Spectra of Supernovae During the Photospheric Phase 31

Stuart A. Sim

Abstract Photospheric-phase spectra have a central role in the classification and study of supernovae of all types. We describe the basic physical picture that can be used to understand photospheric-phase spectra and comment on some of the complications and subtleties associated with studying photospheres in supernovae. We discuss the main radiation processes that contribute to the opacity and emissivity in supernova ejecta of both Type I and Type II events and describe how the relative roles of these processes vary across the ultraviolet to near- infrared spectral regions. We present a simple theoretical framework that can be used to understand the shaping of spectral features in supernovae, in particular the standard P Cygni line profile shape, which often appears in the analysis of supernova spectra. Although we focus on qualitative understanding, we comment on some of the challenges involved in quantitative modelling and discuss common approximations that have been used in supernova spectral synthesis. We conclude with a short overview of applications of spectral modelling and comment on the wide range of approaches that has been used. Contents 1 Introduction...... 770 2 The Photospheric Phase...... 770 2.1 Photospheres in Supernovae...... 771 3 Radiation Processes...... 772 3.1 Opacity in Supernovae...... 773 3.2 Scattering and Fluorescent Emission in Supernovae...... 775 4 Formation of Spectral Features...... 778 4.1 Line Scattering Profile in Expanding Atmospheres...... 778 4.2 Recombination Emission...... 783

S.A. Sim () School of Mathematics and Physics, Queen’s University Belfast, Belfast, UK e-mail: [email protected]

5 Widely Used Approximations in Spectral Modeling...... 784 5.1 The “Core-Halo” Approximation...... 784 5.2 Spherical Symmetry...... 785 5.3 Time Independence...... 785 6 Some Applications of Spectral Modeling...... 786 6.1 Line Identiﬁcation...... 786 6.2 Empirical Determination of Composition...... 787 6.3 Synthetic Spectra for Explosion Models...... 789 6.4 Distance Determination...... 790 7 Conclusions...... 790 8 Further Reading...... 791 9 Cross-References...... 791 References...... 791

1Introduction

Supernova (SN) spectra are most often recorded during the photospheric phase, which can last from a few days to many weeks post-explosion. Analysis of such spectra is one of the most important means to study SNe: they are used to classify SNe, to constrain kinematics and composition, and to test explosion models. In this article, we will overview the basic physical picture that can be used to understand the formation of photospheric phase spectra and summarize the underlying physical processes. We will also brieﬂy discuss example applications of spectral modelling for SN photospheric spectra and highlight the variety of approaches that is adopted.

2 The Photospheric Phase

The photospheric phase in the spectral evolution of a SN refers to epochs at which the spectrum can be broadly understood in terms of a (pseudo-)continuum, usually with discrete spectral features imprinted (see, e.g., Fig. 1). Such spectra are often discussed on the basis of an underlying physical picture in which the ejecta are sufficiently optically thick that photons do not directly escape from the inner regions: in the inner ejecta, conditions are close to equilibrium with a radiation field that is near isotropic and approximated by the Planck function at the local temperature (i.e., local thermal equilibrium, LTE, applies to both the material and radiation field). Proceeding outward in this picture, a region is necessarily encountered where photons start to escape, and it is about this region (and above) that we can directly learn by observing and modeling spectra. This apparent emitting surface – the photosphere – is a familiar concept in astrophysics: photospheres are of central relevance to our understanding of observations of many optically thick astronomical objects. The concept of a well-defined surface from which the bulk of the continuum radiation escapes is widely used as a simple/pragmatic approximation in several of the strategies used to interpret SN spectra (see Sects. 5.1 and 6). However, the notion of a photosphere as a well-defined surface below which LTE becomes a good approximation is not always appropriate; 31 Spectra of Supernovae During the Photospheric Phase 771

Fig. 1 Spectra of the SN 1994D (SN Ia) for epochs Si II Ca II ranging from 12 to 1 days (as −12 labelled) before maximum light (Figure reproduced from Branch et al. 2005, −11 Publications of the Astronomical Society of the −10 Ca II Paciﬁc, 117, 545. DOI: −9 10.1086/430135.©The Astronomical Society of the −8 Paciﬁc. Reproduced with −7 permission. All rights reserved)

LOG FLUX −4

−3 −2

−1

4000 5000 6000 7000 8000 WAVELENGTH indeed, SN ejecta are a good case study to illustrate some of the complications and limitations associated with deﬁning and utilizing the concept of a photosphere.

2.1 Photospheres in Supernovae

The photosphere is often defined as the surface at which the optical depth (integrated from the outer boundary of an object inwards) reaches D 2=3. This definition has its origin in the theory of gray stellar atmospheres where, under the Eddington approximation, D 2=3 is the surface at which the temperature is equal to the effective temperature (Hubeny and Mihalis 2015). However, it is an over simplification to suppose that the continuum simply “forms” at this point in the atmosphere, or that the overall spectral-energy-distribution (SED) shape is uniquely (or even predominantly) determined in one well-defined region. In realistic radiation transfer modeling, the precise definitions used for the photosphere (or photospheric radius) can vary. For example, some authors use the term to mean the region at which the radiation field comes into equilibrium with the local gas temperature (widely known as the thermalisation depth), while others retain a definition with the total optical depth of D 2=3. This distinction is important given that scattering (or fluorescent) opacities dominate over thermalization processes under many circumstances of relevance to SNe (see Sect. 3 and 772 S.A. Sim references therein). That is, the thermalization depth can be significantly deeper than the surface D 2=3, and in some cases true thermalization may never be achieved in the ejecta. As discussed by Pinto and Eastman (2000), the ejecta of SNe Ia are not sufficiently optically thick to achieve full thermalization at maximum light: instead, the pseudo-blackbody SED is achieved by fluorescence. Furthermore, since the opacity is a strong function of wavelength, a formal photospheric definition requires that a suitable optical depth scale is chosen. As illustrated by Dessart and Hillier (2005a) for Type II SN models, if one considers a monochromatic optical depth, the radius at which D 2=3 can be almost a factor of two larger at blue wavelengths than in the red, if line opacity is taken into account (for relatively cool models). Ultimately, all such complications and subtleties in defining the photosphere have no meaningful consequence in the most sophisticated studies of spectrum formation in SN ejecta. Contemporary radiative transfer codes (including those of, e.g., Höflich et al. 1998, Hauschildt and Baron 1999, Dessart and Hillier 2005b, Sauer et al. 2006, Kasen et al. 2006, Kromer and Sim 2009, and Wollaeger et al. 2013) do not depend on any imposed definition of a photosphere, and provided that the analysis is made consistently, the results of such calculations can be directly compared to observations without explicit reference to photospheric definitions. For high precision, quantitative work on photospheric phase spectra, such methods are the current state of the art. Nevertheless, significant physical insight into some problems can still be derived from the idealized notion of a relatively thin photospheric region from which a near-blackbody continuum emerges and the assumption that observed spectral features then form in, or above, this region. For pragmatic reasons, this simplified picture is still used – either directly or in slightly varied forms – in the interpretation and modeling of SN spectra. For example, the popular SYNOW / SYN++ code (Fisher 2000; Thomas et al. 2011) and the Monte Carlo code developed by Mazzali (2000) both impose a blackbody “photospheric” boundary condition at the base of a line-forming region. Although an idealization, such approaches have the flexibility to explore the very wide parameter space of possible SN ejecta properties at modest computational expense and thus continue to play an important part in the modeling of SN spectra (see Sects. 5.1 and 6). They also allow a straightforward way to identify which ionic transitions produce the observed features. This is an essential first step to constrain the composition of the ejecta, which is much more challenging in supernovae than in stellar atmospheres due to line blending.

