity (v) divided by the local Keplerian velocity (vK) with respect to mass coordinate (M). Depicted are model sequences lm10810-3 and lm10806-3. Solid lines represent the facc = 1.0 case (lm10810-3), dotted lines the facc = 0.6 case (lm10806-3). The color bar indicates the time-coordinate of the profile. The black lines indicate the mass coordinate of the point of ignition for the respective model. The solid black line is valid for facc = 1.0 and the dotted line for facc = 0.6.

In the text

	Fig. 12 Density at the point of ignition (ρign) plotted against mass accretion rate. The black dotted line indicates the minimum density for a supersonic runaway according to Woosley & Weaver (1994). Only model sequences with facc = 1.0 included.
In the text