supernova explosion models: predictions versus observations

New Astronomy Reviews 48 (2004) 615–621

www.elsevier.com/locate/newastrev

Supernova explosion models: predictions versus observations

W. Hillebrandt

Max-Planck-Institut f€ur Astrophysik, Karl-Schwarzschild Strasse 1, Garching D-85741, Germany

Abstract

Recent progress in modeling type Ia supernovae by means of 3-dimensional hydrodynamic simulations as well as

several of the still open questions are addressed in this review. It will be shown that the new models have considerable

predictive power which allows us to study observable properties such as light curves and spectra without adjustable

non-physical parameters. Finally, first results obtained by the European Supernova Collaboration (ESC) for a sample

of nearby SNe Ia and their implications for constraining the models and systematic differences between them are also

briefly discussed.

� 2003 Elsevier B.V. All rights reserved.

PACS: 97.60.Bw; 98.80.Es; 26.50.+x

Keywords: Supernovae; Observational cosmology; Nuclear physics aspects of supernovae

1. Introduction

The most popular progenitor model for the

average type Ia supernova is a massive white

dwarf, consisting of carbon and oxygen, which

approaches the Chandrasekhar massMchan by a yet

unknown mechanism, presumably accretion from

a companion star, and is disrupted by a thermo-

nuclear explosion (see, e.g., Hillebrandt and Nie-meyer (2000) for a recent review). The general

picture is that first carbon burns rather quietly in

the core of the contracting white dwarf. Because

this core is convectively unstable temperature

fluctuations will be present and they may locally

E-mail address: [email protected] (W. Hille-

brandt).

1387-6473/$ - see front matter � 2003 Elsevier B.V. All rights reserv

doi:10.1016/j.newar.2003.12.032

reach run-away values. In this generally acceptedscenario explosive carbon burning is ignited either

at the center of the star or off-center in a couple of

ignition spots, depending on the details of the

previous evolution. After ignition, the flame is

thought to propagate through the star as a sub-

sonic deflagration wave which may or may not

change into a detonation at low densities (around

107 g/cm3), disrupting the star in the end.Numerical simulations of any kind of turbulent

combustion have always been a challenge, mainly

because of the large range of length scales in-

volved. In type Ia supernovae, in particular, the

length scales of relevant physical processes range

from 10�3 cm for the Kolmogorov-scale to several

107 cm for typical convective motions. As was al-

ready mentioned, in the currently favored scenariothe explosion starts as a deflagration near the

ed.

mail to: [email protected]

616 W. Hillebrandt / New Astronomy Reviews 48 (2004) 615–621

center of the star. Rayleigh–Taylor unstable blobs

of hot burned material are thought to rise and to

lead to shear-induced turbulence at their interface

with the unburned gas. This turbulence increases

the effective surface area of the flamelets and,

thereby, the rate of fuel consumption; the hope isthat finally a relatively fast deflagration might re-

sult, in agreement with phenomenological models

of type Ia explosions investigated in the past

(Nomoto et al., 1984; Hillebrandt and Niemeyer,

2000).

Despite considerable progress in the field of

modeling turbulent combustion for astrophysical

flows (see, e.g., Hillebrandt and Niemeyer, 2000),the correct numerical representation of the ther-

monuclear deflagration front is still a weakness of

the simulations. Therefore, in the following section

we present the main ideas of a new approach that

is based on a level-set prescription of nuclear

flames together with a sub-grid model for turbu-

lent flame velocities. Once this flame model has

been fixed numerical simulations of the thermo-nuclear explosion of a given white dwarf can be

done by just choosing ignition conditions, the only

remaining parameters. Moreover, we will present

results of 3D simulations which demonstrate that

these ideas, in principle, do work if one performs

calculations with sufficiently high spatial resolu-

tion. It is very encouraging that all our models lead

to explosions, with explosion energies in the rangeof those observed for type Ia supernovae, and that

these models predict light curves which fit the

observations. In addition, also their nucleosyn-

thesis products are in reasonable agreement with

expectations. In the last section we will present

first results of a new European initiative to study

the physics of type Ia supernovae by means of

both, systematic and targeted observations and byincreasing the quality and the predictive power of

theoretical models at the same time.

