A&A 465, 263-269 (2007)
DOI: 10.1051/0004-6361:20066510
W. Schmidt
Lehrstuhl für Astronomie, Institut für Theoretische Physik und Astrophysik, Universität Würzburg, Am Hubland, 97074 Würzburg, Germany
Received 5 October 2006 / Accepted 5 January 2007
Abstract
In thermonuclear supernovae, intermediate mass elements
are mostly produced by distributed burning provided that a
deflagration to detonation transition does not set in. Apart from
the two-dimensional study by Röpke & Hillebrandt (2005, A&A, 429, L29), very little
attention has been payed so far to the correct treatment of this
burning regime in numerical simulations. In this article, the
physics of distributed burning is reviewed from the literature on
terrestrial combustion and differences which arise from the very
small Prandtl numbers encountered in degenerate matter are pointed
out. Then it is shown that the level set method continues to be
applicable beyond the flamelet regime as long as the width of the
flame brush does not become smaller than the numerical cutoff
length. Implementing this constraint with a simple parameterisation
of the effect of turbulence onto the energy generation rate, the
production of intermediate mass elements increases substantially
compared to previous simulations, in which the burning process was
stopped once the mass density dropped below
.
Although these results depend on the
chosen numerical resolution, an improvement of the constraints on the
the total mass of burning products in the pure deflagration scenario
can be achieved.
Key words: stars: supernovae: general - hydrodynamics - turbulence - methods: numerical
In the course of the last few years, observational indications in favour of a delayed detonation in type Ia supernovae (SNe Ia) have mounted. For example, calculations of the X-ray spectrum of the Tycho supernova remnant assuming various hydrodynamical models appear to support a deflagration to detonation transition (DDT) (Badenes et al. 2006). Furthermore, an investigation of near-infrared emission lines of three branch-normal supernovae by Marion et al. (2006) implies very little carbon residuals at radial velocities less than . Even the most advanced three-dimensional simulations of thermonuclear supernovae assuming pure deflagrations (Schmidt & Niemeyer 2006; Röpke et al. 2006) fail to satisfy this constraint. In models with delayed detonations (Gamezo et al. 2005), on the other hand, the supersonic propagation of burning fronts dispose of virtually all carbon except for the outermost layers. Furthermore, a gravitational confined detonation was suggested as an alternative scenario (Plewa et al. 2004).
However, a recent numerical study by Maier & Niemeyer (2006) demonstrated that detonation waves fail to penetrate processed material stemming from the initial deflagration phase. Therefore, pockets of unburned material are likely to survive even a delayed detonation. Apart from that, Niemeyer (1999) pointed out several theoretical arguments against DDTs. In addition, accommodating the observed variability of SNe Ia (Stritzinger et al. 2006) within the DDT scenario appears to be difficult, because delayed detonations tend to produce at least a solar mass of iron group elements and explosion energies in excess of , which are typical values characterising bright SNe Ia. But there are also SNe Ia of moderate or low luminosity producing much smaller masses of iron-group elements and less explosion energy. Apart from that, Jha et al. (2006) inferred non-negligible amounts of carbon at low expansion velocities from the late-time spectroscopy of the SN 2002cx, a peculiar supernova with very low luminosity. If this result was confirmed by further observations, one might conceive of a sub-class of SNe Ia originating either from Chandrasekhar mass white dwarfs which are only partially burned by pure deflagrations or from sub-Chandrasekhar mass progenitors. In this article, we shall be concerned exclusively with the Chandrasekhar mass scenario.
Schmidt & Niemeyer (2006) showed that the explosion energy and mass of iron group elements in thermonuclear supernova simulations with pure deflagration can be varied over about an order of magnitude if non-simultaneous point ignitions are applied. To that end, the simple MLT model of the pre-supernova core proposed by Wunsch & Woosley (2004) was adopted for the implementation of a stochastic ignition procedure. In a numerical case study, models with a total number of ignitions ranging from a few up to several hundred events per octant were investigated. The main result is that the total mass of iron group elements can be adjusted to any value smaller than with a maximal explosion energy of roughly . Therefore, these models are feasible candidates for less energetic SNe Ia.
