MULTI-DIMENSIONAL MODELS FOR DOUBLE DETONATION IN SUB-CHANDRASEKHAR MASS WHITE DWARFS

Taking the (0.801 + 0.143) M_☉ WD model A as an initial condition, we did a series of 2D simulations that cover a local region including the core–shell interface. The domain range in most cases is 0 < R₈ < 1 and 3.3 < z₈ < 5.3 in cylindrical coordinates (R, z), the core–shell interface being at the spherical radius r₈ = 4.08. The grid resolution in most cases is 0.977 km (no AMR). The gravitational field is taken to be static and the boundary conditions are usually reflecting on all sides (as the detonation proceeds supersonically, the choice of boundary conditions is not critical). All simulations employed a full 19 isotope network that covers helium, carbon, and oxygen burning. The detonation spots used to start the detonation had a radius of 50 km, a temperature of 2 × 10⁹ K at the center and 1.8 × 10⁹ K at the outer edge (linearly decreasing), and a radial velocity of 8000 km s⁻¹ at the outer edge (increasing linearly from zero) and twice the ambient density.

Table 2 contains a list of setups and results, and Figure 2 depicts snapshots of several cases. In the standard case (a), the helium detonator is centered at the radius at which the temperature is maximal, 186 km above the core–shell interface. While the detonation creates an indentation in the interface, and some of the carbon at the outer edge of the core burns as a result of the entering shock, the burning does not support a detonation in the core. Direct drive clearly fails in this case.

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With the helium detonation initiated 200 km higher, case (b), the detonation penetrates into the core. The front is unstable though, developing dents filled with compressed material that fails to ignite. The formation of these instabilities appears to be the result of complex interactions of reflecting shock waves. Figure 3 shows snapshots of the temperature in the detonation front in case (b). There are small indentations in the front, seemingly related to waves in the post-detonation region (top panel, indicated by cyan arrows). A few hundred kilometers below the core–shell interface, a big dent forms (at R ⩽ 200 km in the plots). While it is later consumed by a secondary detonation front, a new dent forms (middle and bottom panels, olive-green arrows). The vortical structures visible in the plots on a high zoom level are indicative of Richtmyer–Meshkov instabilities (RMIs; e.g., Brouillette 2002). The presence of RMIs is supported by the fact that the pressure gradient behind the detonation is normal to the front, whereas the density gradient across the interface is radial with respect to the center of the star, i.e., the pressure and density gradients are misaligned, as is typical for RMIs. It is possible, however, that RMIs here emerge as a secondary phenomenon. Global simulations, presumably in 3D, would be needed to determine whether such an unstable detonation eventually blows out or consumes the core, but they are too expensive at this resolution. Recognizing that these are marginal cases, we opted not to follow such pathological detonations in this study and dismissed them as failures.

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With the helium detonation initiated 400 km above the temperature maximum, case (c), the detonation stably penetrates into the core. This strengthens our assumption that the instabilities described above are indicative of marginal cases. To make sure that the detonation stays stable, we followed it in a global simulation that encompasses half of the star, with the same resolution (≈1 km) as in the local simulation in a region that extends from the detonation spot to the center of the star (R₈ < 1 and z₈ < 5). This detonation stably proceeds all the way to the center. Oxygen burning starts 830 km below the interface (at 80% of the core radius).

In a simulation similar to case (b) with half resolution (1.95 km), case (d), the front is stable. The formation of indentations in the front is completely suppressed. We followed this detonation until a depth of 1780 km beneath the core–shell interface. Oxygen starts to burn 1130 km below the interface (at 72% of the core radius; see the lower dashed line in panel (d)), where the density exceeds 5.3 × 10⁶ g cm⁻³.

