CONDITIONS FOR SUCCESSFUL HELIUM DETONATIONS IN ASTROPHYSICAL ENVIRONMENTS

The study of Type Ia supernovae (SNe Ia) occupies a critical position with regard to our understanding of the universe. SNe Ia have already delivered powerful insights into several areas of physics, especially cosmology and nucleosynthesis. Yet, despite their immense value to astrophysics, how SNe Ia actually explode remains a vexing problem. It has been generally accepted that these violent explosions are the result of the thermonuclear disruption of compact carbon and oxygen (C/O) white dwarf (WD) stars that have neared, or exceeded, the Chandrasekhar mass limit. Beyond that, however, little is certain. For instance, the mechanism(s) by which SN Ia progenitors reach their critical states is currently unknown, and more fundamentally, we remain ignorant as to what these critical conditions are. Complicating the issue further, WDs only need to be close to the Chandrasekhar limit to be capable of auto-igniting at their centers, and external mechanisms can lead to the ignition of even less massive WDs. These mechanisms include collisions between WDs in dense stellar environments or in multiple stellar systems (Rosswog et al. 2009a; Raskin et al. 2009; Hawley et al. 2012; Katz & Dong 2012), tidal encounters with moderately massive black holes (BHs; Luminet & Pichon 1989; Rosswog et al. 2008; Ramirez-Ruiz & Rosswog 2009; Rosswog et al. 2009b; Haas et al. 2012), mergers of WDs with neutron stars (NSs) or stellar-mass BHs (Fryer & Woosley 1998; Fryer et al. 1999; Lee & Ramirez-Ruiz 2007; Metzger 2012; Fernández & Metzger 2013), and the accretion of dense material from a WD companion (Nomoto 1982; Woosley et al. 1986; Bildsten et al. 2007; Fink et al. 2007, 2010; Guillochon et al. 2010; Shen et al. 2010; Woosley & Kasen 2011; Dan et al. 2011, 2012; Sim et al. 2012). SN Ia initiation and the diversity of possible outcomes once ignition occurs are therefore pressing issues to the astrophysics community that call for further study.

Several models have been proposed for the sequence of occurrences that lead to SN Ia-like events, but all share the common feature that a self-sustained detonation powered by nuclear reactions propagates through a large fraction of the interior of the host object (or objects). In the absence of hydrogen, the fuel that powers this detonation consists of either pure helium (He) or the products of He burning, typically a mixture of carbon and oxygen. This detonation must be triggered by an increase in density, temperature, or both. The conditions for the initiation of detonations is a topic that has been discussed at length in the context of C/O burning (Arnett & Livne 1994; Niemeyer & Woosley 1997; Röpke et al. 2007; Seitenzahl et al. 2009), but only to a limited extent for the ignition of pure He (Khokhlov & Ergma 1986; Khokhlov 1989). In this paper, we present a parameter space study to find the critical temperature gradient length scales necessary to successfully initiate a detonation in He media for a given combination of density and temperature.

A detonation will occur when the nuclear timescale is on the order of the local dynamical timescale. However, the definition of both of these timescales is ambiguous; the effective nuclear timescale depends on a number of reactions with different rates, and the effective dynamical timescale depends on the size of the region being heated. Furthermore, the heated region is most likely not uniform in either composition or temperature at any particular time, and even less so over the timescale in which the heating occurs. We can analytically estimate the length scale ξ that must be heated to produce a detonation by calculating the speed at which the detonation will travel u and then multiplying by the timescale of the corresponding nuclear reactions τ_nuc, ξ = uτ_nuc. Both u and τ_nuc can be calculated by using simple one-dimensional models of the detonation in combination with a nuclear network (Khokhlov 1989). This estimate is an upper limit on the size required, as heating a region larger than this scale would result in an overdriven detonation that would eventually slow to the steadily propagating solution. However, while this method can establish the length scale of the resulting detonation structure, it is unclear if a region smaller than ξ may be capable of yielding a detonation, especially given that the initial conditions are unlikely to resemble the assumed steady-state detonation structure. For instance, the hydrodynamical simulations of He detonations in WD atmospheres of Townsley et al. (2012) show that steadily propagating weak detonations can often strengthen toward a steady-state solution, though only after the detonation has traveled some distance.