Due to the artificially chosen termination of thermonuclear burning at mass densities below , however, the prediction of the carbon and oxygen residuals in the models with stochastic ignition by Schmidt & Niemeyer (2006) was not reliable. The transition from the flamelet regime to the regime of distributed burning is expected to occur at a mass density of about (Niemeyer & Kerstein 1997; Niemeyer & Woosley 1997). Since the level set method - as implemented by Reinecke et al. (1999b) - was applied for the numerical flame front propagation, the treatment of the distributed burning regime remained unclear. As a first approximation, distributed burning was mostly suppressed by introducing the aforementioned density threshold. This resulted in an overestimate of the amount of unburned material. Apart from that, a delayed detonation might be triggered in the late burning phase (Khokhlov et al. 1997; Niemeyer & Woosley 1997). However, the arguments put forward by Niemeyer (1999) and the numerical results by Bell et al. (2004) shed serious doubt on this proposition. For this reason, it appears even more important to consider the possibility of the subsonic distributed burning mode at lower densities.
A first attempt to include distributed burning in deflagration models of SNe Ia was made by Röpke & Hillebrandt (2005). They carried out two-dimensional simulations demonstrating that the explosion energy increases and the fraction of carbon and oxygen at low radial velocities is reduced. However, their prescription of the turbulent burning speed in the distributed regime suffered from a misconception of the relevant scales. In this article, we will argue that the existing treatment of burning fronts with the level set method can be carried over to the distributed regime provided that the burning time scale is smaller than the eddy turn-over time scale associated with the numerical cutoff length. Consequently, this constraint has to be observed in numerical simulations of thermonuclear supernovae with the level set method.
After reviewing the physics of distributed burning in Sect. 2, we will show in Sect. 3 that the turbulent burning speed continues to be determined by the magnitude of turbulent velocity fluctuations at unresolved scales after the onset of distributed burning. In Sect. 4, we will discuss results from numerical simulations of thermonuclear supernovae with stochastic ignition, where the burning process was terminated based on a comparison of the burning and unresolved eddy turn-over time scales. Thereby, it was possible to increase the production of intermediate mass elements significantly compared to previous simulations with a density threshold of for thermonuclear burning. In particular, we found that the total mass of intermediate mass elements appears to be independent of the details of the ignition process, although slower ignition implies a larger amount of fuel at the transition from the flamelet to the distributed burning regime.
In combustion physics, two different regimes of deflagration are
distinguished: on the one hand, the flamelet regime, in which
the microscopic flame propagation speed is solely determined by the
thermal conductivity of the fuel. In the distributed burning
regime, on the other hand, the transport of heat and mass is
influenced by turbulence even at scales comparable to the width of
the flame brush. As argued by Niemeyer & Kerstein (1997), the correct
criterion for the transition from burning in the flamelet regime to
distributed burning is that the flame width
is about the
Gibson length
.
The Gibson length is implicitly
defined by the equality of the velocity
associated with
turbulent eddies of size
and the laminar flame speed
(Peters 1988):
The conductivity
can be related to the viscosity
by
,
where the Prandtl number is a
dimensionless characteristic number of the fuel. In stark contrast
to most terrestrial fluids which have a Prandtl number of the
roughly unity,
for degenerate carbon and oxygen
in white dwarfs (Nandkumar & Pethick 1984). Introducing the turbulent
viscosity
,
the turbulent
diffusivity can be expressed as
,
where
accounts for the effect of turbulent eddies in
the range of length scales from the Kolmogorov scale
to the
.
Thus,
Eq. (5) can be written in the form
Assuming approximate local equilibrium of the energy flux through
the cascade of turbulent eddies from size
down to
the Kolmogorov scale, we have
.
Equation (6) for the enhanced flame
propagation speed then implies
In terms of time scales, the thin-reaction-zones regime is
characterised by
(10) |
Reinecke et al. (1999a) introduced the level set method for the
modelling of burning fronts in thermonuclear supernova simulations.
The basic idea of this approach is to represent the flames by the
zero level set of a distance function that is determined by a
partial differential equation. The intrinsic propagation speed in
the flamelet regime is the laminar burning speed
.
Because simulations of SNe Ia are essentially large eddy
simulations, however, the effective propagation speed is
asymptotically proportional to the magnitude of unresolved turbulent
velocity fluctuations, i.e.
,
where
is the
resolution of the numerical grid (Niemeyer & Hillebrandt 1995). For this
reason, the effective propagation speed of the numerically computed
flame fronts is called the turbulent flame speed and denoted by
.
For the calculation of
,
we have
adopted Pocheau's model (Pocheau 1994; Schmidt et al. 2006b):
According to Kim & Menon (2000a), the above model for the turbulent
flame speed can be readily extended into the thin-reaction-zones
regime:
(13) |
(14) |
As the exploding white dwarf expands further and the density drops
below
,
the nuclear time scale rises
rapidly and, inevitably, the
condition (11) for the
broken-reaction-zone regimes will be met. The broadened flame will
then dissolve into a flame brush in which fuel and ash are mixed by
turbulent eddies. Although there is no well defined width
of the flame brush, we can specify a turbulent
mixing length
which corresponds to the
typical size of eddies with turn-over time of the order of the
effective reaction time
:
(15) |
Since the width of the flame in the thin-reaction-zones regime is
very small compared to typical grid resolutions in simulations of
thermonuclear supernovae, the flame brush in the
broken-reaction-zones regime will initially not be resolved either,
i.e.
