The spectral types of stars and Hertzsprung–Russell diagram

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Numerical simulations with V1D

The MESA simulations are usually stopped when the maximum infall core velocity reached 1000 km s􀀀1. At that time, we remap the MESA model into V1D (Livne, 1993; Dessart et al., 2010a,b). The model was resampled onto a grid with a mass resolution m of 10􀀀4 􀀀 10􀀀3M at the base, increasing to 10􀀀2M at and beyond 2M. Within a few percent of the stellar surface, the mass resolution is progressively increased to have a surface resolution of 10􀀀6 􀀀 10􀀀5M. At the progenitor surface, we go down to a density of 10􀀀12 g cm􀀀3 but ideally one should use an even lower value in order to have optically thin shells at the outer boundary where the shock breaks out. The explosion is in all cases triggered by moving a piston at 10 000 km s􀀀1 at the inner boundary, which we place at the location where the entropy rises outward from the center to 4 kB baryon􀀀1 (see, e.g., Ugliano et al. 2012). This location is typically in the outer part of the Si-rich shell, just below the O-rich shell, and located around 1.55M in models presented in current work. The explosion models are done iteratively until we obtained an ejecta with the desired 56Ni mass. Iteration is needed because the 56Ni mass is sensitive to the piston properties (location, speed) and to the magnitude of fallback. The asymptotic ejecta energy also depends on fallback. In most of our simulations, we enforce a chemical mixing using a boxcar algorithm (see Dessart et al. 2012 for discussion) that affects all species. In Section 4.3.2, we also explore the impact of mixing only 56Ni (and substituting it with H to keep the mass fraction normalized to unity at each depth).

Explosive nucleosynthesis

Explosive nuclear burning is often treated in the CCSN simulations (see, e.g., Ugliano et al. 2012; Ertl et al. 2016; Sukhbold et al. 2016; Kozyreva et al. 2017; Janka et al. 2017). In our work, we also calculate explosive nucleosynthesis. In Figure 2.5 we show the chemical stratification and velocity versus mass at t = 0 s and at t = 2 s after explosion in model m12mlt3du2p5a12. In the Si-rich shell, the shock changes the composition to produce some 56Ni and intermediate-mass elements (largely independent of the initial composition).
This means that the heavy elements (A > 28) are made primarily by the shock, while below they come primarily from the nuclear burning prior to explosion (the material in the O-rich shell and further out in mass space).

SN radiation modelling: approaches and codes

Present work involves sophisticated numerical simulations of almost the entire life of the star from the main sequence until core collapse and the explosion as a SN. A little percent of studies published in the literature cover the whole chain of events leading to the SN explosion. A lot of studies are dedicated to the stellar evolution alone and do not include calculations of the ejecta evolution in case of SN explosion. Other studies, on the contrary, start simulations with progenitor models, obtained without stellar evolution, and calculate only ejecta properties and its evolution. The reason why most of the studies do not combine stellar evolution modelling with an explosion and ejecta evolution modelling is that the physics of stellar evolution and of ejecta  evolution is very different. One code cannot provide such a vast field of application. The basic point is that different equations are solved. Stellar evolution modelling does not have to treat shocks, which are essential in hydrodynamics of explosions. Similarly, nuclear burning is key for producing a massive star model at death but it is not critical to model the explosion. In the present work, the approach is to start a simulation with the 1-D stellar evolution with MESA from the main sequence till the core collapse. As a result, we obtain pre-supernova models with different mass, radius, chemical composition, density and temperature distribution. All these parameters are important input data for the explosion, which is performed with V1D code. Starting at post-explosion time of 11 days, when the ejecta reach homologous expansion, we perform the radiative-transfer modelling using the non-Local-Thermodynamic-Equilibrium time-dependent radiative transfer code CMFGEN. One way to study the impact of the certain input parameter on the resulting ejecta is to explode different pre-supernova models with the same kinetic energy. Note that different pre-supernova models can be obtained from the identical main sequence progenitors, evolved with different prescriptions (e.g., mass loss). The other way is to explode identical pre-SN models with different kinetic energy or to apply different mixing to these explosions. Both ways are used in the present work. A number of codes is used in the community to calculate the radiative transfer in the expanding ejecta of SN explosion.

