1. Type Ia Supernova Explosion Mechanisms or How to Blow Up a White Dwarf Jeffrey M. Silverman Astro 254: HEAp 5/1/2006

2. What are Type Ia Supernovae? White Dwarf (WD): an evolved star held up by electron degeneracy pressure and no longer producing energy through fusion reactions. Type Ia supernovae (SNe Ia): the result of thermonuclear disruptions of WDs. Origin of SNe Ia:  A lowmass WD, made primarily of C and O, accretes matter from a non-degenerate companion star until it reaches the Chandrasekhar mass (M Chan ≈ 1.4M  ).  The WD then becomes unstable and the entire star undergoes thermonuclear runaway on the order of 30 to 300s. Hillebrandt, W., & Niemeyer, J. C. 2000, ARA&A, 38, 191: summarizes all 20 th c. knowledge of SNe Ia and is cited by pretty much every paper on the subject.

3. Requirements for SNe Ia Explosion Models Any viable scenario for the explosion mechanism must satisfy the following (necessary but probably not sufficient) requirements: 1. Agreement of the ejecta composition and velocity with observed spectra and light curves.  Explosion must be sufficiently energetic (about 10 51 ergs).  It must produce a substantial amount of high-velocity intermediate mass elements (e.g. Mg, Ne, S).  Isotopic abundances must not deviate significantly from solar values. 2. Robustness of the explosion mechanism.  Most SNe Ia are extremely homogeneous, thus a model should not give rise to widely different outcomes depending on fine-tuning of model parameters or initial conditions.

4. Requirements for SNe Ia Explosion Models 3. Intrinsic variability.  The basic model should be robust with respect to small fluctuations, but must contain at least one parameter to account for the observed differences between SNe Ia (i.e. explosion strengths). 4. Correlation with progenitor system.  The explosion strength parameter must be logically connected to the progenitor WD in order to explain observed variations as a function of the host stellar population.

5. Equations and Simulations To very good approximation, the exploding WD material can be described as a fully ionized plasma with varying degrees of electron degeneracy. Governing equations are the hydro equations for mass, species, momentum, and energy transport including gravitational acceleration, viscosity, heat and mass diffusion, and nuclear energy generation. Must be supplemented by an equation of state for an ideal gas of nuclei, an arbitrarily relativistic and degenerate electron gas, radiation, and electron-positron pair production and annihilation. Size scales involved in simulations range from ≤1cm at the beginning of the explosion, to about the size of the WD radius (~a few thousand km) by the end of the explosion. This range of involved length scales spans about 8 orders of magnitude, thus numerical simulations are difficult.

6. More on Simulations Some 2D simulations have successfully blown up a WD, however they all output less energy than predicted. 3D models have had more success, and more energy output, but many have been restricted to one spatial octant and assume symmetry in the other octants. This approach imposes an artificial symmetry constraint on the SNe Ia. To explore asymmetries of the explosion process Röpke & Hillebrandt (2005) modeled a full star. They noted their new asymmetries alone did not introduce any additional asymmetries into the model. This result validates previous single-octant simulations since it shows that the symmetry constraint imposed there did not suppress any noticeable physical effects.

7. Hydrodynamical Effects The best studied and probably most important hydrodynamical effect is the Rayleigh-Taylor (RT) instability. RT results from the buoyancy of hot, burned fluid (intermediate mass elements and Fe-group elements) or “ash,” with respect to the dense, unburned material or “fuel” (the original C and O of the WD). Subject to the RT instability, small surface perturbations of burned ash grow until they form bubbles or mushrooms that begin to float outward while spikes of dense fuel fall inward toward the WD’s center.

8. Carbon mass fraction for a 10 7 g/cc WD shown every 2.55×10 -4 s from the beginning of the explosion. The fuel is red (50% C, 50% O), the center of the star is toward the bottom of each panel, and each panel represents an area of about 150cm by 250cm. This shows a balance between the RT growth and the burning. Spikes of fuel push into the hot ash, forming well defined mushroom caps. Before these caps can travel too far into the ash, they burn away, turning yellow/green, and finally leaving a light blue outline in the ash as the burning completes. (Bell 2004)

9. Initial Conditions Evolution of the thermonuclear runaway immediately prior to explosion is crucial for determining initial conditions. Using a simple toy model, García-Senz & Woosley (1995) found that the RT instability can often cause mushrooms of ash to rise a few hundred km with velocities of up to 100 km/s before the actual explosion begins. This suggests a high probability for off-center ignition at multiple, unconnected points. Röpke et al. (2006) demonstrated that the location and number of ignition spots have a significant influence on the final explosion energetics and nucleosynthesis. They also concluded that this “multi-spot ignition” achieves better agreement with observations. However, the initial sizes, positions, and number of ignition points are currently free parameters in most simulations.