To determine the critical temperature gradient length scale for a given density and floor temperature, while simultaneously resolving the initiation region and subsequent shock wave front, we perform one-dimensional reactive hydrodynamical calculations that begin with a hot spot in a He medium and check if they lead to detonations. Our measurement of these critical length scales allow further constraints to be placed on SN Ia models as well as other more exotic thermonuclear explosions involving the disruption of He WDs. For many systems, we find that the spatial scales relevant for the initiation of these He detonations are unresolved in even the most computationally expensive multidimensional simulations, with a continuum of critical lengths ξ_crit that ranges from 1 cm to 10¹⁰ cm.

This paper is structured as follows. In Section 2, we present the SN Ia initiation models in question and the density and temperature conditions in which they subsist. In Section 3, we describe the numerical methods and initial conditions used for the study and present the results. In Section 4, we discuss the results and their application to thermonuclear explosion models.

WDs, the end point in the evolution of the vast majority of stars in the universe, are extremely common, accounting for about 65% of the total heavy element abundance (i.e., heavier than He, Fukugita & Peebles 2004). Because WDs experience no mass loss through single stellar evolution, their material can only be ejected through dynamical interactions with other objects and/or by the rapid release of nuclear binding energy. Although at present the reservoir is dominated by C/O WDs, which possess a thin layer of He (Lawlor & MacDonald 2006), a non-negligible fraction of the mass lies within pure He WDs (with masses in the range of 0.17–0.45 M_☉, ∼6% of WDs; Brown et al. 2010; Kleinman et al. 2013) and hybrid WDs consisting of small C/O cores and thick He envelopes with total masses of ∼0.5 M_⊙ (Iben & Tutukov 1985; Nelemans et al. 2001b). Pure He WDs cannot be the result of the evolution of single stars within the lifetime of our Galaxy; these WDs can only be produced via interacting binary systems (Nelemans et al. 2001a, 2001b; Rappaport et al. 2009) and approximately half of all WDs in close binaries are expected to be either He-core or hybrid WDs (Ruiter et al. 2011). In addition to systems in which both stars are WDs, binary systems in which the accretor is either an NS or a BH are also possible. As a result, the conditions under which He may be ignited are diverse, with a variety of accretion rates, initial entropies, and surface gravities.

In Figure 1, we present the density-temperature parameter space relevant to He ignition. If the timescale on which the He material can react, its dynamical timescale τ_dyn = (Gρ)^−1/2 s, is much shorter than the burning timescale $\tau _{3\alpha }=9.0\times 10^{-4} T_9^3 \exp {(4.4/T_9)}\rho _6^{-2}$ s (Khokhlov & Ergma 1986), the material can expand rapidly enough to quench burning (T = 10⁹T₉ K and ρ = 10⁶ρ₆ g cm⁻³). This is because in a pure He environment the combustion rate is limited by the triple-α reaction. Appreciable burning will therefore only take place if τ_3α ≲ τ_dyn. This comparison of timescales can be used as a simple estimate for whether or not substantial He burning can occur in particular physical systems, but the onset of dynamical burning does not guarantee that a self-sustained detonation will form, as the conditions for detonation are non-trivial (Khokhlov 1989). However, we do not expect that detonations will occur in systems where τ_3α ≫ τ_dyn, because burning would likely be quenched by expansion. Above temperatures of approximately 8 × 10⁹ K, energy loss by neutrino emission begins to overtake the energy production due to triple-α reactions (the gray-shaded region in Figure 1). We have therefore restricted our simulations to the region in which τ_3α ⩽ 10τ_dyn and in which the relation $|\dot{E}_{\nu }|<\dot{E}_{3\alpha }$ holds, where $\dot{E}_{\nu }$ includes contributions from pair annihilation, plasma, photoneutrino, recombination, and bremsstrahlung processes (Itoh et al. 1996). For comparison, we have included in Figure 1 the trajectories for virial hydrostatic He WDs; that is, we have calculated the parametric density-temperature relation as a function of radius for WDs that are raised to temperatures such that their degeneracy and radiation pressures are roughly equal. To model these WDs, we solve a polytropic equation of state with n = 3/2 and with masses up to 0.45 M_☉. In addition, we also display in Figure 1 the trajectories for double WD mergers, collisions of WDs, tidal compression events of WDs orbiting BHs, and the disruption of He WDs orbiting NSs. These trajectories all fall within the region of interest, and thus are suitable candidates for systems in which successful He detonations can occur.