3 Radiation Processes

Here we overview the main radiation processes at work in the formation of the photospheric-phase spectra of SNe at ultraviolet to near-infrared wavelengths. We assume the reader has a basic familiarity with principles of atomic physics and radiation-matter interactions (see Sect. 8 for references to background material). 31 Spectra of Supernovae During the Photospheric Phase 773

3.1 Opacity in Supernovae

A range of physical processes is responsible for the opacities in and above the photospheres of SNe. The relative contributions of different opacity sources depend on a number of factors including the composition, density, and degree of ionization in the ejecta. However, in essentially all cases, electron scattering provides an important contribution. Across the wavelength regions typically observed (i.e., rest frame ultraviolet and redder), electron scattering is well approximated by the Thomson limit in which the cross section is gray and the opacity depends only on the free electron density. Thus, electron scattering effectively sets a minimum continuum opacity coefficient across the entire spectrum, to which the role of more complex processes can be compared. It is worth noting, however, that electron scattering is inefficient in wavelength redistribution or thermalization: thus, even when comparatively weak, it is always other interaction processes that dominate the coupling between different wavelength intervals in the spectrum and radiative heating/cooling of the gas. Additional continuum opacity is provided by bound-free and free-free processes. In Fig. 2, the contributions of the bound-free continua of hydrogen and free- free continua of hydrogen and helium are shown (and compared to the electron scattering opacity) for a simplified calculation applicable to the photospheric

Fig. 2 Illustrative line (solid), electron scattering (dashed), H bound-free (dotted), and free-free (dot-dashed) opacities calculated for an LTE toy model of Type IIP ejecta at a photospheric phase (using the TARDIS code of Kerzendorf and Sim 2014). Specifically, the opacity per gram is shown for a zone with velocity around 11;000 km s1 at an epoch of 15 days post-explosion in a model with a power-law density profile Œgcm3 D 1:271012.v=11;000 km s1/12 and a luminosity 8:9 of 10 Lˇ. The line opacity shown has been calculated from LTE Sobolev optical depths using the expansion opacity formalism of Eastman and Pinto (1993) (see also Blinnikov et al. 1998) with a wavelength bin size of = D 0:012. Solar metal abundances have been adopted. Hydrogen bound-free opacity from excited states is computed assuming LTE level populations 774 S.A. Sim phase in a hydrogen-rich SN. Bound-free absorption (i.e., photoionization) is the process by which an electron is liberated from an atom or ion by absorption of a photon. This process can occur directly or, for complex ions, by photoabsorption followed by autoionization (see, e.g., Tennyson 2011). Free-free absorption (inverse Bremsstrahlung), which becomes increasingly important at long wavelengths, is associated with the absorption of photons by free electrons in the presence of an ion. Although typically subdominant to electron scattering in the near ultraviolet and optical wavelength ranges (see, e.g., Fig. 2), bound-free and free-free absorption are important thermalization processes and key to determining the continuum shape in SNe II. Arguably, the most important source of opacity to study in SN ejecta is that due to bound-bound transitions (i.e., “line” opacity). The study of spectral lines, and what they revel about the ejecta, is often one of the main objectives in the modeling of photospheric spectra, as will be elaborated in Sects. 4 and 6 below. Thanks to the large velocities in SN ejecta, individual spectral lines affect the observed spectrum across a substantial range of wavelengths. Particularly at ultraviolet and blue wavelengths, this means that blending of spectral lines is common and the identification of a “continuum” becomes practically impossible: in such regions the apparent SED is determined by the pseudo-continuum of overlapping lines. The relative importance of line opacity, compared to the continuum processes described above, is shown for the hydrogen-rich toy model in Fig. 2. Specifically, we show the line opacity computed as a function of wavelength in the expansion opacity formulation of Eastman and Pinto (1993) (adopting a solar composition for the metals). In this case, we see that the cumulative effect of overlapping lines makes the effective line opacity dominate over the electron scattering opacity for wavelengths shorter than 3000 Å. This phenomenon, known as line blocking / line blanketing, is a critical part of interpreting the ultraviolet/blue SED of SNe, and its importance can vary considerably depending on properties of the ejecta. For H-rich ejecta, as discussed, e.g., by Dessart and Hillier (2005a), the temperature (or, more specifically the ionization state) is very important: cooler models show stronger line blanketing, primarily owing to a low ionization state of the metals (a critical regime occurs around effective temperatures of 8000 K, when Fe II starts to contribute significant opacity; Dessart and Hillier 2005b). Thanks to the dominance of metal lines in forming the line blanketing opacity, the ultraviolet SED is rather sensitive to the ejecta composition (Dessart and Hillier 2005b) and potentially provides a diagnostic for metallicity. In the metal-rich ejecta of Type I SNe, line opacity is even more important. As discussed by Pinto and Eastman (2000, see Fig. 1; see also Sauer et al. 2006), the line opacity is orders of magnitude higher than the electron scattering (and bound- free/free-free) opacities throughout the ultraviolet and much of the optical, if a composition rich in iron-group elements is considered. Moreover, in most Type Ia explosion models, the composition is expected to vary within the ejecta, leading to substantial differences in the opacity distribution through the different regions that are probed during photospheric phases. This is illustrated in Fig. 3, which compares the line (and electron scattering) opacity distribution in two different regions of the 31 Spectra of Supernovae During the Photospheric Phase 775

Fig. 3 Line (solid) and electron scattering (dashed) opacity as calculated for the SN Ia W7 explosion model (Nomoto et al. 1984) with TARDIS (Kerzendorf and Sim 2014). Results are shown for two velocity shells in the ejecta at around maximum light: 11,500 km s1, which is close to the midpoint of the Si/S-rich ejecta, and 14;000 km s1, which is in O-rich ejecta layers. The line opacity shown has been calculated from the TARDIS Sobolev optical depth using the expansion opacity formalism of Eastman and Pinto (1993) (see also Blinnikov et al. 1998) widely used W7 model (Nomoto et al. 1984) for Type Ia SNe. These illustrate both the complexity and importance of the line-opacity distribution in metal-rich SNe – in particular, it is very difﬁcult to identify spectral regions that are not affected by line opacity, making it all but impossible to analyze or interpret ultraviolet/optical SN Ia photospheric spectra without modeling line opacity. We do note, however, that there is one important exception in the wavelength region around 7000 Å, which is relatively devoid of strong lines. Consequently, this region is often used, particularly in analysis and interpretation of spectropolarimetry, as a relatively good probe of the “electron scattering photosphere’’ (Patat et al. 2009).