.
In terms of time scales,
.
As long as this constraint is satisfied, it appears to be a sensible
approximation to propagate the unresolved turbulent flame brush with
the level set method. For
The extension of the level set method into the distributed burning regime as outlined in this Section markedly differs from the flame speed model applied by Röpke & Hillebrandt (2005), which is based on Damköhler's hypothesis (5) and, consequently, is restricted to the thin-reaction-zones regime. Essentially, their model is based on the assumption or, equivalently, . It should become clear from the above discussion, however, that this assumption does not hold.
We investigated the effect of including burning with the extended
level set prescription in a follow-up study of Schmidt & Niemeyer (2006). To
that end, we modified the criterion for the termination of the
burning process in the code as follows. From the values of
and
calculated by Timmes & Woosley (1992), we
obtained
and linearly
fitted
as a function of
,
where
is the mass density in units of
,
for
.
The power law
resulting from this fit is
To begin with, we computed a series of single-octant simulations on grids consisting of N=128^{3} cells with the code described in Schmidt et al. (2006b). Due to the hybrid geometry of these grids, with a fine-resolved uniform inner part and exponentially growing cell size in the outer part, in combination with the co-expanding grid technique introduced by (Röpke 2005), it was possible to study trends even with a relatively small number of grid cells. If we had chosen a higher resolution, the applicability of the level set method would have been constrained even further. This can be seen from the criterion (16) with the scaling law . Since the nuclear time scale gradually increases as the explosion progresses, lowering the cutoff length implies that the termination point will be met earlier. On the other hand, a certain resolution is mandatory for capturing the large-scale production of turbulence by Rayleigh-Taylor instabilities. From the resolution study in Schmidt et al. (2006b) and the influence of doubling the resolution of thermonuclear supernova simulations with stochastic ignition discussed by Schmidt & Niemeyer (2006), we conclude that 128^{3} cells per octant are sufficient for arriving at a sensible estimate of the total amount of burning products.
Table 1: Power-law fit of the nuclear energy timescale in the range . The numerical data are taken from Timmes & Woosley (1992).
Figure 1: Single-octant simulations with N=128^{3}, and various choices of . Plotted are the evolution of the total energy and the integrated mass of intermediate mass elements, respectively. Also shown is a reference simulation, in which the burning process was stopped at the density threshold . | |
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Applying the stochastic ignition procedure formulated by Schmidt & Niemeyer (2006), we selected a reference model with , where the parameter adjusts the time scale of Poisson processes generating the ignition events in thin adjacent spherical shells. Depending on the choice of , the outcome of the explosion, in particular, the release of nuclear energy and the total mass of iron group elements is varying greatly. For , the total number of ignition events per octant is roughly 100 and the yield of nuclear energy becomes nearly maximal (see Table 1 and Fig. 1 in Schmidt & Niemeyer 2006). We repeated this simulation with the same initial white dwarf model and parameter settings, except for the termination of the burning process as explained above, using three different values of .
Table 2: Total energy release and masses of burning products at for the simulations shown in Fig. 1.
Figure 2: Evolution of the total energy and the integrated mass of intermediate mass elements, respectively, for several full-star simulations with N=256^{3}. For each value of the exponentiation parameter , which controls the number of ignitions events, either or (corresponding to ) was set for the termination of the burning process. | |
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The time evolution of the total amount of intermediate mass elements (represented by ^{24}Mg) plotted in the right panel of Fig. 1 clearly shows the influence of the burning termination criterion. Assuming , the total magnesium mass at increases by compared to the reference simulation (see Table 2, in which burning ceases for . One should note that this is only a lower bound, because indicates that the broken-reaction-zones regime has just been entered and volume burning at resolved scales, which cannot be treated with the present code, would consume even more carbon and oxygen. The mass of iron group elements (^{54}Ni), on the other hand, remains unaffected which, of course, reflects nickel being produced entirely within the flamelet regime at densities while magnesium is stemming largely from distributed burning at lower densities. Even for , meaning that the extended level set prescription breaks down well within the broken-reaction-zones regime, the yield of intermediate mass elements is still higher than in the case of the density threshold . If , on the other hand, becomes slightly smaller. Since implies a greatly reduced energy generation rate, the physical quenching of the flame brush would be nearly reached in this case and, as a consequence, only little volume burning could possibly follow. In this case, we can be fairly sure that the actual amount of burning products would be more or less what is seen in the simulation.