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3.1. Numerical Methods and Initial Conditions

To study the detonation of He, we have carried out a set of one-dimensional Cartesian simulations using the adaptive mesh refinement code Mezcal. The code has been tested extensively against standard test problems, and has been used to simulate magnetohydrodynamic jets (e.g., De Colle & Raga 2006; De Colle et al. 2008) and gamma-ray burst afterglows (De Colle et al. 2012a, 2012b). In Mezcal, the equations of hydrodynamics are integrated in time using an explicit second-order Runge–Kutta method, while the fluxes are calculated using the Harten–Lax–van Leer (HLL) method, with a second-order spatial reconstruction of the primitive variables at the cell interfaces. A thirteen element "α-chain" nuclear network, which includes all alpha process elements from ⁴He to ⁵⁶Ni (Timmes et al. 2000) and energy losses due to neutrino cooling, is integrated in time by solving the system of equations:

$$\begin{equation} {\partial \rho _i\over \partial t}+\nabla \cdot (\rho _i \boldsymbol v)= \Gamma _i, \end{equation} \tag{ 1 }$$

where ρ_i is the mass density of the element i (with i = 1, ..., 13), v is the velocity vector and Γ_i is the total reaction rate for the element i. The system of equations (1) is coupled to the hydrodynamics by an operator splitting method (LeVeque 2002). We first integrate Equation (1) by setting Γ_i = 0, and then we use the new ρ_i values to integrate the set of ordinary differential equations (Timmes & Swesty 2000):

$$\begin{equation} {d\rho _i\over dt} = \Gamma _i. \end{equation} \tag{ 2 }$$

Finally, the original internal energy within a cell e₀ is updated based on the energy loss and gain due to the nuclear reactions and the neutrino cooling to obtain the new internal energy e₁.

To properly couple the hydrodynamics with energy injected by burning, and removed by neutrinos, the time step is limited such that the energy change is less than 0.5% of the internal energy for all cells. This is achieved by setting the time step $\Delta t_{\rm nuc}^n = 5 \times 10^{-3} \Delta t^{n-1} e_0/ \vert e_1-e_0\vert$, where e₀ and e₁ are the thermal energies before and after the update from the nuclear routine. Otherwise, the time step is set by the Courant condition, defined as the minimum among all the cells of Δx/(v + c_s), where Δx, v and c_s are the size of the cell, the velocity, and the sound speed respectively. As in other hydrodynamical studies, both the energy release calculated by a reduced network and the fraction of ⁵⁶Ni produced can vary by ∼30% when compared to a 489 isotope network (Timmes et al. 2000), or when using a coupling constant that is larger than what is used in this study (Hawley et al. 2012). Therefore, while we expect that this reduced network should capture the general nucleosynthetic trends, the normalization of these trends may differ from what is presented here.

In the Mezcal code, a basic Cartesian grid is built at the start of the simulation, and it is refined based on the initial conditions and the subsequent evolution of the flow (see De Colle et al. 2012a). In the simulations presented here, we use a refinement (and derefinement) criterion based on the ensuing density gradients. In particular, when the fractional increase in density, computed by comparing a cell with its two neighbor cells, is larger (smaller) than 0.5 (0.05), the cell is refined (derefined) and two new cells are created (or the old cells are eliminated).

For our study we initialized linear temperature gradients ("hot spots") of a particular size ξ in a pure He environment of density ρ, where the maximum temperature T_max occurs at x = 0. We assume one-dimensional Cartesian geometry with reflecting boundary conditions at x = 0 and outflow boundary conditions at x = x_max. For consistency, we always chose the temperature floor T₀ = 10⁻²T_max. We employed the method of bisection to determine the minimum size ξ_crit of the hot spot that resulted in a self-sustained detonation for a given T_max and ρ.