3.2 Scattering and Fluorescent Emission in Supernovae

In addition to providing the opacity, the radiation processes described above (or, more precisely, their inverse processes) determine the effective emissivity in the SN ejecta. The thermalization continuum processes (free free and bound free) are relatively simple in this regard: their emissivities depend on the local plasma conditions (electron temperature and particle densities) and not directly on the radiation field itself. As noted in Sect. 2, however, much of the emissivity in SN ejecta during the photospheric phase is due to scattering, meaning that strongly non-local effects can be introduced into the radiative transfer problem. In the Thomson limit, electron 776 S.A. Sim scattering can be treated as elastic in a frame co-moving with the ejecta at the point of interaction. This makes electron scattering extremely easy to implement with Monte Carlo radiative transfer methods and was one of the motivating factors for the use of Monte Carlo methods in early (and continuing) efforts to study the formation of SN spectra (following, e.g., Mazzali and Lucy 1993). Despite elastic scattering being a good approximation for electron scattering in a co-moving frame, however, the Doppler shifts associated with frame transformations mean that Thomson scattering does alter photon frequency distributions in the observer frame. Specifically, as discussed by Auer and van Blerkom (1972) and Hillier (1991), electron scattering in a spherically expanding flow leads to the development of red- skewed spectral features (the net effect of electron scattering on any emission feature is to broaden and “stretch” the profile to longer wavelengths). The development of electron scattering wings can be particularly prominent in SNe of Type IIn, where a substantial optical depth due to scattering can be present in the cold dense shell of material at the interface between the ejecta and swept-up circumstellar material (Chugai 2001; Dessart et al. 2009). The most complicated of the major contributors to the emissivity is due to the multitude of overlapping bound-bound (line) transitions. As noted in Sect. 3.1, line opacity is often dominant at ultraviolet wavelengths and maintains an important role across much of the optical spectrum, particularly for metal-rich SNe. To properly understand the emissivity associated with lines, it is necessary to model the atomic processes governing the excitation and deexcitation of atomic/ionic species. It is reasonable to assume that statistical equilibrium is a good approximation for most states excited by line absorption: the natural lifetimes of atomic states with permitted radiative decay channels are typically 106 s or less, much shorter than the timescale of evolution in the ejecta. Consequently, each time an atomic/ionic state is excited by absorption of a photon, it will quickly relax. In principle, either collisional or radiative processes can deexcite an excited state – the former transfers the energy to the particle kinetic pool (constituting a thermalization event) and the latter constitutes a direct contribution to the emissivity. For typical conditions in SN ejecta, however, the collisional deexcitation probability is small (. 104 Pinto and Eastman 2000, see also Kasen et al. 2006), and radiative deexcitation dominates. One might hope that this simplifies the problem: if radiative reemission were in the same atomic transitions as responsible for excitation, then the bound-bound component of the emissivity could be treated as effective resonance scattering. However, for the complex metal ions, which dominate the overlapping line opacity, deexcitation is predominantly expected to occur via atomic levels other than that from which an original photoexcitation occurs (Pinto and Eastman 2000). Thus, photoabsorption via a particular bound-bound line will often lead to reemission in a different line (or lines): the process known as fluorescence (when the emitting lines are redder than the absorbing line; inverse fluorescence when bluer). Treatment of bound-bound emissivity is further complicated when recombination is effective in populating excited states. This effect is most prominent in the optical spectra of H-rich (Type II) SNe when the H emission line series can show substantial net 31 Spectra of Supernovae During the Photospheric Phase 777

Fig. 4 Redistribution of photon packets in a Monte Carlo simulation of line ﬂuorescence in a white-dwarf detonation model (see Kerzendorf and Sim 2014 for details). The plot shows the co-moving frame wavelength of photons prior to line interaction (horizontal axis) versus post interaction (vertical axis); the color indicates relative numbers of events in the Monte Carlo simulation. Strong wavelength redistribution is particularly apparent at ultraviolet wavelengths (Figure reproduced from Kerzendorf & Sim 2014, Monthly Notices of the Royal Astronomical Society, 440, 387. DOI: 10.1093/mnras/stu055)

emission, thanks to the role of radiative recombination to excited states followed by radiative cascades (Kirshner and Kwan 1975, see also Sect. 4.2). To illustrate the complexity of line emissivity, Fig. 4 shows the wavelength redistribution of energy packets in a Monte Carlo radiation transport simulation for a white-dwarf detonation model. In this redistribution plot, line interactions that leave the wavelength unchanged appear along the diagonal, but it is apparent that many interactions lead to coupling between very different parts of the spectrum. In particular, absorption in the ultraviolet can lead to emission across much of the spectral region shown. To accurately model this complicated cross-talk between different wavelength regions is challenging and can be one of the more difficult issues to consider when attempting quantitative spectral modeling, particularly for metal-rich ejecta. In the simplest approaches, it is sometimes assumed that the complicated atomic physics of fluorescence can be ignored and a resonance- scattering assumption adopted for bound-bound lines (e.g., Fisher 2000; Mazzali and Lucy 1993). Although simple, this assumption actually works reasonably well for the optical spectra of metal-rich SNe (Lucy 1999; Mazzali 2000) in codes that operate by imposing a photospheric boundary condition. However, modeling flux redistribution is vital for a good description of ultraviolet (Lucy 1999; Mazzali 2000) or infrared (Kasen 2006; Kromer and Sim 2009) spectral regions, or if the simulation 778 S.A. Sim encompasses formation of the photospheric SED (see Kasen et al. 2006). This redistribution is most reliably described by approaches that simulate the detailed atomic physics of nonresonant scattering (e.g., Baron et al. 2006; Dessart and Hillier 2005b; Kromer and Sim 2009). However, such methods can be computationally expensive, and some approximations of intermediate complexity can be used. For example, (Mazzali 2000, see also Lucy 1999) allow excited atomic/ionic states to decay via all channels to lower energy states, but neglect the possibility of further excitation (as argued by Lucy 1999, this is expected to be a useful approximation for modeling most hydrogen-poor SNe). Alternatively, for large atomic datasets, Kasen et al. (2006) simulate fluorescence /inverse fluorescence via an effective thermalization approach: following bound-bound absorption, it is assumed that the collective emissivity of many lines can be approximated via a combination of thermal line emissivity at the local conditions and a minor contribution from resonance scattering. Provided that the adopted fraction of resonance scattering is small, the synthetic spectrum is fairly insensitive and close to that obtained with a detailed treatment of fluorescence (Kasen et al. 2006).

4 Formation of Spectral Features

Having outlined the range of physical processes that contribute to the formation of SN spectra at photospheric epochs (Sect. 3), we now turn our attention to speciﬁc considerations related to the shape of spectral line proﬁles: much of the analysis of photospheric phase spectra relates to inferring ejecta kinematics and composition based on interpretation and modeling of spectral features. As indicated above, the variety of processes at work leads to considerable complexity in the spectrum formation process. In particular, many features (particularly at short wavelengths) are formed from blends of multiple transitions such that approaches that account for multiple overlapping lines across a range of wavelengths are often needed for any quantitative interpretation of a spectrum. Nevertheless, it is instructive to focus on how an individual spectral transition forms: this can be useful for interpreting cases where a single transition (or close multiplet) is dominant (e.g., the characteristic Si II 6335 line of SNe Ia) and also provides the basis for more complete analyses in which multiple overlapping lines contribute.

4.1 Line Scattering Proﬁle in Expanding Atmospheres

Although a significant simplification (see Sect. 3.2), we will focus on the illustrative case of a line profile formed purely by resonance scattering in SN ejecta. For simplicity, we will consider only leading-order relativistic effects when needed (i.e., first order in ˇ D V=c, where V is ejecta velocity; for relativistically complete approaches, see, e.g., Jeffery 1995a,b). When considering radiative transfer in expanding media, it is often convenient to use a mixed-frame approach (Hubeny and Mihalis 2015). Accordingly, in the following we will distinguish between quantities 31 Spectra of Supernovae During the Photospheric Phase 779 measured in a reference frame that is locally co-moving with the ejecta (primed quantities) and those in the fixed reference frame of an observer (unprimed), which can be taken as the center-of-mass frame of the SN ejecta.