The fronts of detonations started at different altitudes have different curvatures when they hit upon the core–shell interface. This raises the question as to whether the curvature of the detonation front plays a key role. Apart from altitude, the curvature is determined by the shape of the detonation spot, provided the spot is sufficiently close to the interface (for isotropic expansion velocities, any explosion eventually becomes spherical). A detonator in the form of an oblate spheroid fares better than a spherical one, compare panels (e) and (a). However, the detonation front also starts out considerably hotter than in the spherical case ($T_9^{\rm max}=3.24$ instead of 2.88 at t = 0.01 s), indicating that part of the observed difference is possibly related to the details of how the detonation was started, rather than curvature alone.¹ The detonation spot in case (e) is stretched by a factor of three in the radial direction, its extent in the vertical direction being the same as that of the spherical spots. Shortly before it reaches the core–shell interface, the curvature of the underside of the oblate detonation front roughly corresponds to a sphere centered 200 km above the hottest layer, i.e., it is comparable to the detonation in case (b). A simulation with a detonator stretched by a factor of four (i.e., even more oblate) gives a very similar result (not shown in Figure 2). The curvature in this case corresponds to a sphere centered more than 400 km above the hottest layer, i.e., one would expect from case (c) that the detonation should enter the core without problems, yet it does not. We therefore surmise that curvature is not as important as altitude.

We also considered the possibility of a layer where helium mixes with core material. In case (f), we start with a mixture of 70%/20%/10% ⁴He/¹²C/¹⁶O in the region 3.98 < r₈ < 4.18, i.e., 100 km above and below the original core–shell interface. The detonation stably transcends into the core in this case. We surmise that a mixed layer on both sides of the interface effectively increases the distance from the detonator to the core, thus facilitating direct drive.

In cases (g) and (h), we start with a mixture of 70%/20%/10% ⁴He/¹²C/¹⁶O and 50%/30%/20% ⁴He/¹²C/¹⁶O, respectively, in a 100 km wide region above the original interface only. Although the detonation changes when it enters the mixed region, becoming hotter and faster, it fails to penetrate the core and stay intact in both cases. Even with half resolution, the detonation in case (g) shows the same instability (unlike case (d), for which the coarser resolution helped to suppress instabilities in the detonation front). A mixture of 80%/20% ⁴He/¹²C in a 100 km wide region also leads to failure (not shown in Figure 2).

The above discussion shows that direct drive is not likely for model A if the helium detonation starts where the gas is hottest, i.e., the most natural location for a detonator. Models B and C both have a denser 1.0 M_☉ core which in principle should detonate more easily than the 0.8 M_☉ core of model A. Model C is slightly more compact than model B.

Following a similar procedure, we tested for the possibility of direct ignition with detonators at different altitudes for these models. The radius of the detonation "hot spots" was 20 km and the distance between the core–shell interface and the temperature maximum was 149 km and 106 km in models B and C, respectively. In both cases, direct ignition failed if the detonation was started at this altitude.

If we started the helium detonation 100 km above the hottest layer of model B, however, the detonation front smoothly transcended into the core. If ignited 50 km above the hot layer, the detonation fragmented and died upon passing into the core. In model C, the direct drive worked with detonators at altitudes of 100 km, 50 km, and even 25 km above the hottest layer. At 12 km, however, it failed.² These results for direct drive with detonators at different altitudes are summarized in Table 3.

To explore the relevant physics underlying the existence of a critical altitude for directly driving a detonation into the CO core beneath the ignition point in the helium layer, a series of idealized calculations were carried out using the KEPLER code. The hypothesis to be explored was that successful propagation requires a critical amount of momentum in the downwardly moving shock wave. That momentum should be greater than or equal to that of a "critical mass" (Niemeyer & Woosley 1997; Röpke et al. 2007; Seitenzahl et al. 2009) of carbon and oxygen at the same density as exists at the helium–carbon interface. In addition, that momentum should be focused into an area comparable to the geometrical size of the critical mass. Stated another way, the altitude of the initial helium detonation should exceed the radius of the critical mass required for initiating a successful carbon detonation evaluated at a density equal to that at the helium–carbon interface.