2. Turbulent thermonuclear burning in degenerate

C+O matter

Due to the strong temperature dependence of theC-fusion reaction rates nuclear burning during the

explosion is confined to microscopically thin layers

that propagate either conductively as subsonic de-

flagrations (‘‘flames’’) or by shock compression as

supersonic detonations. Both modes are hydrody-

namically unstable to spatial perturbations as can

be shown by linear perturbation analysis. In the

nonlinear regime, the burning fronts are eitherstabilized by forming a cellular structure or become

fully turbulent, and the total burning rate increases

as a result of flame surface growth. Neither flames

nor detonations can be resolved in explosion sim-

ulations on stellar scales and therefore have to be

represented by numerical models. From basic

principles one cannot rule out pure detonations as

cause of type Ia supernovae. However, such ex-plosions would incinerate the entire white dwarf

into Fe-group elements, in contradiction with the

observations of intermediate-mass elements in the

spectra. Therefore thermonuclear burning should

at least start as a deflagration front.

The best studied and probably most important

hydrodynamical effect for modeling SN Ia explo-

sions is the Rayleigh–Taylor (RT) instability re-sulting from the buoyancy of hot, burned fluid

with respect to the dense, unburned material

(M€uller and Arnett, 1982; Niemeyer and Hille-

brandt, 1995). Subject to the RT instability, small

surface perturbations grow until they form bub-

bles (or ‘‘mushrooms’’) that begin to float upward

while spikes of dense fluid fall down. In the non-

linear regime, bubbles of various sizes interact andcreate a foamy RT mixing layer whose vertical

extent grows with time. Secondary instabilities

related to the velocity shear along the bubble

surfaces (Niemeyer and Hillebrandt, 1995) quickly

lead to the production of turbulent velocity fluc-

tuations that cascade from the size of the largest

bubbles (�107 cm) down to the microscopic Kol-

mogorov scale, lk � 10�4 cm, where they are dis-sipated. Since no computer is capable of resolving

this range of scales, one has to resort to statistical

or scaling approximations of those length scales

that are not properly resolved. The most promi-

nent scaling relation in turbulence research is

Kolmogorov�s law for the cascade of velocity

fluctuations, stating that in the case of isotropy

and statistical stationarity, the mean velocity v ofturbulent eddies with size l scales as v � l1=3

(Kolmogorov, 1941).

W. Hillebrandt / New Astronomy Reviews 48 (2004) 615–621 617

Given the velocity of large eddies, e.g. from

computer simulations, one can use this relation to

extrapolate the eddy velocity distribution down to

smaller scales under the assumption of isotropic,

fully developed turbulence. Turbulence wrinkles

and deforms the flame. These wrinkles increase theflame surface area and therefore the total energy

generation rate of the turbulent front. In other

words, the turbulent flame speed, defined as the

mean overall propagation velocity of the turbulent

flame front, becomes larger than the laminar

speed. If the turbulence is sufficiently strong the

turbulent flame speed becomes independent of the

laminar speed, and therefore of the microphysicsof burning and diffusion, and scales only with the

velocity of the largest turbulent eddy (Clavin,

1994).

As the density of the white dwarf material de-

clines and the laminar flamelets become slower and

thicker, it is plausible that at some point turbu-

lence significantly alters the thermal flame struc-

ture (Niemeyer and Woosley, 1997). So far,modeling this so-called distributed burning regime

in exploding white dwarfs has not been attempted

explicitly since neither nuclear burning and diffu-

sion nor turbulent mixing can be properly de-

scribed by simplified prescriptions. However, it is

this regime where the transition from deflagration

to detonation is assumed to happen in certain

phenomenological models.