4.1.1 Photon Red Shifting Many of the complications (e.g., extensive line blending in the observed spectrum), but also the simplifications, in SN radiation transport come from the large velocity gradients that arise in the ejecta. At the heart of this is the relationship between the frequency of a photon in the observer’s frame, , and the evolution of the frequency in reference frames that are locally co-moving with the ejecta, 0. Considering this relationship is useful because all the local atomic physics processes can be relatively easily described in a locally co-moving reference frame (i.e., based on knowledge of 0) while predicted observables are required in the observer’s rest frame (i.e., related to ). In the presence of a radial velocity field (V D ˇc), the Doppler formula relates photon frequencies () between the two frames: to first order,

0 D .1 ˇ/ (1) where is the cosine of the angle between the photon direction of travel and the radial (r) direction. To identify the physical processes with which a photon can interact as it prop- agates though the ejecta, we need to follow the evolution of 0. By differentiating along the photon spatial trajectory (s) for the speciﬁc case of homologous expansion (i.e., V D r=t, where t is the time since explosion), it can be shown that, to ﬁrst order (Jeffery 1995b),

d0 0 D .1 ˇ/: (2) ds ct

Since jˇj <1, this derivative is always negative: i.e., the frequency of a propagating photon, as viewed from a sequence of reference frames that are co- moving with the ejecta, is continuously redshifting. Moreover, since in the bulk of SN ejecta ˇ . 0:1, to a good approximation the rate of co-moving frame redshift is independent of direction and can be estimated knowing only the time since explosion (d0=ds =ct).

4.1.2 The Sobolev Approximation for Line Opacity That 0 is continuously redshifting is a useful simplification in itself, but it is when this fact is combined with the Sobolev approximation (Lamers and Casinelli 1999; Sobolev 1960) that it can lead to a major conceptual (and computational) simplification in the treatment of bound-bound transitions. To illustrate the principle of the Sobolev approximation, we begin with a general statement of the absorption coefficient for a bound-bound transition (of rest frequency 0) between two atomic states, denoted l (lower state) and u (upper state) 780 S.A. Sim Â Ã Bluh0 nu gl 0 D nl 1 . / 4 nl gu where Blu is the Einstein coefficient and n and g denote atomic level population number densities and statistical weights for the two bound states ( is the absorption crossR section per gram and the mass density). .0/ is a normalized function 1 0 1 .x/ dx D 1 that describes the absorption line profile as a function of D 0 0, the frequency difference of the photon from line center. For the present qualitative discussion, we need not concern ourselves with the detailed shape of .x/, except to note that it is expected to be sharply peaked around x 0.Here we treat simulated emission as negative absorption and assume that the relevant absorption and emission profiles are equal (see, e.g., Hubeny and Mihalis 2015). Since optical depth, , along a trajectory, s, is defined by d D ds, we can obtain the total optical depth due to the l ! u transition along a ray (say from point s0 to s1) by integrating along s Z Â Ã s1 Bluh0 nu gl 0 lu D nl 1 . / ds (3) s0 4 nl gu

Thanks to the global red-shifting property of photons (see above), it is easy to change variables (s ! 0) and express as an integral over co-moving frame frequency

Z 0 Â Ã .sDs1/ Bluh0 nu gl ds 0 0 lu D nl 1 . / d : (4) 0 0 .sDs0/ 4 nl gu d

0 Then, provided .0/ is strongly peaked around 0 and that d is large (compared ds to the spatial gradients of the level populations), we may consider the limit in which .x/ ! ı.x/ so that S if 0 D between s D s and s D s D lu 0 0 1 (5) lu 0 otherwise where the Sobolev optical depth Â Ã ˇ ˇ ˇ ˇ S Bluh0 nu gl ˇ ds ˇ lu D nl 1 ˇ 0 ˇ: (6) 4 nl gu d

Thus, when considering absorption of photons by spectral lines, we expect that the region in which interactions can happen will be small (conﬁned to a small volume 0 around the Sobolev point at which D 0) and, furthermore, that the total optical S depth associated with passing through the Sobolev resonance (lu) can be calculated from local properties at the Sobolev point. 31 Spectra of Supernovae During the Photospheric Phase 781

Using this approximation greatly simplifies the treatment of line profile formation into a two-step process: first, for each monochromatic ray directed toward the observer, we merely identify the Sobolev point for the spectral line(s) of interest. Until photons on the ray propagate to that point, the line contributes no optical depth. The entire optical depth of the line is then encountered within a small region around the Sobolev point. Photons that are not absorbed within the Sobolev region will continue to propagate and, thanks to their continuous red-shifting, will never encounter that particular line again. Under a resonance-scattering assumption, radiation absorbed within the Sobolev region will be reemitted and may eventually escape the Sobolev resonance (potentially following multiple reabsorptions and emissions) with a new direction of propagation (however, generalization to treatment of nonresonant scattering can also be achieved quite easily; see, e.g., Lucy 1999). Since the Sobolev approximation allows for a substantial simplification, it has been widely used in SN modeling tools/codes: it can be employed directly in codes that treat opacity on a line-by-line basis (e.g., Kromer and Sim 2009; Mazzali 2000; Thomas et al. 2011) or indirectly as part of an expansion opacity formalism (see, e.g., Blinnikov et al. 1998; Eastman and Pinto 1993;Jeffery1995a;Karp et al. 1977). In practice, however, there are important issues associated with use of the Sobolev approximation. In SN ejecta, velocity gradients are large and the approximation to the integral made between Eqs. 4 and 5 (see above) is expected to be good. However, the simple Sobolev treatment neglects the possibility that multiple sources of opacity overlap, as discussed by Baron et al. (1996). In particular, for metal-rich ejecta at short wavelengths, the density of strong spectral lines is sufficiently high that significant overlap is expected. Thus the assumption that lines can be treated as isolated (implicit in the discussion above) is not well justified. Consequently, the most sophisticated codes used for quantitative modeling avoid the Sobolev approximation and treat overlapping line opacity in detail. Nevertheless, the simplicity of the Sobolev approach makes it useful for qualitative understanding of spectrum formation (see Sect. 4.1.3), and the approximation continues to be used, in the interests of computational expediency, for many applications (see Sect. 6).

4.1.3 The P Cygni Line Profile With reference to the Sobolev treatment of line opacity, it is relatively easy to understand the origin of the characteristic P Cygni profile shape (see, e.g., Lamers and Casinelli 1999) for a scattering line that forms in expanding SN ejecta. The key elements of the profile formation are illustrated in Fig. 5: continuum photons emerge from a photospheric region surrounding the optically thick core of the ejecta. As these photons propagate through the surrounding layers, their co-moving frame frequency (0) is continuously redshifting (in accordance with Sect. 4.1.1) and those 0 with >0 at the photosphere will have the possibility of Doppler shifting into Sobolev resonance with a spectral line of rest frequency 0. Depending on the line optical depth, a fraction of such photons will be scattered. From the observer’s perspective (Fig. 5), this scattering will remove some photons and add others. Specifically, photons that were initially directed toward the observer can be scattered 782 S.A. Sim