Of course, this is quite approximate. The density and pressure are not constant in either the helium layer or the carbon underneath, and the energy yield from helium detonation is different from that of carbon detonation. We do not know if the helium detonation ignites at a point, which implies spherical propagation, or in an extended region which might have a more planar geometry. These complexities can be partly compensated for, however. The effective radius of the helium is one that encloses a critical mass (evaluated at the density of the interface), even though the density within that sphere varies. The detonation can be studied in a two-phase medium with a helium detonator surrounded by a thick shell of carbon and oxygen. The detonation should occur in spherical geometry so long as the size of the region in the helium layer that initially runs away is small compared with its altitude above the interface. We note that if, pending further study, the latter assumption were violated, the required altitude would be reduced (compare Tables 1 and 4 in Seitenzahl et al. 2009).

In the first study, spheres of 50% carbon and 50% oxygen were prepared with a constant density of 1.93 × 10⁶ g cm⁻³, as appropriate to the interface in model B. In a 1D spherical region of variable radius, the temperature was given a constant gradient (with respect to radius, not mass, even though the code is Lagrangian) with values ranging from 2.8 × 10⁹ K at the center to 1.0 × 10⁸ K at the edge. This 200 zone region was surrounded by 300 zones of carbon and oxygen at 10⁸ K. All zones had the same radial thickness. The radius of the region with the temperature gradient was varied and its ability to generate a self-sustaining detonation was determined. The central value of 2.8 × 10⁹ K was sufficiently high to guarantee that carbon burning propagated with a phase velocity that was initially supersonic. This procedure is very similar to the one previously followed by Niemeyer & Woosley (1997) and Röpke et al. (2007).

For a "detonator" radius of 150 km, an initially strong detonation decayed away after traversing an additional 70 km of "cold" carbon and oxygen. For a radius of 200 km, however, the detonation propagated successfully to the edge of the grid at 500 km. This is sufficiently far, considering that in the real star the density would have become higher and the detonation more robust. These detonators enclosed masses of 2.73 × 10²⁸ g (150 km) and 6.47 × 10²⁸ g (200 km), respectively.

A more precise descriptor of the critical detonation size may be the temperature gradient rather than the critical mass. For the two runs described here, the temperature gradients are 120 K cm⁻¹ (failed detonation) and 90 K cm⁻¹ (successful), respectively. Seitenzahl et al. (2009) studied the initiation of carbon detonation in spherical geometry and found a critical gradient of 4360 K cm⁻¹ at a density of 5 × 10⁶ g cm⁻³ (their Table 4). Our value is consistent with an admittedly uncertain extrapolation of their Figure 9 to 1.93 × 10⁶ g cm⁻³.

More realistically, though, the detonator consists of helium, not carbon and oxygen, and the initial background is better approximated as isobaric rather than assuming constant density. An initial temperature of 2.8 × 10⁹ K in the carbon also implies an unrealistically high energy density, not likely to be achieved at these low densities. A pure helium sphere with variable radius was surrounded by a thick shell of 50% ¹²C and 50% ¹⁶O again at 1.93 × 10⁶ g cm⁻³ with a temperature of 1.0 × 10⁸ K. The helium core was comprised of 200 Lagrangian mass shells of equal mass with a central temperature of only 1.0 × 10⁹ K and a temperature at the edge of 1.0 × 10⁸ K, and a constant gradient (with respect to radius) in between. As with the carbon detonator, the large value of central temperature assured a phase velocity for the helium burning that was initially supersonic and the well-zoned gradient implied the existence of a region where it was sonic. The density within the helium core was varied so as to provide a constant pressure equal to the pressure in the cold CO layer. Thus, pressure on the entire grid was initially 7.25 × 10²² dyne cm⁻². Gravity was neglected and the pressure was initially held constant by applying a boundary pressure equal to the pressure on the grid. If the radius of the helium sphere was 211 km enclosing 6.37 × 10²⁸ g, the detonation successfully propagated into the surrounding carbon. Another run with a helium sphere radius of 159 km (2.69 × 10²⁸ g) failed to detonate the carbon shell, however.