3. A numerical model for turbulent combustion

It is straight forward to convert the ideas pre-

sented in the previous section into a numerical

scheme. The basic ingredients are a finite-volume

method to solve the fluid-dynamics equation, afront-tracking algorithm which allows us to

propagate the thermonuclear flame (assumed to be

in the flamelet regime), and a model to determine

the turbulent velocity fluctuations on unresolved

sub-grid scales. Since the details of the method

have been published elsewhere (Reinecke et al.,

1999, 2002a,b) we only repeat the basic ideas here.

The central aspect of our code is a fronttracking method based on a level set function G

which is determined in such a way that the zero

level set of G behaves exactly as the flame. This can

be obtained from the consideration that the total

velocity of the front consists of two independent

contributions: it is advected by the fluid motions at

a speed~v and it propagates normal to itself with aburning speed s.

This front tracking algorithm is implemented as

an additional module for the hydrodynamics code

PROMETHEUS (Fryxell et al., 1989). In all sim-

ulations presented here a simple implementation

was used which, however, describes the basic

physics quite well (Reinecke et al., 1999, 2002a). It

assumes that the G-function is advected by thefluid motions and by burning and is only used to

determine the source terms for the reactive Euler

equations. Nuclear burning can now be computed

provided the normal velocity of the burning front

is known everywhere and at all times. In compu-

tations discussed in the following it is determined

according to a flame-brush model of Niemeyer and

Hillebrandt (1995).

4. Some results of supernova simulations

We have carried out numerical simulations in

2D and 3D, for a variety of different initial con-

ditions, and for different numerical resolution.

Details of these models, including convergencetests, are given in a series of papers (Reinecke

et al., 1999, 2002a,b). Again, only some of the

essential results are repeated here.

In most of our simulations the white dwarf,

constructed in hydrostatic equilibrium for a real-

istic equation of state, has a central density of

2:9� 109 g/cm3, a radius of 1:5� 108 cm, and a

mass of 2:8� 1033 g, identical to the one used inNiemeyer and Hillebrandt (1995). The initial mass

fractions of C and O are chosen to be equal, and

the total binding energy is 5:4� 1050 erg. At low

densities (q6 107 g/cm3), the burning velocity of

the front is set equal to zero because the flame

enters the distributed regime and our physical

model is no longer valid. However, since in reality

some matter may still burn the energy release ob-tained in the simulations is probably somewhat

too low. An extended parameter study, varying the

Fig. 1. Snapshots of the flame front for a scenario with nine ignition spots per octant in 3D. The fast merging between the leading and

trailing bubbles and the rising of the entire burning region is clearly visible. One ring on the coordinate axes corresponds to 107 cm. The

snapshots are at t ¼ 0:0, 0.12, and 0.36 s, respectively (from Reinecke et al., 2002a).


chemical composition as well as the ignition den-

sity, is presently under way and will be published

elsewhere.

A first and important result is that we do find

numerically converged solutions. Although an in-crease in spatial resolution gives more structured

burning fronts with larger surface area, the cor-

responding increase of fuel consumption is com-

pensated by the lower values of the turbulent

velocity fluctuations on smaller length scales. So

the net effect is that, for identical initial conditions,

the explosion energies are independent of the nu-

merical resolution, demonstrating that the level-setprescription allows to resolve the structure of the

burning front down almost to the grid scale, thus

avoiding artificial smearing of the front which is an

inherent problem of front-capturing schemes.

Fig. 2. Energy evolution of several three-dimensional (3d) ex-

plosion models (dashed and dashed-dotted). For comparison

we also show the centrally ignited (‘‘three fingers’’) model of

Reinecke et al. (2002a) (solid line). The other labels give the

number of initial ‘‘bubbles’’ (bn) and the number of grid points

per dimension (ijk).

In our approach, the initial white dwarf model

(composition, central density, and velocity struc-

ture), as well as assumptions about the location,

size and shape of the flame surface as it first forms

fully determine the simulation results. A plausibleignition scenario suggested by Garcia-Senz and

Woosley (1995) is the simultaneous runaway at

several different spots in the central region of the

progenitor star. Therefore, in the following we will

concentrate on such initial conditions. Fig. 1

shows snapshots of a typical example and Fig. 2

the energy generated for a series of models in-

cluding, for comparison, one centrally ignitedmodel (3c_3d_256).