Fig. 5 Idealized picture of line formation in SN ejecta during the photospheric phase in a “core- halo” approximation. The photosphere (white region) surrounds an optically thick core and is assumed to be geometrically thin. Continuum photons travel through the surrounding expanding ejecta in which they can undergo line scattering. Three example photons that undergo line scattering are illustrated (scattering locations indicated with white crosses). Photon 1 is scattered out of the observer line of sight (los), contributing to blueshifted absorption in the proﬁle. Photons 2 and 3 are scattered into the line of sight, yielding redshifted and blueshifted emission, respectively (see text). The envelope material is color coded to qualitatively indicate the line-of-sight velocity for the observer

out of the line of sight (e.g., photon 1 in Fig. 5). Since such photons necessarily scatter in the approaching side of the ejecta (i.e., they reach Sobolev resonance in the ejecta layers between the photosphere and the observer), their observer-frame frequencies must be greater than 0 and so their removal will give rise to blueshifted absorption. In contrast, photons can be scattered into the observer’s line of sight from both the approaching and receding regions of the ejecta (e.g., photons 2 and 3inFig.5): thus the emission component of the profile is expected to contain both blueshifted and redshifted photons produced across the range of velocities present in the ejecta. This combination of blueshifted absorption and broad emission is generic for scattering-dominated line profiles in SN ejecta, and the shape and strength of the profile are controlled primarily by the variation of optical depth with velocity in the ejecta. Four example, (idealized) line profiles are shown in Fig. 6, each calculated from a spherical model with a different optical depth distribution. 31 Spectra of Supernovae During the Photospheric Phase 783

Fig. 6 Example calculations of scattering line profiles in SNe (flux versus apparent redshift velocity). Each panel shows the profile shape for an isolated pure-scattering spectral line formed in homologously expanding ejecta in which the Sobolev optical depth depends on velocity as 1 n S D 0.v=8000 km s / . The values adopted for 0 and n are indicated in each panel. In each case, the blue line shows the complete line profile; the black line shows the attenuated continuum spectrum; and the gray-shaded region indicates the radiation contributed by scattering into the line of sight. All calculations were made using an idealized “core-halo” approximation (see Sect. 5.1) in which a geometrically thin blackbody photosphere is assumed to be located at 8000 km s1. The blueshift velocity corresponding to this imposed photosphere is indicated by the vertical line in each panel. Note that for all the cases shown with modest optical depth at the photosphere (0 D 2), the deepest absorption occurs close to the photospheric velocity, but there is a notable offset for the higher optical depth (0 D 200) case. Calculations were made using the TARDIS code (Kerzendorf and Sim 2014).

4.2 Recombination Emission

Although the characteristic profile with blueshifted absorption and redshifted emission is common to many classes of SNe, emission components can be enhanced by additional processes (Sect. 3.2). Particularly notable is the role of recombination line emission in hydrogen-rich SNe: the emission component of, e.g., H˛ in SNe IIP often significantly outweighs the absorption (see, e.g., Kirshner and Kwan 1975). Modeling of this recombination is thus necessarily included in theoretical work focused on the quantitative study of SNe IIP, and this is one of the reasons why codes 784 S.A. Sim that adopt a scattering-dominated approach to line profile formation (e.g., Fisher 2000; Kerzendorf and Sim 2014; Mazzali 2000) cannot be quantitatively applied to the modeling of H profiles in Type II spectra. Although a complete discussion is beyond the scope of this article, we note that recombination emission is also key to understanding of Type IIn spectra: these objects form a very diverse class, but have in common early-phase spectra that contain emission line profiles with narrow cores (formed in circumstellar material) and broad bases (shaped by electron scattering; Chugai 2001; Dessart et al. 2009, see Sect. 3.2).

5 Widely Used Approximations in Spectral Modeling

As noted in Sect. 2, the most sophisticated modeling is done with radiative transfer codes that simulate thermalization and scattering in detail and that incorporate non- LTE physics in the calculation of atomic level populations for a wide range of ions. In such calculations, artiﬁcial photospheric boundary conditions are avoided, and a self-consistent ultraviolet-to-infrared spectrum can be produced. Clearly, such high-quality calculations are important for precision work. However, these complicated approaches have the drawback of being relatively computationally expensive. Consequently, it remains the case that many studies continue to employ signiﬁcant approximations. We have already discussed the Sobolev approximation for line opacity in Sect. 4.1.2. In the subsections below, we comment on several more approximations that are used, their utility, and their drawbacks.

5.1 The “Core-Halo” Approximation

The most basic picture of the photospheric phase (Fig. 5) lends itself to a simple model in which the SN ejecta are divided into an optically thick “core” surrounded by a “halo” of expanding ejecta in which the spectral features are formed. A major approximation used by several of the most flexible spectral modeling codes is to assume that the “core” region effectively emits as a continuum source (usually assumed to be a black body) and that the spectrum formation problem can be addressed by modeling the transport of this radiation through the “halo.” Although crude, this has practical advantages over more sophisticated methods that avoid this division. Most importantly, it allows for the development of codes that are sufficiently computationally inexpensive to explore the large parameter space of compositions (and density/degree of ionization) that may be needed to match an observation. This approach also easily allows for modelling in which the luminosity can be directly inputted (and, e.g., fixed to match observations) without the need to simulate photon diffusion from high optical depths. Such utility, however, does come at the cost of having an oversimplified description for the continuum formation and a relatively poor model for the overall SED, particularly at long wavelengths. For example, as discussed for SNe Ia by 31 Spectra of Supernovae During the Photospheric Phase 785

Mazzali (2000, see also Sauer et al. 2006), although adopting a simple photospheric boundary condition can provide a good match at blue wavelengths, it can lead to increasingly poor agreement in the red part of the optical spectrum (and beyond) owing to the low thermalization opacities (see Sect. 3.1). This limitation must always be borne in mind when modeling SN spectra using a “core-halo” approximation, with particular caution applied to interpreting the full SED shape.

5.2 Spherical Symmetry

It is possible to incorporate full 2D/3D geometries in radiative transfer calculations for SNe (see, e.g., Dessart and Hillier 2011; Hauschildt and Baron 2010; Höflich et al. 2006; Kasen et al. 2006; Kromer and Sim 2009; Wollaeger et al. 2013). Such studies have shown that global departures from spherical symmetry can lead to spectra whose properties vary significantly with observer orientation (effects can include changes in individual line profile shapes and widths, but also overall SED color due to, e.g., altered line blocking). Departures from symmetry on relatively short length scales (“clumping”) can also affect the strength of spectral features, if sufficiently strong (e.g., Chugai and Utrobin 2014; Thomas et al. 2002). It would be very challenging to fully explore the role of departures from spherical symmetry when empirically modeling observed spectra: the substantial increase in the size (and degeneracy) of the parameter-space to be explored would be prohibitive for all but the simplest of modeling approaches. Consequently, it remains a common practice in many analyses to assume that the ejecta are smooth and spherically symmetric. This approach can be motivated by the generally low levels of linear polarization generally observed in the majority of SN explosion (specifically SNe IIP and SNe Ia) and, empirically, by the success of 1D models in quantitatively matching observed photospheric spectra. However, particularly for SNe Ib/c, there is observational evidence in favor of large-scale departures from spherical symmetric (Maund et al. 2007; Tanaka et al. 2008). Aside from modeling spectropolarimetry, the main application of multidimensional radiative transfer calculations for SNe at photospheric phases is in making predictions for hydrodynamical explosion models. Particularly for SNe Ia, there is now a range of explosion scenarios for which the explosion (and associated nucleosynthesis) has been simulated in 2D or 3D (see, e.g., Röpke et al. 2011). Most such models predict departures from spherical symmetry, although the nature and consequence of these asymmetries can vary substantially. Synthetic spectra obtained from multidimensional radiative transfer calculations therefore have a role in establishing the validity of proposed models and distinguishing between them.