The energy yield from carbon detonation at this density, taken from the calculation, is 3.3 × 10¹⁷ erg g⁻¹. Oxygen does not burn here, but carbon and neon do, to magnesium and silicon. The yield from helium detonation, on the other hand, is only (5...6) × 10¹⁷ erg g⁻¹ because only about one-third of the helium burns to ⁵⁶Ni immediately behind the shock (the helium burning reaction rate saturates at high temperature). Some helium continues to burn well after the shock has passed, eventually raising the energy yield to 7 × 10¹⁷ erg g⁻¹, but this energy is not so important for sustaining the shock wave. The critical radius is also the cube root of the critical mass. This explains the similarity between the runs using a CO detonator and a helium detonator at this density.

These results qualitatively agree with those obtained using a 2D representation of the full star in the CASTRO code (Table 3). They also agree reasonably well with the pure CO study at constant density described above. Apparently, the critical altitude (or detonator radius) is not very sensitive to the different energy yields of carbon and helium burning. This is somewhat surprising, but confirms the hypothesis that the critical altitude in the helium layer is one that would enclose the critical mass of carbon required for detonation at the interface density.

Here we consider the case of a single, spherical detonation spot setting off a helium detonation where the hottest layer (at r₈ = 4.27) intersects with the positive z-axis in our coordinate system. This axisymmetric problem was investigated with different resolutions in three 2D runs (1, 4, and 5), and one 3D run (11) in a quarter-star-sized domain and mirror symmetries at the inner boundaries.

The helium detonation wave wraps around the star (see Figure 4(a)) and reaches the antipode of the detonator after about 1.2 s. For comparison, a sound wave in the helium shell would need about 5.6 s to reach the other side of the star. This corresponds to a front velocity of ∼1.1 × 10⁹  cm s⁻¹. The polar (latitudinal) velocity of the ash directly behind the front and the sound speed in the ash both are roughly 6.9 × 10⁸  cm s⁻¹. Their sum is larger than the actual front velocity, consistent with incomplete burning rather than a Chapman–Jouguet detonation (cf. Sim et al. 2012). The ⁵⁶Ni fraction far (∼1000 km) behind the detonation front is about 70%.

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The sliding detonation induces an internal compressional wave that converges off-center inside the core on the z-axis (in general, the axis of symmetry in the problem). The average velocity with which the perturbation propagates in a negative z-direction from the helium ignition point to the final convergence point is ∼4.0 × 10⁸  cm s⁻¹, a little larger than the mean sound speed inside the core (3.4 × 10⁸  cm s⁻¹).

The temperatures and densities in some zones of run 1 are high enough to potentially³ trigger a core detonation where the internal wave finally converges (Niemeyer & Woosley 1997; Röpke et al. 2007; Seitenzahl et al. 2009). However, it is likely that the core ignites earlier, at a larger distance from the center and close to the core–shell interface on the z-axis, where the internal wave front converges in the polar direction (but not in the radial direction), and the converging helium detonation drives a hot, thin inflow of ⁵⁶Ni in a positive direction along the z-axis. As a numerical experiment, we restarted run 1 without suppressing the burning of carbon-rich zones at times t = 1.2 s (when the helium detonation converges), 1.3 s, 1.4 s, and 1.5 s (when the internal wave converges). In all of these cases, a detonation wave in the core forms immediately.

The values of the density and temperature in the hot spot are dependent on the spatial resolution Δx of the grid as well as on the temporal separation Δt of the considered snapshots (for practical reasons, we do not edit and analyze the results at every time step), cf., the values of temperature and density in the hot spots of runs 1, 4, 5, and 11 listed in Table 4. However, all results unequivocally indicate that the core would detonate. We expect that a higher resolution would only increase the likelihood for detonation.