As a test of the implementation of the nuclear network module in the Mezcal code, we first reproduced the resolution study for C/O detonations as presented in Section 3.3 of Seitenzahl et al. (2009), and also did not find convergence until the minimum grid cell size Δx_min ≲ 2 cm. We then performed a resolution study for a pure He composition, with T_max = 10⁹ K, T₀ = 10⁷ K, and ρ = 10⁶ g cm⁻³, and found convergence for Δx_min ≲ 10⁴ cm, which is approximately equal to the estimate of the width of the burning region presented in Table I of Khokhlov & Ergma (1986).

We determined which simulations successfully produced detonations by identifying shocks that propagated at a near-constant velocity, indicating that they were powered by the continual release of energy through sustained nuclear reactions. In Figure 2 we show the results of two different simulations, where the initial conditions differ only in the extent ξ of their hot spots. In the left column of Figure 2, the temperature gradient quickly evolves into a fully-formed shock wave, rapidly converting the majority of the He fuel into Ni (bottom two panels). The right column shows a case where ξ has been reduced by a factor of ∼2. In this case, the burning that is initiated is unable to outpace the hydrodynamic expansion of the fluid, even though triple-α reactions initially inject a significant amount of energy into the system. The boundary between successful and unsuccessful detonation is sometimes signified by the inability to produce a shock. This is in contrast to what Seitenzahl et al. (2009) found for the initiation of C/O detonations; they found a shock is produced even in sub-critical conditions.

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In Figure 3, we show the time evolution of T and X_Ni for a successful detonation. For ρ > 5 × 10⁶ g cm⁻³ it is common for the initial conditions to lead to multiple runaways that only produce a stably-propagating detonation through their mutual interactions. These multiple runaways come as the result of the strong density dependence of the triple-α reaction. The expansion of the hot spot introduces upstream (pre-shock) perturbations to the density, which can potentially experience dynamical burning if the temperature in these regions is ∼10⁹ K. Each runaway produces a pair of left- and right-traveling shock waves that interact with the hot spot as it expands, and collide both with shocks produced from other runaway regions, and with the reflecting boundary condition at x = 0. While this behavior is somewhat pathological, our criteria for the boundary between success and failure is still applicable, as we ensure that these multiple interactions eventually produce a single shock that propagates at a constant speed.

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The choice to use a linear temperature profile is admittedly arbitrary, so we determined the critical size for a Gaussian temperature distribution and compared the results to a linear temperature profile with the same T_max, T₀ and ρ. Figure 4 shows the time evolution of a critical detonation for these two cases where ξ = ξ_crit (for the linear case) or R = R_crit (for the Gaussian case, where R is equal to the standard deviation). While the delay between the start of the simulation and the time at which a self-propagating detonation forms is different between the two simulations, the resulting detonation structure is almost identical.

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3.2. Critical Size Estimates

Past efforts in determining minimal length scales for the initiation of detonations have been concerned with C/O WDs (Arnett & Livne 1994; Niemeyer & Woosley 1997; Röpke et al. 2007; Seitenzahl et al. 2009). In all of the aforementioned works, these critical sizes were determined through the use of one-dimensional hydrodynamical simulations, but while these studies have been performed by numerous groups for C/O mixtures, they have never been performed for pure He. In Moll & Woosley (2013) the critical size required for the successful detonation of a helium envelope with mass 0.045 M_☉ on the surface of a 1 M_☉ C/O WD was recently determined through the use of multidimensional simulations. However, a systematic determination of the critical length scales for all possible combinations of core and envelope mass would be computationally prohibitive in multiple dimensions.