Radioactive decay power from unstable istopes

SN LCs are different at early times, but become similar after 150 days since explosion (see Figure 1.3). At that time, LC is powered primarily by the two-step radioactive decay 56Ni!56Co!56Fe: of the atomic datasets is given in text. Nf refers to the number of full levels, Ns to the number of super levels, and Ntrans to the corresponding number of bound-bound transitions. The last column refers to the upper level of a given ion included in our treatment. At the bottom of the table, we give the total number of full levels treated, and the corresponding number of transitions explicitly included.

MV at plateau – MV at nebular phase relation

In Figure 3.5, we compare the plateau and nebular phase luminosities. The relation, although the data scatter is significant, is well visible in the figure. It indirectly connects the explosion energy and the mass of ejected 56Ni. More energetic explosions tend to produce more 56Ni. This makes sense since strong explosions have a stronger shock and have a more efficient explosive nucleosynthesis (Müller et al., 2017).

MV at plateau – expansion rate relation

In Figure 3.6 we show Vabs in Fe II 5169 Å, derived from the position of the absorption maximum  of the line. This gives a measure of the expansion rate. This measure is quite homogeneous and remarkably lower than for the standard archetypal Type II-P SN 1999em.
Hamuy & Pinto (2002) noticed that in their sample of 17 Type II SNe objects with brighter plateaus have higher envelope expansion velocities (see their Fig. 1). We show the absolute magnitude versus expansion rate measured by the position of the maximum absorption for the bound-bound transitions at Fe II 5169Å (both at 50 days since explosion, which is roughly the middle of the plateau) in the Figure 3.7. Thirteen objects from our sample have both photometric and spectroscopic observations close to 50 days; values have been interpolated to the value at 50 days if enough information is available.
It is difficult to find a strict relation between the luminosity and the expansion rate for our sample alone. It could be a result of rather high uncertainties in the distances, redshifts and reddening due to host galaxies. However, when our sample is plotted alongside with other moderate luminosity and standard SNe II-P and the data from Hamuy (2003), it is visible that all the objects in the plot follow the same relation, though the scatter for low-luminosity SNe is a bit higher. Fits to data from Hamuy (2003) and to all SNe listed in legend in Figure 3.6 show different slopes. These fits take into account errors along the vertical axis (uncertainties in distance moduli and V-band photometry errors). The main outlier is SN 2010id, for which the observations in the V-band seem problematic (see Appendix A.1.14).

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Stripped-envelope SNe

Stripped-envelope supernovae (SE-SNe, Clocchiatti & Wheeler 1997) are a subset of CCSNe where the progenitor star has lost much of its outer layers of H and in some cases He. Type IIb SNe show evidence of HI lines in their spectra during few weeks, however, later HI lines vanish from the spectra (see Figure 3.13). Type Ib SNe are classified as H-poor and He-rich, and Type Ic SNe as H-poor and He-poor (see, e.g., Filippenko 1997). This suggests placing SNe in a sequence of increasing mass loss from the progenitor star: II-P – II-L – IIb – Ib – Ic (e.g., Nomoto et al. 1995). Binarity is also thought to be essential for the production of SNe IIb=Ib=Ic by Roche lobe overflow. It is often difficult to distinguish between SNe Ib and Ic, and these objects are often referred to as Ib=c or Ibc SNe. Characteristic feature of SNe Ibc is almost symmetrical bell-shaped LC from explosion to 40–50 d after with the maximum light roughly in the middle of the period. After 40–50 d since explosion the decline rate of the brightness changes normally to the smaller one, around 􀀀0:02 mag per day, whereas decline rate from maximum to 40–50 d is around 􀀀0:05 mag per day for most of SE-SNe. From this point of view, LCs of Type IIb SNe are indistinguishable from Ibc light curves (see Figure 3.14). The LC is powered by 56Ni, like in SNe Ia, but the amount of 56Ni is much lower (about 0.1M for CCSNe).
In Figure 3.15, we show the spectral evolution for Type Ib=Ic SNe. Note SN 1998bw with very broad lines (20 d spectrum). This SN is associated with GRB980425. Compared to spectral evolution of other CCSNe (Figures 3.8, 3.9, 3.12, 3.13), SNe Type Ib=Ic show wider lines and complete absence of H lines. More detailed discussion on photometric and spectral evolution of SE-SNe can be found in, e.g., Dessart et al. (2011); Modjaz et al. (2014).