10. Flame Ignition As the WD grows close to M Chan, the energy near the core is governed by neutrino losses (the dominant effect) and compressional heating. At a central density of ~10 9 g/cc, electron screening of nuclear reactions enhances the energy generation rate until it begins to exceed the neutrino losses. This “smoldering” of the core marks the beginning of the thermonuclear runaway. During the next ~1000 years, the core convects with progressively smaller turnover time scales,  c. Simultaneously, typical time scales for thermonuclear burning,  b, drops even faster as a result of the rising core temperature and the steep temperature dependence of the nuclear reaction rate (~T 12 ).

11. Flame Ignition (cont’d) At T≈7×10 8 K, ~1 hour before explosion,  c and  b become comparable: convective plumes burn at the same rate as they circulate. At T≈1.5×10 9 K, essentially when the explosion begins,  b becomes extremely small compared with  c : C and O virtually burn in place. A new equilibrium between energy generation and transport is found on length scales ~10 −4 cm, where thermal conduction by degenerate electrons balances nuclear energy input. This thin “reaction zone” forms at the interface between the ash and the fuel: the “flame” is born. The flame can propagate into the surrounding fuel by one of two mechanisms allowed by the Rankine- Hugoniot jump conditions: a detonation or a deflagration.

12. Evolution of the flame fronts in a model that matches observations fairly well at times t = 0 s, t = 0.6 s, and t = 2.0 s. The first two plots are ~10 7 cm on a side, while the third plot is ~10 9 cm on a side. (Röpke 2006)

13. Detonation The more powerful and less controllable of the two burning mechanisms. If the overpressure created by the flame’s heat is sufficiently high, a hydrodynamical shock wave forms that ignites the fuel by compressional heating. This self-sustaining combustion front that propagates by shock-heating is called a “detonation.” Detonations generally move supersonically and therefore do not allow the unburned medium to expand before it is burned. Their speed depends mainly on the total amount of energy released per unit mass.

14. Deflagration The less powerful and more controllable burning mechanism. If the overpressure created by the initial flame’s heat is weak, the temperature gradient at the fuel-ash interface steepens to stabilize the equilibrium between heat conduction and nuclear energy generation. The resulting combustion front, called a “deflagration,” consists of a diffusion zone to heat up the fuel, followed by a thin reaction layer to consume the fuel and generate the energy. Moves subsonically with respect to the ash and thus may be strongly affected by turbulence in the fuel. The deflagration flame speed is about 10 4 to 10 7 cm/s for   ≈10 7 to 10 9 g/cc and the deflagration flame thickness (width of the diffusion zone) is 10 −4 to 1 cm.

15. Prompt Detonation The first hydrodynamical simulation of an exploding WD was done by Arnett in 1969. He employed an explosion mechanism called “prompt detonation.” It assumed spherical symmetry (i.e., 1D) and thermonuclear combustion commenced as a detonation wave, consuming the entire star at the speed of sound. Given no time to expand prior to being burned, fuel is transformed almost completely into iron-peaked nuclei. This fails to produce significant amounts of intermediate mass elements, in contradiction to SNe Ia observations (Filippenko 1997). It is for this reason that prompt detonations are generally ruled out as viable candidates for the SN Ia explosion mechanism.

16. Pure Turbulent Deflagration Flames in a WD can have Reynolds numbers as high as 10 14, thus one expects strong turbulence to develop. The turbulence will interact with the flame, wrinkling and stretching its surface. Since the fuel consumption rate of deflagration flames is determined by the flame surface area, turbulence will significantly accelerate the flame propagation. If the flame is accelerated but remains a deflagration, then we have “pure turbulent deflagration” of the WD. Many studies agree that very good agreement with observations is obtained if the turbulent flame speed is accelerated up to roughly 30% of the sound speed. However, it is also possible for the deflagration to either transition into a detonation or become quenched by expansion of the WD’s outer layers.

17. Delayed Detonation Turbulent deflagrations can sometimes undergo spontaneous transitions to detonations (“deflagration- detonation transitions” or DDTs). These “delayed detonations” of a WD provide an explanation for the initial slow burning required to pre- expand the star, followed by a fast combustion to produce high-velocity intermediate mass elements. Many 1D simulations have shown this scenario to accurately reproduce SN Ia spectra and light curves, as well as nucleosynthesis products consistent with solar abundances. Most models predict the transition from deflagration to detonation at a density of   ≈10 7 g/cc. This transition density is a convenient parameter to explain the observed range of explosion strengths.

18. Delayed Detonation (cont’d) The actual mechanism to trigger this transition has been in question for many years. Some speculate that DDTs rely on rare, strong turbulent fluctuations, larger than what is observed in most simulations. Thus, there may be events that fail to ignite a detonation following the slow deflagration phase which, on its own, cannot give rise to a viable SN Ia explosion. Therefore this scenario could end up as an unobservably dim, as yet unclassified explosion or possibly a “pulsational delayed detonation.”