As a limiting case of multi-point ignition, we at first consider two detonators at opposite points in the helium shell (run 2). While it is unlikely for such detonators to be synchronous, this poses a computationally inexpensive 2D problem which can be regarded as a numerical experiment. The internal waves in this case collide on the z = 0 plane (the detonators being centered on the z-axis at z₈ = ±4.27), beginning at the outer edge of the core at t = 0.60 s and reaching the center at t = 1.05 s. The final wave collision in the central part of the star takes place on a ∼1400 km wide disk without generating a discernible hot spot. However, temperatures ≳ 10⁹ K occur on the collision plane at large radii (r₈ ∼ 1.8–3.2). While the material is not compressed there, we observe a radially inward moving core detonation in a run where the burning of carbon is allowed for t ⩾ 0.65 s. If carbon burning is suppressed, the original core waves are reflected at the z = 0 plane. The reflected waves collide on the z-axis (symmetrically on both sides of the z = 0 plane), creating dense hot spots at z₈ = ±1.45 that would very likely have ignited the core if ignition had failed earlier.

When one of the detonators (in our case the one on the negative side of the z-axis) is delayed by 0.30 s, the two helium detonation fronts collide at t = 0.75 s at an angle of 112° with respect to the positive z-axis; see Figure 5. As in the synchronous case, the collision of the primary internal waves, which here happens on a slightly curved disk at z ≈ −600 km, does not generate a hot spot. However, a wave reflected toward the negative z-axis produces a hot spot that is sufficiently dense to trigger core detonation.

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Before discussing 3D cases of two-point ignition in the helium, it is worth noting that a single, small aspherical detonator is virtually equivalent to a spherical one, regardless of how strongly deformed it is (as long as it sets off a detonation). As the detonation expands with constant velocity in every direction, the original asphericity is quickly lost. To assess the effects of non-axisymmetry on the focusing of the internal waves, we therefore consider cases with two detonators at varying but significant separation from one another. For two synchronous detonators, it suffices to perform a 3D simulation of a quarter of the star, with mirror symmetry at the inner boundaries.

In cases with two moderately separated, synchronous detonators (we consider angular separations of 36°, 54°, and 90°), the initially separate helium detonation fronts collide halfway between the two spots on the geodesic connecting them on the core surface, in our case the positive z-axis. While the detonations merge, the combined front proceeds toward the antipode of this collision point. The front initially has a head start on the plane that connects the two spots with the center of the star (the x = 0 plane in our simulations), but the deficit becomes smaller as the detonations continue to merge.⁴ As it approaches the antipode, the detonation front has the shape of a pointed ellipse. This shape can easily be understood considering the combined detonation as a superposition of two detonations: the pointed ellipse is the intersection between two circles (or, more precisely, lines of constant latitude, if the ignition points are regarded as poles) whose centers are the respective antipodes of the ignition points. In a simulation, the detonation finally converges when the width of the pointed ellipse approaches the size of a grid zone. Theoretically, the ratio of the length of the ellipse compared to its width increases indefinitely, i.e., the wave always converges in a line rather than a point and the difference between having one and two detonators should become larger with increasing grid resolution or diffusion length.

In cases with 36° and 54° detonator separation, the region in which the internal waves converge becomes sufficiently hot and dense to trigger a core detonation; see Table 4 and Figure 4(b). We explicitly confirmed this for run 13 by switching on carbon burning for t > 1.40 s. When two synchronous detonators are 90° apart (run 8), there is no hot spot at the site where the two primary waves converge at the z-axis (in general, the intersection of the two planes of symmetry in this problem). However, the waves induced by the two detonators pass through one another and form two hot spots at a considerable distance (660 km) from the z-axis; see Figure 4(c).

We next consider a case of two asynchronous detonators with an angular separation of 54°, run 7. Note that this scenario is less symmetric than the case of two synchronous detonators, there being only one mirror symmetry in the problem, across the plane which intersects both detonators. We therefore use a 3D domain that comprises half of the star and assume mirror symmetry at the inner boundary. One of the detonators—the one on the +y side in our simulations—is set off 0.15 s (roughly half the time needed for the first detonation to reach the second detonator) after the first one.⁵ Shortly before converging, the helium detonation front assumes the shape defined by the intersection of two circles with different radii; see Figure 6. The same shape can be seen in the internal wave front on concentric spherical surfaces. Just like in the synchronous case (run 6), a dense hot spot forms when the internal wave fronts converge; see Figure 7 and the rightmost snapshot in Figure 6.

