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The detailed theoretical understanding of Type Ia Supernovae is still limited. Two very complicated physical processes are at work in SNe Ia explosions. First there is the explosion mechanism itself, which is still debated and several possibilities are proposed and then there is the complicated, highly non-thermal process of the radiation escape which leads to the observed phenomenon. A recent review of SN Ia theory is presented by Hillebrandt & Niemeyer (2000).

4.1. Explosion models

In general it is agreed that SNe Ia are the result of thermonuclear explosions in compact stars. White dwarfs are favored by their intrinsic instability at the Chandrasekhar mass and the fuel they provide in carbon and oxygen. All the arguments for this scenario have been already clearly laid out before 1986 (Woosley & Weaver 1986 and references therein). Other fuels could be imagined, but all of them have some problems. They either do not provide enough (explosive) energy (like hydrogen) or can not synthesize the intermediate-mass elements (like helium, which detonates). Higher elements are in principle possible, but it is well known that O-Ne-Mg white dwarfs would rather collapse to a neutron star than explode because of the large electron capture effects (e.g. Nomoto & Kondo 1991, Gutiérrez et al. 1996). The initiation of the burning in the degenerate star is, however, a puzzle. For many years it was clear that a detonation (supersonic burning front) would lead to an overabundance of iron-group elements and not enough of the intermediate-mass elements observed in the spectral evolution during the peak phase. A deflagration (subsonic burning) seemed more appropriate, but it was not clear how to prevent the explosions to turn into a detonation. The phenomenological model W7 (Nomoto et al. 1984, Thielemann et al. 1986) or similar explosions (Woosley & Weaver 1986, 1994b) enjoy a great popularity as the explosive input model for spectral calculations since they seemed to reproduce the element distribution fairly accurately (e.g. Harkness 1991, Jeffery et al. 1992, Mazzali et al. 1993, 1995, 1997, Yamaoka et al. 1992, Shigeyama et al. 1994). The burning speed in this model has, however, never been understood in physical terms. Possible alternatives are the pre-expansion of the white dwarf to lift the degeneracy by a slow deflagration first and have the detonation start later (Khokhlov 1991). The critical parameters in these models are the density at the transition from deflagration to detonation, the pre-explosion density, the chemical composition (mostly C/O ratio), and the deflagration speed at the beginning of the burning. The transition density has been proposed as the critical parameter for the nucleosynthesis and hence the amount of Ni produced in the explosion. These delayed-detonation models can reproduce some of the observations (Höflich 1995, Höflich & Khokhlov 1996, Höflich et al. 1996). However, their consistency has been questioned recently (Niemeyer 1999, Lisewski et al. 1999a, b). Another possibility is that the first explosion in the center fizzles and as the star contracts again, the density and temperatures rise high enough to re-ignite carbon near the center and lead to the explosion (Arnett & Livne 1994a, b, Höflich et al. 1995). There are hence several theoretical possibilities to ignite the white dwarf, but it is still not clear which ones are realized in nature. With the variety of SN Ia events observed now, it is possible that SNe Ia come from different burning processes. However, the observed correlations must then be valid across different explosion mechanisms.

Once the explosion has started, the flame has to continue burning enough material to unbind the star. In many calculations this has not occurred and the flame has fizzled. Only recently have some three-dimensional calculations led to weak explosions (Khokhlov 1995, Niemeyer et al. 1996, Reineke et al. 1999).

An altogether different explosion mechanism on sub-Chandrasekhar mass white dwarfs has been explored (Nomoto 1982, Livne 1990, Livne & Glasner 1991, Woosley & Weaver 1994a, Livne & Arnett 1995). In this model, the explosion is generated at the surface of the white dwarf due to a detonation of He at the bottom of the accretion layer. This model solved the progenitor problem by allowing explosions well below the Chandrasekhar mass near the peak of the white dwarf mass distribution. Difficulties here are the initiation of the explosion and the subsequent ignition of the whole star by a pressure wave. Many of these calculations are still parametric and the details have to be worked out (cf. Woosley 1997).

It is customary nowadays to explore several of these explosion models to explain the observations (e.g. Leibundgut & Pinto 1992, Höflich et al. 1995, Höflich & Khokhlov 1996, Höflich et al. 1996).