Table of contents :

1 Introduction 
1.1 Historical background
1.2 Stellar evolution theory
1.2.1 The spectral types of stars and Hertzsprung–Russell diagram
1.2.2 CNO cycle
1.2.3 The triple-alpha process
1.2.4 Carbon, neon and oxygen burning
1.2.5 Si burning
1.2.6 Photodisintegration and core collapse
1.3 Supernovae
1.3.1 SN classification
1.3.2 Thermonuclear Type Ia SNe
1.3.3 Core-collapse SNe
1.3.3.1 Mass loss
1.3.3.2 Red supergiants
1.3.3.3 Type II-P/L SNe
1.3.3.4 Type IIb SNe
1.3.3.5 Type Ibc SNe
1.3.4 Other rare SNe types and proposed models
1.3.4.1 SNe Type Ibn
1.3.4.2 SNe Type IIc
1.3.4.3 Peculiar Type II SNe
1.3.4.4 SN impostors
1.3.4.5 Super-luminous SNe
1.3.4.6 Pair-instability SNe
1.3.4.7 GRB-associated Type Ic-BL SNe
1.4 The need to better understand the diversity of CCSNe
2 Supernova modelling 
2.1 1-D stellar evolution with MESA
2.1.1 13–25M model grid
2.2 Massive star explosions
2.2.1 Introduction
2.2.2 Numerical simulations with V1D
2.2.2.1 Explosive nucleosynthesis
2.2.2.2 Fallback
2.2.2.3 Mixing
2.3 SN radiation modelling: approaches and codes
2.3.1 STELLA
2.3.2 SEDONA
2.3.3 ARTIS
2.3.4 PHOENIX
2.3.5 CMFGEN
2.4 Radioactive decay power from unstable istopes
3 Observational properties 
3.1 Type II-P SNe
3.1.1 Light curves
3.1.2 Color evolution
3.1.3 MV at plateau – MV at nebular phase relation
3.1.4 MV at plateau – expansion rate relation
3.1.5 Spectral evolution
3.2 Type II-L
3.3 Stripped-envelope SNe
4 SN 2008bk — a low-luminosity Type II-P supernova 
4.1 Introduction
4.2 Observational data
4.3 Numerical setup
4.3.1 Pre-SN evolution
4.3.2 Piston-driven explosions
4.3.3 Radiative-transfer modelling
4.4 Properties of our best-match model to the observations of SN 2008bk
4.4.1 Ejecta temperature and ionization
4.4.2 Photometric properties
4.4.3 Spectroscopic properties
4.4.4 Ba II lines and the structure seen in H
4.4.5 Additional remarks
4.5 Sensitivity to progenitor and explosion properties
4.5.1 Radius
4.5.2 Mass
4.5.3 Mixing
4.6 Conclusions
4.7 Line identifications for model X at early and late times in the photospheric phase
5 Low-luminosity Type II-P SNe 
5.1 Modelling
5.1.1 Pre-SN evolution with MESA
5.1.2 Piston-driven explosion with V1D
5.1.3 Radiative-transfer modelling with CMFGEN
5.2 Bolometric and multi-band light curves
5.2.1 Results from simulations
5.2.2 Comparison to observations
5.3 Spectra
5.3.1 Results from simulations
5.3.2 Comparison to observations and spectral line identifications
5.4 Comparison to other work
5.5 Conclusions
6 Kinetic energy variation 
6.1 Model YE1 and comparison to SN 2005cs
6.2 Model YE2 and comparison to SN 2012ec
6.3 Model YE3 and comparison to SN 1999em
6.4 Conclusions
Conclusions
Acknowledgements
Appendix A Observational data 
A.1 A sample of low-luminosity Type II SNe
A.1.1 SN 1994N
A.1.2 SN 1997D
A.1.3 SN 1999br
A.1.4 SN 1999eu
A.1.5 SNe 1999gn, 2006ov and 2008in
A.1.6 SN 2001dc
A.1.7 SN 2002gd
A.1.8 SN 2003Z
A.1.9 SN 2004eg
A.1.10 SN 2005cs
A.1.11 SN 2008bk
A.1.12 SN 2009N
A.1.13 SN 2009md
A.1.14 SN 2010id
A.1.15 SN 2013am
A.1.16 Other candidates in low-luminosity SNe II-P
A.1.16.1 SN 1991G
A.1.16.2 SN 2003ie
A.1.16.3 SN 2014bi
A.1.17 Archetypical Type II-P SN 1999em
A.2 A sample of core-collapse SNe
Appendix B Contributions of indiviudal ions to model spectra 
B.1 Models m12, m25 and m27
Bibliography 

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