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With 120° separation and a delay of 0.30 s, run 9, there is no hot spot where the internal waves from the two detonators converge. As in the case with synchronous detonators at 90° discussed above, the waves from the two detonators pass through one another. The wave from the first detonator is the first to converge and trigger a core detonation.

Unlike the above cases, a setup with three or more detonators is in general free of symmetries and requires a 3D simulation of the full star. In run 10, we again consider two detonators (A and B) with a separation of 120°, one of which (B) is delayed by 0.30 s, exactly as in run 9. In addition, a third detonator C, delayed by 0.20 s, is placed 90° from detonator A and 100° from detonator B on the positive x side. In this case, when the helium detonation converges toward the last patch of unburnt helium, it roughly has a triangular shape. The same shape is present in the internal wave on concentric spherical surfaces. There is a distinct hot spot when the internal waves converge. Shortly after, a secondary hot spot arises as the wave induced by the first detonator converges again with itself. Figure 8 shows snapshots of both hot spots (right-hand panels) and the converging wave fronts from which they originate (left-hand panels).

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Here we consider axisymmetric detonations (one detonator or two opposite detonators) for progenitor models other than A. A detonation in a 0.078 M_☉ helium shell on top of a solar mass core (model E, run 14) is set off by a 25 km spherical detonator centered on the hottest helium layer. The sliding helium detonation yields a distinct hot spot where the internal wave converges at t ≈ 1.40 s. With carbon burning turned on from the time when the helium detonation converges (t ≈ 1.10 s), the core immediately detonates near its surface. This detonation is triggered by a thin, hot radial inflow of ⁵⁶Ni. As shown in the upper panel of Figure 9, temperatures of 2–4 × 10⁹ K and densities on the order of 10⁷ g cm⁻³ are generated along this jet. In a simulation with two synchronous, opposite detonators, run 15, temperatures >10⁹ K are generated neither when the wave fronts converge at the center, nor along the z-axis when the reflected waves collide (in contrast with model A, run 2). The only region hot enough to light the core in this case is located at the equatorial plane near the core's surface, where the collision of the internal waves begins. Although there is no hot inflow comparable to the case with one detonator (note that a radial inflow in this case is not a jet; see the lower panel in Figure 9) the core there ignites in the simulation.

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It is difficult to set off a detonation in a 0.045 M_☉ helium shell surrounding a one solar mass core (model D). We did not succeed with a spherical detonator centered on the hottest helium layer, whose density is 7.17 × 10⁵ g cm⁻³. A detonation does not start even with a (perhaps unrealistically) strong initial detonator: a central temperature of 3 × 10⁹ K, linearly decreasing to 1.5 × 10⁹ K at the outer edge, a density three times that of the ambient medium, and a radius of 25 km (ending 1 km above the core–shell interface). Large shock velocities (we tried 1.6 × 10⁹  cm s⁻¹ instead of our usual 8 × 10⁸  cm s⁻¹) at the outer edge of the detonation spot did not help either.

A detonation in the 0.045 M_☉ shell of model D can be set off, however, by a large spherical detonator of radius 50 km (which is about twice the distance between the core–shell interface and the hottest helium layer) centered 25 km above the hottest helium layer. The resulting detonation front is feebler than in all cases with heavier helium shells, blowing out at too coarse a resolution (≲2 km works, 6.5 km does not) and requiring the use of AMR in our simulated helium shell. However, once started, the detonation propagates around the star and converges after ≈1.40 s (run 16). The corresponding front velocity is ∼9.4 × 10⁸  cm s⁻¹ ≈ 4.3 times the sound speed in the hottest helium layer. When the internal wave induced by the helium detonation converges 0.40 s later, it forms a distinct hot spot. Without suppression of the carbon burning, the core detonates immediately after the convergence of the helium detonation, starting at the z-axis near the core–shell boundary. In a simulation with two synchronous, opposite detonators, run 17, no hot spot is formed in the core. With carbon burning turned on, the core does not detonate in the simulation.