19. Pulsational Delayed Detonation If the initial deflagration of a WD is weak, there are still a few possible scenarios that can blow the thing up. “Pulsed detonations of the first type”  The initial turbulent deflagration fails to release enough energy to unbind the WD.  The star then pulses and triggers a detonation upon recollapse. “Pulsed deflagrations”  Re-ignition occurs as a second deflagration rather than a detonation.  May or may not produce a healthy explosion, depending on the speed of the rekindled flame. “Pulsed detonations of the second type”  The burning again re-ignites as a second deflagration but this time it is able to transition to a detonation.  This model closely resembles the standard delayed detonation scenario. Note that these pulsational models are all in conflict with multidimensional simulations that predict an actual explosion after the first deflagration phase.

20. New Explosion Paradigms Bravo (2005) pointed out that the most successful 3D models start from a large number of ignition points near the center of the WD. However, it is not clear how many hot spots can actually exist at the beginning of a SN Ia explosion. This uncertainty has led to two relatively new explosion models that are based on a small number of initial ignition points:  Pulsating Reverse Detonation (PRD) and  Gravitationally Confined Detonation (GCD)

21. Pulsating Reverse Detonation (PRD) Some simulations have shown that deflagration is insufficient to unbind a WD. However, since hot bubbles of ash can float to large radii in 3D models, most thermal and kinetic energy resides in the outer parts of the star while the C+O (fuel) core is stabilized. In PRD simulations (García-Senz & Bravo 2005), an accretion shock forms at the outer edge of the nearly hydrostatic core as hot ash cools and falls inward. The temperature at the core’s surface can then increase above 2×10 9 K, giving rise to a highly explosive scenario. If a detonation were ignited at this point, it would probably propagate all the way through the core, burning most of it and producing a healthy explosion with a calculated kinetic energy of about 0.9×10 51 erg.

22. Gravitationally Confined Detonation (GCD) Similar to PRD, however the explosion starts with a single hot bubble of ash moving toward the WD’s surface (Plewa et al. 2004). The bubble then breaks through the surface of the WD and ash is spread around the surface where it experiences a strong lateral acceleration while remaining gravitationally bound to the star. Finally, the material focuses at the pole opposite to the point of breakout, providing a high compression and attaining temperatures >2.2×10 9 K. A detonation will probably form at the point of maximum temperature and propagate through the entire WD, burning it to Fe-group and intermediate mass elements. Note that these results are from a 2D calculation and that it has yet to be confirmed by full 3D simulations.

23. (Plewa et al. 2004)

24. Summary SNe Ia explosion mechanisms are very complex and involve understanding the physics leading up to the explosion as well as the processes which couple the interior physics to observable quantities. “In conclusion, we seem to be lucky and Nature was kind to us and singled out from all possibilities the simplest solution, namely a Chandrasekhar-mass C+O white dwarf and a nuclear deflagration wave, to make a Type Ia supernova explosion.” –Hillebrandt & Niemeyer (2000)

25. Summary (cont’d) “Since Hoyle and Fowler proposed the white dwarf scenario for Type Ia SNe, the ideas about the way the star explodes have evolved. The 70’s were the epoch of pure detonations. The 80’s witnessed a flourishing of the deflagrations. The 90’s knew about delayed detonations in its various flavors. Nowadays, at the beginning of the 21st century, the future of SN Ia modeling resides most probably on new paradigms, like Pulsating Reverse Detonation and Gravitationally Confined Detonation.” –Bravo et al. (2005) Results from “the deflagration scenario of SNe Ia in its current implementation (i.e. applying a multi-spot ignition with a Gaussian radial distribution within about180km around the center of the WD)…fall in the range of observational expectations.” –Röpke et al. (2006)

26. References Arnett, W. D. 1969, Ap&SS, 5, 180 Bell, J. B., Day, M. S., Rendleman, C. A., Woosley, S. E., & Zingale, M. 2004, ApJ, 608, 883 Bravo, E., Badenes, C., García-Senz, D. 2005, in AIP Conf. Proc. 797, Interacting Binaries: Accretion, Evolution, and Outcomes, 453 Filippenko, A. V. 1997, ARA&A, 35, 309 García-Senz, D., & Bravo, E. 2005, A&A, 430, 585 García-Senz, D., & Woosley, S. E. 1995, ApJ, 454, 895 Hillebrandt, W., & Niemeyer, J. C. 2000, ARA&A, 38, 191 Plewa, T., Calder, A., & Lamb, D. 2004, ApJ, 612, L37 Röpke, F. K., & Hillebrandt, W. 2005, A&A, 431, 635 Röpke, F. K., Hillebrandt, W., Niemeyer, J. C., & Woosley, S. E. 2006, A&A, 448, 1

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