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Local Ignition in Carbon+Oxygen White Dwarfs

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The details of ignition of Type Ia supernovae remain fuzzy, despite the importance of this input for any large-scale model of the final explosion. Here, we begin a process of understanding the ignition of these hotspots by examining the burning of one zone of material, and then investigate the ignition of a detonation due to rapid heating at single point.

We numerically measure the ignition delay time for onset of burning in mixtures of degenerate material and provide fitting formula for conditions of relevance in the Type Ia problem. Using the neon abundance as a proxy for the white dwarf progenitors metallicity, we then find that ignition times can decrease by ~ 20% with addition of even 5% of neon by mass. When temperature fluctuations that successfully kindle a region are very rare, such a reduction in ignition time can increase the probability of ignition by orders of magnitude. If the neon comes largely at the expense of carbon, a similar decrease in the ignition time can occur.

We then consider the ignition of a detonation by an explosive energy input in one localized zone, e.g. a Sedov blast wave leading to a shock-ignited detonation. Building on previous work on curved detonations, we confirm that surprisingly large inputs of energy are required to successfully launch a detonation, leading to required matchheads of ~ 4500 detonation thicknesses - tens of centimeters to hundreds of meters - which is orders of magnitude larger than naive considerations might suggest. This is a very difficult constraint to meet for some pictures of a deflagration-to- detonation transition, such as a Zel'dovich gradient mechanism ignition in the distributed burning regime.

Fig. 1 - Temperature evolution for burning a zone at constant pressure with an initial state of X(12C)=1.0, T=1e9 K, ρ=5e8 g/cc. Because of the strong temperature dependence, a runaway takes place and most of the burning happens `all at once'.

Fig 2 - Contour plot of ignition time as a function of initial density and temperature for a constant-pressure ignition of a mixture of half-carbon, half-oxygen by mass.

Fig. 3 - Fit results vs. calculated results for constant-pressure (left) and constant-volume (right) ignition delay times, for the full range of densities (0.1 ≤ ρ₈ ≤ 50), temperatures (0.5 ≤ T₉ ≤ 7), and carbon mass abundances (0.4 ≤ X(12C) ≤ 1.0) considered.

Fig. 4 - Woosley et al. 2004 ignition time results vs. calculated results for constant-pressure ignition delay times for a mixture half-carbon half-oxygen by mass (X(12C) = 0.5), over a truncated range (1 ≤ ρ₈ ≤ 50), (0.5 ≤ T₉ ≤ 1.5).

Fig. 5 - The difference in ignition time for a constant-pressure ignition of X(12C) =0.5, X(O16)=0.5 when some of the Oxygen is replaced by Neon-20. On the top left is shown the fractional difference in ignition time with the addition of X(20Ne)= 0.05 as a function of the ignition time with X(20Ne)= 0.0. On the top right, bottom left, and bottom right are contour plots in ρ-T space of the percent difference in ignition time with X(20Ne)=0.05, 0.1, 0.2$, respectively. The ignition time for the base case is shown in Fig. 2.

Fig. 6 - Mass fraction evolution in constant-density burning, with an initial state of ρ₈=10, T₉=1, X(12C)=0.5, with X(20Ne)=0.0 (left) and X(20Ne)= 0.1 (right). The burning here was calculated with a 513-isotope network. Because of the removal of the α-chain bottleneck at neon on the right, burning proceeds faster and the generation of higher intermediate-mass elements is raised by orders of magnitude at early times.

Fig. 7 - As in Fig. 5, the difference in ignition time for a constant-pressure ignition of X(12C)=0.5$, X(16O)=0.5 when (left) a mass fraction of 0.005 of each of the carbon and oxygen is replaced by Neon-20 and (right) when 0.05 of each is replaced by Neon. Note that in this case, for X(20Ne)=0.1, with the exception of a small region in ρ,T) the ignition time is increased by the same magnitude that it is decreased in the case when all of the neon comes from oxygen, e.g. the X(12C)=0.5,X(16O)=0.4,X(20Ne)=0.1$ case of Fig. 5 in the bottom right panel.