5.3 Time Independence

Many of the spectral modeling methods applied to SNe have their origin in the study of steady-state stellar atmospheres. However, in expanding SN ejecta, a steady-state 786 S.A. Sim approximation is not exact, and the role of time dependence in the formation of SN spectra has been considered. As explained by Hillier and Dessart (2012), time dependence affects SN spectral synthesis in two ways: time dependence (i) in the radiation field and (ii) in the rate equations governing excitation and ionization. Time dependence in radiation transport has been incorporated in several of the current generation of radiative transfer codes, including (Hillier and Dessart 2012; Jack et al. 2009; Kasen et al. 2006; Kromer and Sim 2009). It is particularly easy to incorporate in codes that use Monte Carlo methods (Lucy 2005), where it introduces no substantial complications. Indeed, as argued by Lucy (2005), explicitly accounting for the finite speed of light actually simplifies matters since it can help alleviate efficiency issues associated with Monte Carlo techniques at high optical depths. Although this time dependence is clearly critical to the modeling of light curves, Kasen et al. (2006) show that the direct influence on the strength and shape of spectral features is SNe Ia models that are relatively modest: i.e., spectra obtained from“snapshot” calculations (in which photons do not diffuse in time) are remarkably similar to those obtained from full calculations. However, as discussed by Hillier and Dessart (2012), the effects are larger in SNe with more massive ejecta (e.g., SNe IIP). The role of time dependence in the rate equations is also significant, particularly for the modeling of SNe IIP (Dessart and Hillier 2008; Utrobin and Chugai 2005). Specifically, as discussed by Dessart and Hillier (2008), the effective recombination time for both H and He in the SN envelope may become comparable to the flow timescale, meaning that time-dependent terms will become significant in the rate equations. This influences the modeling of key spectral lines, including H˛ across a wide range of epochs, and can be critical to explain the strength of H˛ at relatively late epochs (see discussion by Dessart and Hillier 2008).

6 Some Applications of Spectral Modeling

Over the decades of observation of photospheric phase spectra for SNe, there have been many different studies that utilize a variety of models for spectrum formation to interpret data. To overview all studies that have been made is far beyond the scope of this article. Instead, we will only attempt to highlight some of the common approaches that are currently used and provide references for further reading.

6.1 Line Identiﬁcation

The ﬁrst, and arguably most important, information that can be inferred from the analysis of spectra is qualitative determination of which ions are present in the SN ejecta. Owing to the large velocities and associated blending of features (see Sect. 3.1), this line-identiﬁcation problem can be challenging, particularly for metal- rich SNe I, and often relies on a combination of experience and use of spectral modeling tools. 31 Spectra of Supernovae During the Photospheric Phase 787

The SYNOW / SYN++ code (Fisher 2000; Thomas et al. 2011) is widely used for line identification in most classes of SNe. This code combines a core-halo approximation (Sect. 5.1) with the Sobolev approximation (Sect. 4.1.2) to calculate the strength and shape of line features across the spectrum for comparison to observations. The code does not impose any particular ionization approximation; it allows the user to vary the contribution of each ion to the spectrum in order to find a good match. This empirical approach gives maximum flexibility when searching for consistent line identifications and determining an approximate photospheric velocity (or identifying features associated with ions that are detached: i.e., appear to be present only at range of velocities that does not extend down to the photosphere). This approach has been widely used in the interpretation, and empirical subclassification, of SNe Ia (see, e.g., Branch et al. 2005, 2009, and references therein), in the characterization of SNe Ibc (e.g., Millard et al. 1999) and, more recently, in the emerging class of superluminous SNe (e.g., Nicholl et al. 2014). The major advantage of this class of spectral modeling is the ease with which calculations can be carried out (runtimes of seconds), which makes it practical to use algorithms to systematically explore parameter spaces (Thomas et al. 2011). However, the lack of a self-consistent treatment of ionization (or thermal properties) mean that important physical quantities (such as absolute elemental abundances or densities) cannot be easily extracted and considerable care must be taken if information beyond a qualitative estimate of composition (and/or velocity) is required.

6.2 Empirical Determination of Composition

To quantitatively infer elemental composition, it is necessary to use more sophisticated methods that estimate self-consistent ionization/excitation states in the ejecta. A variety of approaches can be used, but for semiempirical modeling, codes that retain a “core-halo” photospheric boundary condition (see, e.g., Mazzali 2000, and similar codes) have proved effective. Their modest computational expense makes it relatively easy to search for good spectral ﬁts despite the need to explore large parameter spaces. In particular, while early semiempirical modeling typically considered modeling single spectra using single-composition ejecta models, modern studies attempt to develop semiempirical stratiﬁed models that match time-series of spectra (Sasdelli et al. 2014; Stehle et al. 2005, and references therein). The results of such analyses place constraints on the required elemental yields, velocities, and degrees of (effective) mixing in metal-rich SNe and thus inform the development of explosion/progenitor theories. As one example, Fig. 7 shows a comparison of the observed spectrum of the peculiar Type I supernova SN 2015H to a simple spectral model developed by Magee et al. (2016)usingtheTARDIS radiative transfer code (Kerzendorf and Sim 2014). This illustrates that good matches to data can be achieved with relatively simple methods. It also highlights the complexity of spectrum formation in metal- 788 S.A. Sim

Fig. 7 Spectral modeling of the peculiar supernova SN 2015H. The lower panel shows a comparison of the model spectrum (blue) to the observed spectrum (black) at an epoch of 6 days after peak brightness. The model is based on a power-law density profile with elemental abundances varied to match the data (see Magee et al. 2016). The upper panel illustrates how the synthetic calculation can be used to understand the formation of the spectral features. The color coding in the positive region (above white dashed line) indicates which elements are responsible for the last interaction of escaping photons in the simulation (the area under the spectrum is shaded proportional to the contribution of each element; elements are identified in the color bar). The colors below the white line indicate which elements are responsible for absorbing (or scattering) photons out of each wavelength bin. Black-shaded regions indicate contributions to the spectrum that have emerged from the inner boundary of the calculation without interaction; gray indicates regions where only electron scattering has occurred (Figure from Magee et al., Astronomy & Astrophysics, 589, A89. DOI: 10.1051/0004-6361/20152836, 2016, reproduced with permission ©ESO) rich supernovae: the iron-group elements blanket most of the spectrum and few regions are affected by only one species. However, it is recognized that there will always be some level of degeneracy in empirical spectral modeling and limitations imposed by approximations made, which can be difficult to quantify. For example, it is common to impose a specific density profile for the SN ejecta that is based on some known explosion model (or else to adopt a simple analytic form). Also, in the interests of efficiency, semiempirical work often makes use of approximate non-LTE ionization schemes (e.g., Mazzali and Lucy 1993) which can be effective but require that care is taken in the description of, e.g., the ionizing far-UV flux (Pauldrach et al. 1996). In addition, 31 Spectra of Supernovae During the Photospheric Phase 789 whenever a sharp photospheric boundary condition is used, the usual caveats for quantitative interpretation (Sect. 5.1) apply.