In the left panel of Figure 5, we show the minimum size scale ξ_crit for combinations of ρ and T_max that successfully resulted in a detonation. These scales range from ∼1 cm at ∼10⁹ g cm⁻³ to 10¹⁰ cm at 10⁵ g cm⁻³, with a relatively weak dependence on T at each ρ. Overlaid on this panel are the same regions highlighted in Figure 1, where the red line again shows the virial relation for WDs. Within the blue crescent (where dynamical burning occurs) and along the WD relation, the minimum scale ranges from 10⁸ to 10¹⁰ cm, similar to the range of WD sizes. As shown in Figure 5, the critical T necessary to produce a detonation at a given size ξ is comparable to the virial temperature of the WDs themselves, and thus these conditions can potentially be achieved through dynamical processes that are driven by the WD's own gravity (e.g., collisions or the acquisition of material from a companion). The He shell detonations of Moll & Woosley (2013) are consistent, within a factor of 2–3, with our results; for a spherical cap initiation zone of ∼7 × 10⁵ g cm⁻³ in density and a peak temperature of 2 × 10⁹ K, the necessary size for successful detonations was found to be on the order of 10⁷ cm (see "model D" of Moll & Woosley 2013).

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It is apparent from the left panel of Figure 5 that ξ_crit is primarily a function of density, with only a slight dependence on temperature as T approaches the ν cooling limit and where the dynamical and burning timescales are comparable. In the right panel of Figure 5 we show how the initiation scale varies as a function of density, where the peak temperature of the linear gradient T_max is assumed to be 10⁹ K (corresponding to an upstream temperature T₀ = 10⁷ K), and compare it to the Chapman–Jouguet (CJ) length l_CJ ≡ Dτ_3α, where D is the CJ velocity for that particular density and τ_3α is calculated using the corresponding downstream (post-shock) temperature (Khokhlov 1989). We find that the initiation scales at these densities are such that l_CJ ≳ 100ξ_crit, implying that burning may not proceed to completion within the time it takes the detonation to traverse the system.

3.3. Nucleosynthesis

In the simple CJ model of a detonation (Chapman 1899; Jouguet 1905), the burning is assumed to be instantaneous across an infinitely-thin zone behind the shock, and the detonation propagates at a single speed, the CJ velocity D. In reality, the reactions that take place occur in a finite timespan, and thus the region over which the reactions complete has a finite size. Zel'dovich, von Neumann, and Döring (ZND; Zel'dovich 1940; von Neumann 1942; Döring 1943) improved this by modeling a detonation as a one-dimensional leading shock wave moving at the detonation speed trailed by a reaction zone of finite width (Fickett & Davis 1979). This enables one to estimate a variety of length scales, including the region over which the bulk of the energy is injected, the distance over which the fuel is exhausted, and the distance over which the composition approaches nuclear statistical equilibrium (NSE; Timmes & Niemeyer 2000).

For systems in which the CJ and ZND scales are much smaller than the scale of the He fuel region, the burning is expected to yield a composition of material composed primarily of He and nickel, because the timescale for α-capture is short compared to the triple-α timescale (Khokhlov 1989). If the burning is quenched before the detonation has traveled a few CJ lengths, the burning cannot proceed to completion, as is the case for systems in which the initiation scale is comparable to the local fuel scale. Alternatively, the material can expand, thereby quenching the burning before alpha particles have accumulated on each seed nucleon to produce ⁵⁶Ni (Townsley et al. 2012).

Our results are consistent with this picture. In the left panel of Figure 6, we calculate the fraction of ⁵⁶Ni in the region immediately downstream to the detonation within our simulations shortly after the detonation reaches a steadily-propagating state. To do this, we average the nickel abundance within 3ξ_crit ⩽ x ⩽ 4ξ_crit at the moment the shock crosses x = 4ξ_crit. These measurements are compared to the expected fraction of ⁵⁶Ni if the detonation is allowed to propagate a distance long enough for the downstream material to reach NSE (Figure 6, right panel). We find that the two models produce radically different yields of ⁵⁶Ni, indicating that the post-detonation state that is produced shortly after a detonation first forms is not representative of the state it would have reached if it was allowed to expand until NSE was achieved.