Fig. 8 - Two examples of estimating the detonation thickness, l_i = v_s τ_i in a detonation. On the left, plotted energy release rate from nuclear reactions behind the shock of a leftward-traveling ZND detonation into a pure-carbon quiescent medium of ρ₈= 1, T₉=0.05. The shocked state is ρ₈=2.9, T₉=4.2, and the incoming fuel velocity behind the shock is 4.0e8 cm/s. For the shocked material, the predicted ignition time is ~ 3e-11 s. Even in this case, where the shocked temperature is so high that significant burning occurs immediately, and the `square wave' detonation structure does not apply, the predicted l_i=1.2e-2 cm correctly matches the peak of the reaction zone. On the right, pressure (top) and energy release rate (bottom), plotted relative to their maximum values, behind a leftward-traveling slightly overdriven detonation into the same material as in the previous figure, calculated by the hydrodynamics code FLASH. The line above the plotted quantities shows the predicted l_i calculated with the observed values in the shocked state.

Fig. 9 - Example of detonation speed vs. curvature, for a detonation into a quiescent medium of ρ₈=1, T₉=0.05, X(12C)=0.5, X(O16)=0.5.

Fig. 10 - Steady-state detonation velocity as a function of curvature for background densities of, from left to right and top to bottom, ρ₈=(0.5, 1, 2, 5, 10, 20). In each plot, the velocities are shown for initial carbon abundances, top to bottom, of X(12C)=(1, 0.75, 0.5, 0.25)$. So that they could be shown on the same horizontal scale, the curvatures have been scaled to the maximum sustainable curvature for each set of conditions, given by Table 2. Points represent calculated speeds, and solid lines are fits with parameters also given in Table 2.

Fig. 11 - Fit of maximum curvature for a sustainable steady-state detonation; the fit is within 25% for ρ₈ ≥1, and within a factor of 2.5 for the entire range.

Fig 12 - Shock velocity vs. shock position for a Sedov blast wave with burning, propagating through constant density material. The velocity is measured in units of the steady planar detonation velocity, D=1.218e9 cm/s. Input energies are, left to right, 1e27, 1e28, 1e29, 1e30, 1e31, 1e32$ ergs. Note that when the shock velocity approaches the detonation velocity from above, burning begins to effect the shock speed -- with transients robustly occurring when the shock speed falls to about twice the planar detonation velocity -- but the shock fails to sustain a detonation for the case of input energies of less than 1e31 ergs. For the two largest energies, a detonation does successfully propagate for many detonation thicknesses -- which, for comparison purposes, is about 1e-2 cm -- but is disrupted by instabilities. The inset shows a closeup of the highest-energy `detonation' becoming unstable, although it propagates ~ 1e3 l_i.

Fig. 13 - Minimum scales for detonation. Plotted is κ_max^-1 from Table 2, with dashed lines and points, and r_b from Eq. 11 with solid lines. An expending shock must slow to approximately the steady detonation velocity at a radius no less κ_max^-1, and r_b is the radius in which is contained enough potential nuclear energy from carbon burning to produce the energy for a `Sedov' explosion which would would naturally slow down to D at a radius greater than κ_max^-1. In both cases, lines are plotted for, top to bottom, X(12C)= 1/8, 1/4, 1/2, 3/4, 1$. For the cases considered here, r_b > κ_max^-1.

Fig 14 - Plots showing enhancement of probability of ignition for a 30% reduction in ignition time for a fiducial case (T₉=2, ρ₈ = 10, X(12C)=0.5, and requiring ignition within 0.1 ns). On the left is shown the base-10 logarithm of the probability of ignition as a function of the RMS temperature fluctuation (δ T_RMS/T) with and without the 30% reduction in ignition time -- e.g., log₁₀ P(τ_i,cp < 1e-10 sec) and log₁₀ P(0.7 τ_i,cp < 10^{-10} sec) -- for the constant pressure ignition time fit given in Eq. 2 and assuming Gaussian distribution of temperature fluctuations. On the right is shown the base-10 logarithm of the probability enhancement of ignition with the 30% reduction in ignition time, e.g. log₁₀ [P(0.7 τ_i,cp < 1e-10 sec) / P(τ_i,cp < 1e-10 sec) ]. When ignition is very rare (such as δ T_RMS < 0.1 T for this case), the increase in probability of ignition with a 30% ignition temperature decrease can be substantial (2--7 orders of magnitude).