6.3 Synthetic Spectra for Explosion Models

Although the approaches described above can be guided by theoretical explosion models (e.g., by adopting an ejecta density profile or constraining relative elemental yields), their focus is typically on identifying what is required of an explosion in order to match the data. A complementary approach is to start from specific explosion models (provided, e.g., by hydrodynamical simulations) and calculate synthetic observables. In this case, the radiative transfer calculation, in principle, introduces no additional free parameters (i.e., the input model specifies the full density/composition structure) or degeneracies that need to be explored. This is often the style of modeling to which the most sophisticated radiative transfer codes are applied (e.g., Baron et al. 2006; Dessart and Hillier 2005b, 2010; Höflich et al. 1998; Kasen et al. 2006; Kromer and Sim 2009; Sauer et al. 2006). Such calculations yield synthetic spectra, light curves, and/or spectropolarimetry that can be compared to data to assess the effectiveness of the model. Fig. 8 shows example comparisons of synthetic spectra and spectropolarimetry for a 3D white-dwarf thermonuclear explosion model to observations of a normal SN Ia. In this case it can be seen that the overall agreement in the shape of the spectra and the degree of polarization are relatively good, providing support for the model. However, there is also clear disagreement in the shapes and strengths of notable features, such as Si II 6355 and the Ca II infrared triplet (8567 Å). These

3.0 3.0 SN 2001el −7d SN 2001el −2d SN 2001el +7 d

λ − − 2.0 N100-DDT n5 7d N100-DDT n5 2d N100-DDT n5 +7 d 2.0

1.0 1.0 Scaled F

0.0 0.0

0.6 0.6 . .

(%) 0 4 0 4 P 0.2 0.2

0.0 0.0 5000 6000 7000 8000 5000 6000 7000 8000 5000 6000 7000 8000 Wavelength (A)˚ Wavelength (A)˚ Wavelength (A)˚

Fig. 8 Synthetic spectra (upper panels, red) and degree of polarization (lower panels, red) computed for a selected line of sight in the N100-DDT white-dwarf thermonuclear explosion model of Seitenzahl et al. (2013) for three epochs (7 , 2,andC7 days relative to maximum light, left to right). For comparison, observations of the normal SN Ia SN 2001el are shown in black (data from Wang et al. 2003). The error bars drawn in the lower panels indicate the scale of Monte Carlo noise in the simulated spectropolarimetry (see Bulla et al. 2016 for details) (Figure reproduced from Bulla et al. 2016, Monthly Notices of the Royal Astronomical Society, 462, 1039. DOI: 10.1093/mnras/stuw1733) 790 S.A. Sim discrepancies can be used to identify shortcomings of the underlying model and motivate refinement and/or development of alternative explosion simulations. Studies of this type have now been carried out for models of most major classes of SN and provide a powerful means to quantitatively evaluate the successes and failures of explosion models. Analysis of these full simulations also provides the best insight to understand the interplay among physical processes and help refine approximations that can be adopted in more simplified codes. Nevertheless, the substantial computational expense for full explosion model simulations means that such calculations are still relatively rare and it is expensive to systematically “tweak” models to quantitatively match observations. Furthermore, even the most sophisticated codes still make a number of assumptions: currently, the majority of studies made either neglect (or at least significantly simplify) the treatment of non- LTE effects or else are limited to studies of spherically symmetric models.

6.4 Distance Determination

Throughout the history of the modeling and interpretation of SN spectra, there has been continued interest in the use of Type IIP SNe as distance indicators (e.g., Baron et al. 2004; Dessart and Hillier 2006; Eastman et al. 1996; Kirshner and Kwan 1974). Several distinct approaches have been suggested, but the underlying principle of several is that if a physical model for the SN spectrum can be developed, that model will predict the true luminosity, which can then be compared to the observed flux to infer distance. The methods used have been gradually refined, and it has been shown that distances accurate to 10% can be achieved, provided that sufficiently high- quality, detailed spectral models are developed for good quality data sets (Dessart and Hillier 2006). Thus, with modern spectral modeling, the prospects are now good for accurate distance measurements from samples of SNe II.

7 Conclusions

Despite their complexity, the observation and analysis of photospheric-phase spectra remains central to the study of supernovae: such data are relatively easy to obtain (compared to e.g., late-phase spectra, when the supernova is much fainter, or to polarimetry, for which very high signal to noise is needed) and are much more informative than photometry alone. Thus continued advancement in our quantitative ability to interpret and model photospheric phase spectra is essential. In this short article, we have only scratched the surface of this topic and must direct the reader to the references given for more complete, quantitative discussions. We hope, however, to have given the reader a ﬂavor of both the physics at work in the formation of photospheric-phase spectra and how its study can be used to help unravel the mysteries of supernova explosions. 31 Spectra of Supernovae During the Photospheric Phase 791

8 Further Reading

Further background information on the physical processes mentioned in Sect. 3 are given, for example, in the textbooks by Rybiki and Lightman (1985) and Tennyson (2011). Hubeny and Mihalis (2015) provide a substantial introduction to the theory of spectral formation, including for expanding atmospheres (see also Lamers and Casinelli 1999). For details of particular applications, and the radiative transfer codes/techniques currently used, we refer the reader to the references given in Sects. 5 and 6.

9 Cross-References

Introduction to Supernova Polarimetry Observational and Physical Classiﬁcation of Supernovae Spectra of Supernovae in the Nebular Phase

References

Auer LH, van Blerkom D (1972) Electron scattering in spherically expanding envelopes. ApJ 178:175 Baron E, Bongard S, Branch D, Hauschildt PH (2006) Spectral modeling of SNe Ia near maximum light: probing the characteristics of hydrodynamical models. ApJ 645:480 Baron E, Hauschildt PH, Nugent P, Branch D (1996) Non-local thermodynamic equilibrium effects in modelling of supernovae near maximum light. MNRAS 283:297 Baron E, Nugent PE, Branch D, Hauschildt PH (2004) Type IIP supernovae as cosmological probes: a spectral-ﬁtting expanding atmosphere model distance to SN 1999em. ApJL 616:L91 Blinnikov SI, Eastman R, Bartunov OS, Popolitov VA, Woosley SE (1998) A comparative modeling of supernova 1993J ApJ 496:454 Branch D, Baron E, Hall N, Melakayil M, Parrent J (2005) Comparative direct analysis of type Ia supernova spectra. I. SN 1994D. PASP 117:545 Branch D, Chau Dang L, Baron E (2009) Comparative direct analysis of type Ia supernova spectra. V. Insights from a larger sample and quantitative subclassiﬁcation. PASP 121:238 Bulla M, Sim SA, Kromer M, Seitenzahl IR, Fink M, Ciaraldi-Schoolmann F et al (2016) Predicting polarization signatures for double-detonation and delayed-detonation models of Type Ia supernovae. MNRAS 462:1039 Chugai NN (2001) Broad emission lines from the opaque electron-scattering environment of SN 1998S. MNRAS 326:1448 Chugai NN, Utrobin VP (2014) Disparity between H ˛ and H ˇ in SN 2008 in: inhomogeneous external layers of type IIP supernovae? Astron Lett 40:111 Dessart L, Hillier DJ (2005a) Distance determinations using type II supernovae and the expanding photosphere method. A&A 439:671 Dessart L, Hillier DJ (2005b) Quantitative spectroscopy of photospheric-phase type II supernovae. A&A 437:667 Dessart L, Hillier DJ (2006) Quantitative spectroscopic analysis of and distance to SN 1999em. A&A 447:691 792 S.A. Sim