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As described in Section 3.2, the initiation size ξ_crit is comparable to the size of the WD itself for ρ ≲ 10⁶ g cm⁻³. Incidentally, this is the same region in which the fraction of ⁵⁶Ni declines to the point that it is no longer the predominant species within the ash (Figure 7, right panel. Here the titanium abundance is averaged over the hot spot zone, which we define to be 0 ⩽ x ⩽ ξ_crit, at the time the shock reaches x = 4ξ_crit). For densities 10⁵ ≲ ρ ≲ 10⁶ g cm⁻³, and for temperatures typical of adiabatically compressed WD material (red dashed line), the predominant species that is produced is ⁴⁴Ti (Figure 7, left panel). This progression to heavier species in the ash as the density increases is similar to the result seen when He is burned at constant density and temperature (Hansen 1971).

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The stable decay product of ⁴⁴Ti is ⁴⁴Ca, and the production of this particular isotope has been associated with Type II SNe. However, the amount of ⁴⁴Ti produced per Type II SNe may not be sufficient to explain the amount required to produce the ratio of ⁴⁴Ca/⁴⁰Ca observed in the Sun (Timmes et al. 1996). Given the fact that greater than half of the initial He can be burned to ⁴⁴Ti, He burning can produce vastly larger quantities of ⁴⁴Ti than Type II SNe, which on their own may not be able to explain observed abundance (Woosley & Weaver 1994; The et al. 2006). If the rate of WD collisions, disruptions, and SN Ia events involving a He WD are comparable to the type II rate, He burning on low-mass WDs may be the primary source of ⁴⁴Ti in the universe.

We have presented one-dimensional results of the determination of critical (smallest) spatial scales for the gradient initiation of detonations in He matter using the reactive hydrodynamics code Mezcal. In particular, we have shown that the spatial scale ξ_crit required for the initiation of a detonation differs significantly from the CJ size, specifically, it is approximately two orders of magnitude smaller for a wide range of densities. That the initiation length ξ_crit is greatly exceeded by the CJ length implies that certain systems may not reach NSE within the time it takes a detonation to traverse the He material. The existence of a small region in which a detonation can be initiated does not guarantee that a sufficient length of fuel is available to ensure NSE is eventually achieved. Therefore incomplete burning that terminates at an element lighter than ⁵⁶Ni will occur in many physical systems in which the size of the fuel reservoir is comparable to ξ_crit.

In astrophysical systems for which the state of the fluid is initially determined by a balance of pressure and gravitational forces, a compressive event will typically occur on the local dynamical timescale, which naturally results in the material being heated to the local virial temperature. Whether this action leads to a detonation depends on whether or not the region that is being compressed has a scale that is comparable to ξ_crit. In systems where there is a prolonged period in which He experiences repeated compressions, e.g., mass transferring or merging binary systems, the criteria for detonation will be satisfied as soon as the length scale of the region being compressed is comparable to the initiation scale. These repeated compressions are seen in stably-accreting, unstably-accreting, and merging systems, and are driven by convective motions and/or fluid instabilities (Fryer et al. 1999; Shen & Bildsten 2009; Guillochon et al. 2010; Fernández & Metzger 2013).

Within an atmosphere in which the mass of the atmosphere is small compared to the mass of the host object, the density falls off exponentially, so material will only have approximately constant density over the scale height H. As ξ_crit is a strong function of density (Figure 5), the detonation is not expected to be sustained as it propagates upward through such a steep density gradient, resulting in a "blowout" (Townsley et al. 2012) that quenches burning after propagating in the vertical direction a distance ∼H. However, the detonation can continue to propagate in the horizontal direction, as the bulk conditions in the base of the atmosphere of a nearly-spherical body do not vary much with the position, resulting in a continual blowout of partially-burnt material as the detonation propagates around the host object.

We find that for densities typical of WDs, the predominant element that is produced from burning is not necessarily ⁵⁶Ni, and is more commonly an intermediate α-chain element such as ⁴⁴Ti. Similar to the blowout case presented in Townsley et al., the detonation is quenched in such systems after propagating a distance comparable to ξ_crit. Therefore, the results of our simulations may be used to approximate the nucleosynthetic yields resulting from blowouts at a given initial density ρ (Figures 6 and 7), with the caveat that our setup does not account for the density gradient present in the blowout scenario. For a candidate thermonuclear transient resulting from a double-detonation model, a measure of the amount of ⁴⁴Ti produced by the event, in combination with a measure of its peak brightness, which can be used to estimate the mass of ⁵⁶Ni produced (Ruiter et al. 2013) (and thus the mass of the C/O core that exploded, Sim et al. 2010), may be able to uniquely constrain the nature of the progenitor system.