During the first 0.5 s, all models are nearly in-

distinguishable as far as the total energy is con-

cerned (see Fig. 1), which at first glance appears

somewhat surprising, given the quite different ini-

tial conditions. A closer look at the energy gener-

ation rate actually reveals noticeable differences in

the intensity of thermonuclear burning for thesimulations, but since the total flame surface is

initially very small, these differences have no visible

impact on the integrated curve in the early stages.

However, after about 0.5 s, when fast energy

generation sets in, the models with more ignition

spots burn more vigorously due to their larger

surface and therefore they reach higher final en-

ergy levels. Fig. 2 also shows that the centrallyignited model (c3_3d_256) is almost identical to

the off-center model b5_3d_256 with regard to the

explosion energetics. But, obviously, the scatter in

the final energies due to different initial conditions

appears to be small. Moreover, all models explode

with an explosion energy in the range of what is

observed.

Fig. 3. Isotopic abundances obtained for the centrally ignited

3D model 3c_3d_256 in comparison to W7 predictions (from

Travaglio et al., 2003).


5. Predictions for observable quantities

In this section, we present a few results for

various quantities which can, in principle, be ob-

served and which therefore can serve as tests forthe models.

The most direct test of explosion models is

provided by observed light curves and spectra.

According to ‘‘Arnett�s Law’’ light curves measure

mostly the amount and spatial distribution of ra-

dioactive 56Ni in type Ia supernovae, and spectra

measure the chemical composition in real and ve-

locity space.Sorokina and Blinnikov (2003) have used the

results of one of our centrally ignited 3D-models,

averaged over spherical shells, to compute color

light curves in the UBVI-bands. Their code as-

sumes LTE radiation transport and loses reliabil-

ity at later times (about 4 weeks past maximum)

when the supernova enters the nebular phase.

Also, this assumption and the fact that the opacityis not well determined at longer wavelength make

I-light curves less accurate. Keeping this in mind,

their light curves look very promising. The main

reason for the good agreement between the model

and SN 1994D is the presence of high-velocity

radioactive Ni in outer layers of the supernova

model which is not be predicted by spherical

models.A summary of the gross abundances obtained

for some of our 3D models is given in Table 1.

Here, ‘‘Mg’’ stands for intermediate-mass nuclei,

and ‘‘Ni’’ for the iron-group. In addition, the total

energy liberated by nuclear burning is given. Since

the binding energy of the white dwarf was about

5� 1050 erg, all models do explode. Typically one

expects that around 80% of iron-group nuclei areoriginally present as 56Ni bringing our results well

into the range of observed Ni-masses. This success

of the models was obtained without introducing

Table 1

Overview over element production and energy release of typical

supernova simulations

Model name m\Mg" [M�] m\Ni" [M�] Enuc [1050 erg]

c3_3d_256 0.177 0.526 9.76

b5_3d_256 0.180 0.506 9.47

b9_3d_512 0.190 0.616 11.26

any non-physical parameters, but just on the basis

of a physical and numerical model of subsonic

turbulent combustion. We also stress that ourmodels give clear evidence that the often postulated

deflagration-detonation transition is not needed to

produce sufficiently powerful explosions.

Finally, we have ‘‘post-processed’’ several of

our models in order to see whether or not also

reasonable isotopic abundances are obtained. The

results, shown in Fig. 3, are preliminary and

should be considered with care. However, it isobvious that, with a few exceptions, also isotopic

abundances are within the expected range. Ex-

ceptions include the high abundance of (unburned)

C and O, and the overproduction of 48;50Ti, 54Fe,

and 58Ni. We think that this reflects a deficiency of

some of our models which burn to little C and O at

densities too high and temperatures too low, and

which also, because of the low 56Ni, would not givea good light curve.