Dessart L, Hillier DJ (2008) Time-dependent effects in photospheric-phase type II supernova spectra. MNRAS 383:57 Dessart L, Hillier DJ (2010) Supernova radiative-transfer modelling: a new approach using non- local thermodynamic equilibrium and full time dependence. MNRAS 405:2141 Dessart L, Hillier DJ (2011) Synthetic line and continuum linear-polarization signatures of axisymmetric type II supernova ejecta. MNRAS 415:3497 Dessart L, Hillier DJ, Gezari S, Basa S, Matheson T (2009) SN 1994W: an interacting supernova or two interacting shells? MNRAS 394:21 Eastman RG, Pinto PA (1993) Spectrum formation in supernovae – numerical techniques. ApJ 412:731 Eastman RG, Schmidt BP, Kirshner R (1996) The atmospheres of type II supernovae and the expanding photosphere method. ApJ 466:911 Fisher A (2000) Direct analysis of Type Ia supernovae spectra. PhD thesis. University of Oklahoma Hauschildt PH, Baron E (1999) Numerical solution of the expanding stellar atmosphere problem. J Comput Appl Math 109:41 Hauschildt PH, Baron E (2010) A 3D radiative transfer framework. VI. PHOENIX/3D example applications. A&A 509:A36 Hillier DJ (1991) The effects of electron scattering and wind clumping for early emission line stars. A&A 247:455 Hillier DJ, Dessart L (2012) Time-dependent radiative transfer calculations for supernovae. MNRAS 424:252 Höflich P, Gerardy CL, Marion H, Quimby R (2006) Signatures of isotopes in thermonuclear supernovae. New Astron Rev 50:470 Höflich P, Wheeler JC, Thielemann FK (1998) Type Ia supernovae: influence of the initial composition on the nucleosynthesis, light curves, and spectra and consequences for the determination of ˝M and . ApJ 495:617 Hubeny I, Mihalis D (2015) Theory of stellar atmospheres. Princeton University Press, Princeton Jack D, Hauschildt PH, Baron E (2009) Time-dependent radiative transfer with PHOENIX. A&A 502:1043 Jeffery DJ (1995a) An expansion opacity formalism for the Sobolev method. A&A 299:770 Jeffery DJ (1995b) The Sobolev optical depth for time-dependent relativistic systems. ApJ 440:810 Karp AH, Lasher G, Chan KL, Salpeter EE (1977) The opacity of expanding media – the effect of spectral lines. ApJ 214:161 Kasen D (2006) Secondary maximum in the near-infrared light curves of type Ia supernovae. ApJ 649:939 Kasen D, Thomas RC, Nugent P (2006) Time-dependent Monte Carlo radiative transfer calculations for three-dimensional supernova spectra, light curves, and polarization. ApJ 651:366 Kerzendorf WE, Sim SA (2014) A spectral synthesis code for rapid modelling of supernovae. MNRAS 440:387 Kirshner RP, Kwan J (1974) Distances to extragalactic supernovae. ApJ 193:27 Kirshner RP, Kwan J (1975) The envelopes of type II supernovae. ApJ 197:415 Kromer M, Sim SA (2009) Time-dependent three-dimensional spectrum synthesis for type Ia supernovae. MNRAS 398:1809 Lamers HJGLM, Casinelli JP (1999) Introduction to stellar winds. Cambridge University Press, Cambridge Lucy LB (1999) Improved Monte Carlo techniques for the spectral synthesis of supernovae. A&A 345:211 Lucy LB (2005) Monte Carlo techniques for time-dependent radiative transfer in 3-D supernovae. A&A 429:19 Magee MR, Kotak R, Sim SA, Kromer M, Rabinowitz D, Smartt SJ et al (2016) The type Iax supernova, SN 2015H. A white dwarf deflagration candidate. A&A 589:A89 Maund JR, Wheeler JC, Patat F, Baade D, Wang L, Höflich P (2007) Spectropolarimetry of the type Ib/c SN 2005bf. MNRAS 381:201 31 Spectra of Supernovae During the Photospheric Phase 793

Mazzali PA (2000) Applications of an improved Monte Carlo code to the synthesis of early-time supernova spectra. A&A 363:705 Mazzali PA, Lucy LB (1993) The application of Monte Carlo methods to the synthesis of early- time supernovae spectra. A&A 279:447 Millard J, Branch D, Baron E, Hatano K, Fisher A, Filippenko AV, Kirshner RP, Challis PM, Fransson C, Panagia N, Phillips MM, Sonneborn G, Suntzeff NB, Wagoner RV, Wheeler JC (1999) Direct analysis of spectra of the type IC supernova SN 1994I. ApJ 527:746 Nicholl M, Smartt SJ, Jerkstrand A, Inserra C, Anderson JP, Baltay C et al (2014) Superluminous supernovae from PESSTO. MNRAS 444:2096 Nomoto K, Thielemann F-K, Yokoi K (1984) Accreting white dwarf models of type I supernovae. III – carbon deflagration supernovae. ApJ 286:644 Patat F, Baade D, Höflich P, Maund JR, Wang L, Wheeler JC (2009) VLT spectropolarimetry of the fast expanding type Ia SN 2006X. A&A 508:229 Pauldrach AWA, Duschinger M, Mazzali PA, Puls J, Lennon M, Miller DL (1996) NLTE models for synthetic spectra of type IA supernovae. The influence of line blocking. A&A 312:525 Pinto PA, Eastman RG (2000) The physics of type IA supernova light curves. II. Opacity and diffusion. ApJ 530:757 Röpke FK, Seitenzahl IR, Benitez S, Fink M, Pakmor R, Kromer M, Sim SA, Ciaraldi-Schoolmann F, Hillebrandt W (2011) Modeling type Ia supernova explosions. Prog Part Nucl Phys 66:309 Rybiki GB, Lightman AP (1985) Radiative processes in astrophysics. Wiley, Weinheim Sasdelli M, Mazzali PA, Pian E, Nomoto K, Hachinger S, Cappellaro E, Benetti S (2014) Abundance stratification in type Ia supernovae – IV. The luminous, peculiar SN 1991T. MNRAS 445:711 Sauer DN, Hoffmann TL, Pauldrach AWA (2006) Non-LTE models for synthetic spectra of type Ia supernovae/hot stars with extremely extended atmospheres. II. Improved lower boundary conditions for the numerical solution of the radiative transfer. A&A 459:229 Seitenzahl IR, Ciaraldi-Schoolmann F, Röpke FK, Fink M, Hillebrandt W, Kromer M et al (2013) Three-dimensional delayed-detonation models with nucleosynthesis for type Ia supernovae. MNRAS 429:1156 Sobolev VV (1960) Moving envelopes of stars. Harvard University Press, Cambridge Stehle M, Mazzali PA, Benetti S, Hillebrandt W (2005) Abundance stratification in type Ia supernovae – I. The case of SN 2002bo. MNRAS 360:1231 Tanaka M, Kawabata KS, Maeda K, Hattori T, Nomoto K (2008) Optical spectropolarimetry and asphericity of the type Ic SN 2007gr. ApJ 689:1191 Tennyson J (2011) Atomic spectroscopy: an introduction to the atomic and molecular physics of astronomical spectra. World Scientific Publishing Co. Pte. Ltd, Hackensack Thomas RC, Kasen D, Branch D, Baron E (2002) Spectral consequences of deviation from spherical composition symmetry in type Ia supernovae. ApJ 567:1037 Thomas RC, Nugent PE, Meza JC (2011) SYNAPPS: data-driven analysis for supernova spectroscopy. PASP 123:237 Utrobin VP, Chugai NN (2005) Strong effects of time-dependent ionization in early SN 1987A. A&A 441:271 Wang L, Baade D, Höflich P, Kokhlov A, Wheeler JC, Kasen D et al (2003) Spectropolarimetry of SN 2001el in NGC 1448: asphericity of a normal type Ia supernova. ApJ 591:1110 Wollaeger RT, van Rossum DR, Graziani C, Couch SM, Jordan IV GC, Lamb DQ, Moses GA (2013) Radiation transport for explosive outflows: a multigroup hybrid Monte Carlo method. ApJS 209:36