4.2. Radiation transport

Another complicated process stands between the explosion models and the observations. The release of the photons from the explosion is computationally extremely difficult to follow. The reasons are the continuous change of the energy deposition and the detailed physics of the conversion of the gamma -rays injected inside the ejecta from the radioactive decays to the low-energy photons observed. The opacity changes due to the thinning of the expanding ejecta for the high-energy input, but at the same time the high velocities and the abundance of higher elements with their large number of transitions complicates the calculations (Harkness 1991, Höflich et al. 1993, Eastman 1997, Pinto & Eastman 2000). The exact treatment is still debated, but it has become increasingly clear that the old assumption of a thermal input spectrum is not tenable. Even though SNe Ia display a nearly thermal 'continuum' during their peak phase, they are really dominated by the time-dependent photon distribution. The clearest demonstrations of this fact are the lack of photons in the J-band (Spyromilio et al. 1994, Meikle 2000) which is due to the absence of emission lines in this wavelength region and the occurrence of the maximum in different optical filters, which is reversed for most supernovae, i.e. the near-IR filter curves peak before the optical ones (Contardo et al. 2000, Hernandez et al. 2000).

Due to the large opacities in the ejecta the photon degradation proceeds through several channels (e.g. Lucy 1999, Pinto & Eastman 2000). Since the UV region is blocked by many velocity-broadened iron-group lines (Harkness 1991, Kirshner et al. 1993), the photons are progressively redshifted until the optical depth is small enough for them to escape. This occurs first in the near-IR and hence the peak is reached earlier at these wavelengths (Meikle 2000, Contardo et al. 2000). However, only in wavelength regions where plenty of line transitions in the outer layers are available is there any significant flux.

Nevertheless, the optical spectrum has been modeled rather successfully even with thermal input sources (Harkness 1991, Jeffery et al. 1992, Mazzali et al. 1993, 1995, 1997, Nugent et al. 1997). This is possible since the outer layers already encounter a pseudo-thermal input spectrum (Pinto & Eastman 2000). Detailed treatment of the NLTE effects has been included by several groups (Baron et al. 1996, Pauldrach et al. 1996, Höflich 1995, Höflich et al. 1996, Lucy 1999, Pinto & Eastman 2000).

At late phases the ejecta are optically thin for optical and infrared photons and we see a spectrum dominated by collisionally excited Fe and Co lines (Axelrod 1980, Ruiz-Lapuente & Lucy 1992, Spyromilio et al. 1992, Kuchner et al. 1994, Bowers et al. 1997, Mazzali et al. 1998). At these epochs it has been assumed that the energy of the positrons in the 56Co decay is locally deposited. This has recently been questioned because of the increased slope of the light curves (Ruiz-Lapuente & Spruit 1998, Cappellaro et al. 1998, Milne et al. 1999). After about 450 days a thermal instability develops in the ejecta which rapidly cool down from about 3000 K to 300 K. Excitation of optical and near-infrared transitions declines rapidly and the cooling continues by fine-structure lines of Fe in the mid- and far-infrared. This is often referred to as the IR catastrophe. The predictions are that this would happen after about 500 days (Fransson et al. 1996) but it has never been observed so far.

It will take a few more years until these problems can be addressed completely. A closer link between the observations and the models has been pursued by trying to understand the correlations which have been observed. The light curve decline has been modeled (Höflich et al. 1996) and explained as due to differences in the amount of Ni produced in the explosion. Also the color dependence could possibly be explained this way. Other issues like the rise time or the occurrence of the secondary peak in the near-IR remain, however, open. A possible interpretation of the light curve stretching during the peak phase and for the bolometric light curves links the time scales of the Ni decay, the diffusion time (for a constant opacity) and the age of the supernova (Arnett 1982, Arnett 1999).

By comparing the kinetic energy as derived from line widths and the measured Ni masses it should be possible to derive global parameters of the explosion. First such steps have been made by Mazzali et al. (1998), Cappellaro et al. (1998), and Contardo et al. (2000). This alternative route will not replace the detailed modeling of light curves and spectra, but may provide a more direct input for the explosion models.

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