6. A new observational approach

In order to improve our knowledge of type Ia

supernovae through an advance in both observa-

tions and modeling, a group of European astron-omers has recently organized as a European

Research Training Network (RTN), also named

the European Supernova Collaboration (ESC). The

40200

18

16

14

12

Days since B maximumm

ag +

con

st

= CL= CS= JJ= WM= DF= TD= SS= MO= AF= BA= WF

Fig. 5. Color light curves of SN 2002er. A total of 11 telescopes

was used for the imaging of this supernova, indicated by dif-

ferent symbols (from Pignata et al., 2003).


idea is that accurate observations and modeling of

relatively nearby SNe is the only way to under-

stand their nature and the causes of their range of

properties. Participants to the RTN include Ger-

man, British, Italian, French, Spanish and Swedish

institutes, and more recently also groups in Aus-tralia and China. They have successfully applied

for joint observing time on most major European

telescopes, thus maximizing and optimizing the

amount of telescope time allocated for SNe Ia

studies.

In the short-time since the beginning of the

RTN, the ESC has already collected very accurate

data on 6 nearby SNe Ia, including SNe 2002boand 2002er, the latter appearing to be rather

standard. A series of spectra of SN 2002bo are

shown in Fig. 4. It is possible to identify many of

the lines in the spectra, which near maximum are

dominated by elements such as Fe, Si, S and Ca.

The photometry of SN 2002er (Fig. 5) is perhaps

Fig. 4. Spectral evolution of SN 2002bo. Wavelength is in the

observers frame. The earliest spectrum is on scale. The others

are shifted with increments 0.6 (from Benetti et al., 2003).

the most accurate coverage of the evolution of a

SN Ia available to date.

These data (and those still to come) allow us to

constrain the models and to search for the cause of

differences between and the observed correlations

of SNe Ia. Until now, especially early observationsand complete coverage of light curves and spectral

evolution are available only for a small number of

supernovae, but this will change in the near future.

Following the spectral evolution from very early

epochs all the way into the nebular phase allows

‘‘mass tomography’’ and to constrain the models.

Accurate bolometric light curves constructed from

filter light curves and spectra constrain the ener-getics of the explosion and, thus, provide tools to

test the ‘‘standard candle’’ hypothesis.

7. Conclusions

In this paper, we have discussed the physics of

thermonuclear combustion in degenerate densematter of C+O white dwarfs. It was argued that

not all relevant length-scales of this problem can

be numerically resolved and a numerical model to

describe deflagration fronts with a reaction zone


much thinner then the cells of the computational

grid was presented. This new approach was ap-

plied to the simulate thermonuclear supernova

explosions of Mchan white dwarfs in 3D.

All models we we presented here (differing only

in the ignition conditions and the grid resolution)explode. The explosion energy and the Ni-masses

are only moderately dependent on the way the

nuclear flame is ignited making the explosions

robust. However, since ignition is a stochastic

process, the differences we find may even explain

some of the spread in observed SN Ia�s.Based on our models we can predict light

curves, spectra, and abundances, and the first re-sults look promising. The light curves seem to be

in very good agreement with observations, and

also the nuclear abundances of elements and their

isotopes are found to be in the expected range. Of

course, the next step is to compute a grid of

models, with varying white dwarf properties, and

to compare them with the increasing data base of

well-observed type Ia supernovae. The hope isthat, together with a new set of observational data,

this will give us a tool to understand their physics.

Acknowledgements

The author would like to thank, in particular,

Jens Niemeyer, Martin Reinecke, and ClaudiaTavaglio for their contributions to this work,

which also profited a lot from numerous discus-

sions with members of the RTN ‘‘The Physics of

Type Ia Supernova Explosions’’. This work was

supported by the European Commission under

contract HPRN-CT-2002-00303 and the DFG-

SFP 375. Support by ECT* in Trento is also ac-knowledged where this work was completed.

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supernova explosion models: predictions versus observations

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