We also performed full-star simulations with N=256^{3}, for which is initially the same as in the single-octant simulations. Varying the exponentiation parameter , the burning process was terminated once , i.e. . The results in comparison to the corresponding simulations with the termination criterion from Schmidt & Niemeyer (2006) are plotted in Fig. 2. Because of the substantially larger mass of unburned carbon and oxygen that will be left over at the transition from iron group to intermediate mass element production if is smaller and, consequently, the ignition proceeds slower, one would expect to increase with . However, the burning statistics listed in Table 3 clearly demonstrates that is almost constant for varying . Hence, distributed burning appears to consume about the same amount of fuel independent of the state of the exploding star at the end of the flamelet regime.
Table 3: Total energy release and masses of burning products at for the simulations shown in Fig. 2.
Figure 3: Total and fractional mass densities in a central section and the corresponding probability density functions in radial velocity space for a full-star simulation with at . | |
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Figure 4: Total and fractional mass densities in a central section and the corresponding probability density functions in radial velocity space for a full-star simulation with at . | |
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Figure 5: Total and fractional mass densities in a central section and the corresponding probability density functions in radial velocity space for a full-star simulation with at . | |
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Figures 3-5 show contour plots of the total mass density and the partial densities of C+O, Mg and Ni, respectively, in two-dimensional central sections at time . In comparison to Figs. 2-4 in (Schmidt & Niemeyer 2006), which show the corresponding density contours in the simulations with the burning process terminated if , one can see that the explosion ejecta are substantially enriched in intermediate mass elements. This is also illustrated by the plots of the fractional masses as function of radial velocity in Figs. 3-5. Although the residuals of unburned material at low radial velocities, which have been plaguing the deflagration models, are still present in each case, it is likely the a significant further reduction would result from volume burning beyond the break-down of the level set description. Nevertheless, a non-negligible amount of carbon and oxygen at radial velocities larger than about will be left over for any conceivable choice of parameters.
We have argued that the turbulent flame propagation speed in thermonuclear supernova simulations is given by the subgrid scale turbulent velocity even in the distributed burning regime. Moreover, the level set representation of the flame front remains valid beyond the flamelet regime provided that the width of the flame brush is smaller than grid resolution. In terms of time scales, this constraint corresponds the termination criterion (16).
Very little is known about the interaction between turbulence and the burning process in the broken-reaction-zones regime, in which turbulence is mixing fuel and ash faster than the nuclear reactions are progressing. Since the break-down of the level set description possibly occurs in this regime, we introduced the dimensionless parameter which specifies the reduction of the energy generation rate due to turbulent entrainment of fuel and ash. The limiting case corresponds to the thin-reaction-zones regime. In this regime, the flame is broadened due to the enhanced diffusivity in the preheating zone while the reaction zone is not significantly affected by turbulent eddies. As turbulence increasingly disturbs the reaction zone, diminishes. For , burning will be quenched.
Setting to values in the range from 0.01 to 1, we performed several supernova simulations with burning being terminated once (16) was fulfilled. In these simulations, we applied the stochastic ignition procedure described in (Schmidt & Niemeyer 2006). For , a greater fraction of intermediate mass elements was produced as in reference simulations with termination of the burning process at the density threshold . Remarkably, we found that of intermediate mass elements was produced in the case independent of the rapidity of the ignition process. Consequently, the ignition process mainly determines the total mass of iron group elements, whereas the production of intermediate mass elements appears to depend solely on the progression of distributed burning.
Although a substantial increase of the amount of iron group elements might result from volume burning at resolved scales, which cannot be treated within the present methodology, it seems unlikely that the remaining fraction of carbon and oxygen would be consumed completely. For this reason, if clear indications of carbon residuals at least in certain type Ia supernovae were not found, delayed detonations would be an unavoidable conclusion. Even in this case, however, studying the physics of distributed burning in more detail might very well help to clarify the physical mechanism of the putative DDT. On the other hand, if burning was completed in the distributed mode in some SNe Ia, small-scale numerical models of burning in the broken-reaction-zones regime and explicit treatment of volume burning in large-scale supernova simulations would be the prerequisites for further progress.
Acknowledgements
I thank Jens C. Niemeyer, Friedrich Röpke and Wolfgang Hillebrandt for inspiring discussions at Schloss Ringberg. The simulations were run on the HLRB of the Leibniz Computing Centre in Munich. This work was supported by the Alfried Krupp Prize for Young University Teachers of the Alfried Krupp von Bohlen und Halbach Foundation.