In the density regimes that are expected to realistically facilitate He detonations, intermediate-mass elements from ⁴⁰Ca to ⁴⁸Cr are expected to dominate the nucleosynthetic yields. Perets et al. (2010) reported on SN 2005e, an anomalous low-luminosity SN, in which roughly half of the ejected mass was calcium and significant fraction was radioactive titanium. It was concluded, due to the abundance of helium burning products, that helium burning had occurred in this system. Our results are consistent with this conclusion; the average density of pure He WDs tends to be ∼10⁵ g cm⁻³, which coincides roughly with the peak in calcium production (see Figure 7, top left panel). The numerical investigation of Waldman et al. (2011) into He shell detonations on low-mass WDs could produce large amounts of ⁴⁸Cr in fuel densities of ∼5 × 10⁵ g cm⁻³, which is in agreement with our findings (Figure 7, bottom left panel). The production of ⁴⁸Cr is particularly interesting because the energy release and timescale of its decay chain are comparable to those of ⁵⁶Ni. Because of this similarity, equal amounts of ⁴⁸Cr and ⁵⁶Ni would produce events of near-equal brightness and similar temporal evolution.

In some circumstances, such as collisions and tidal disruptions, the material's temperature is increased in a single dynamical episode to super-virial temperatures, as the source of gravity in these systems is external and can be arbitrarily large. In these cases, the scale of the system need not be comparable to the initiation scale. In Figure 8, we transform the critical sizes presented in Figure 5 into the mass M contained within the critical region, and compare this to the integrated mass corresponding to a density ρ within WDs of different masses. We find that the temperature increase is sufficient to produce detonations in fully-virialized He WDs with masses as low as 0.24 M_☉, although detonations in lower mass WDs are possible if the material is raised significantly above the virial temperature, as can be the case in either a tidal disruption or in a collision with a more massive (possibly relativistic) compact object. This idea is given some credence by the results of hydrodynamical simulations illustrating that 0.2 M_☉ He WDs do not detonate after a head-on, equal-mass collision (Rosswog et al. 2009a), but do successfully detonate when colliding with a more massive WD or after being tidally compressed and shock-heated by an IMBH (Rosswog et al. 2009a, 2009b). However, because ξ_crit is exceeded by the CJ length by a factor of ∼100, the initiation scales are rarely much smaller than the size of the physical systems themselves, and thus incomplete burning is observed even in systems which are driven to super-virial temperatures (e.g., Rosswog et al. 2008).

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While this work presents the first systematic study concerning the length of the temperature gradient necessary to initiate a detonation in a pure He environment, the initial conditions we use are a simplification of the physical problem. Therefore, caution should be exercised in the application of these critical radii, which may deviate somewhat from what we have presented when the simplifying assumptions we have made are relaxed. We do find that the critical radii do not depend sensitively on the functional form of the temperature profile, as illustrated in Figure 4, but in reality the density, temperature, and/or composition are unlikely to be homogenous and smooth over the initiation region, as they are assumed to be here. Though we have restricted our study to one-dimensional simulations and optimistically employed a 13-species α-chain network, we have demonstrated that the variety of possible outcomes of He detonation is larger than anticipated, and that these outcomes have a diversity of predictive applications to upcoming transient searches.

We thank M. Dan, D. Kasen, S. Rosswog, and D. Townsley for stimulating discussions and fruitful collaboration. We thank F. Timmes for making the equation of state and reaction network used in this work publicly available. We thank the referee for providing thoughtful and constructive comments. We acknowledge support from the David and Lucile Packard Foundation, NSF grant: AST-0847563, the NASA Earth and Space Science Fellowship (J.G.), and the DGAPA-PAPIIT-UNAM grant IA101413-2 (F.D.C.).