A Global Model of Type II-Plateau Supernovae

A Global Model of The Light Curves and Expansion Velocities of Type II-Plateau Supernovae

Abstract

We present a new self-consistent and versatile method that derives photospheric radius and temperature variations of Type II-Plateau supernovae based on their expansion velocities and photometric measurements. We apply the method to a sample of well-observed, nearby supernovae with published light curves and velocities. We simultaneously fit velocity and magnitude measurements distributed over photometric passbands spanning wavelengths from to m. The light curve differences among the Type II-Plateau supernovae are well-modeled by assuming different rates of photospheric radius expansion, which we explain as different density profiles of the ejecta and we argue that steeper density profiles result in flatter plateaus, if everything else remains unchanged. The steep luminosity decline of Type II-Linear supernovae is due to fast evolution of the photospheric temperature, which we verify with a successful fit of SN1980K. Eliminating the need for theoretical supernova atmosphere models, we obtain self-consistent relative distances, reddenings, and nickel masses fully accounting for all internal model uncertainties and covariances. We use our global fit to estimate the time evolution of any missing band tailored specifically for each supernova and we construct spectral energy distributions and bolometric light curves. We produce bolometric corrections for all filter combinations in our sample. We compare our model to the theoretical dilution factors and find good agreement for the and filters. Our results differ from the theory when the , , , or bands are included. We investigate the reddening law towards our supernovae and find reasonable agreement with standard reddening law in bands. Results for other bands are inconclusive. We make our fitting code publicly available.

Methods: statistical — stars: distances — supernovae: general

1. Introduction

Core-collapse supernovae announce the death of (at least some) stars with initial masses . The gravitational collapse of the Chandrasekhar-mass iron or oxygen-neon-magnesium core of these massive stars rebounds when the strong nuclear interaction causes a stiffening of the equation of state. A shock wave then propagates outwards into the infalling matter, but simulations show that it halts its progress and turns into a quasi-static accretion shock. With the aid of neutrinos emanating from the nascent proto-neutron star and some additional poorly understood mechanism, the shock starts moving out again and produces a supernova explosion (e.g. Janka, 2012; Ugliano et al., 2012; Burrows, 2013). Although exciting, the prospects of directly constraining the supernova explosion mechanism through observations of neutrino emission, gravitational waves or their combination (e.g. Ott et al., 2004, 2012; Yüksel & Beacom, 2007; Müller & Janka, 2014) are extremely uncertain, because the supernova must explode nearby. Instead, constraints on the explosion mechanism can be obtained by trying to understand observed patterns in the explosions, their remnants, and their progenitors.

As the reinvigorated shock propagates through the progenitor, it heats up the swept up material. When the temperatures are higher than about  K, heavy elements such as Ni are produced (e.g. Weaver & Woosley, 1980; Woosley, 1988; Thielemann et al., 1990). For lower temperatures, only lighter elements are synthesized and still lower temperatures cause only ionization of the matter. After the short initial luminosity spike when the supernova shock breaks out of the surface of the star (e.g. Ensman & Burrows, 1992; Chevalier & Fransson, 2008; Soderberg et al., 2008; Katz et al., 2010; Nakar & Sari, 2010; Tominaga et al., 2011), the luminosity increases as the surface area of the ejecta expands. Simultaneously, the effective temperature decreases. If the progenitor star had a substantial hydrogen envelope, the ejecta are optically-thick and the combination of homologous expansion and the decreasing effective temperature results in a phase of relatively constant optical brightness; we observe a Type II-P supernova. When the ejecta becomes optically-thin, the brightness drops and the luminosity evolution becomes dominated by the deposition of energy of the decaying radioactive nuclei. Specifically, the normalization of the exponentialy decreasing luminosity is related to the total mass of Ni that was synthesized during the explosion.

Theoretically, the supernova explosion energy combines the contributions from the binding energy of the progenitor, the neutrino-driven wind, recombination of the dissociated nuclei, and exothermic nuclear burning, all of which happen predominantly deep in the progenitor close to the proto-neutron star (e.g. Scheck et al., 2006; Ugliano et al., 2012). Supernova explosion energies are estimated by comparing the observed fluxes and spectra to theoretical models of expanding supernova atmospheres with assumptions on the progenitor properties and the explosion remnant (e.g. Arnett, 1980; Litvinova & Nadezhin, 1983, 1985; Hamuy, 2003; Utrobin & Chugai, 2008; Kasen & Woosley, 2009). In recent years, a number of red supergiant progenitors to Type II-P supernovae have been identified and their initial masses have been estimated (e.g. Van Dyk et al., 2003, 2012a, 2012b; Smartt et al., 2004, 2009; Li et al., 2006; Mattila et al., 2008; Fraser et al., 2012; Maund et al., 2014). One surprising aspect of these discoveries is that there appears to be a lack of high-mass red supergiants exploding as Type II-P supernovae (Li et al., 2006; Kochanek et al., 2008; Smartt, 2009; Smartt et al., 2009). Although there are several explanations (Yoon & Cantiello, 2010; Walmswell & Eldridge, 2012; Kochanek et al., 2012; Groh et al., 2013), an exciting possibility is that some massive stars do not explode as ordinary supernovae but instead collapse to a black hole, potentially accompanied only by a weak transient (Nadezhin, 1980; Kochanek et al., 2008; Lovegrove & Woosley, 2013; Piro, 2013; Kochanek, 2014a, b; Horiuchi et al., 2014). The supernovae with estimates of progenitor mass, explosion energy and nickel mass can potentially offer strong constraints on the supernova explosion mechanism.

The Type II-P supernovae have also been used to measure distances using either the expanding photosphere method (e.g. Kirshner & Kwan, 1974; Schmidt et al., 1994a; Hamuy et al., 2001; Baron et al., 2004; Dessart & Hillier, 2005, 2006; Jones et al., 2009) or the “standardized candle” method (e.g. Hamuy & Pinto, 2002; Nugent et al., 2006; Poznanski et al., 2009, 2010; D’Andrea et al., 2010; Olivares E. et al., 2010). The expanding photosphere method uses theory-based “dilution” factors to transform the black-body flux of an observed color temperature derived from a given filter set to the total supernova flux. The theory necessary to calculate the dilution factors includes line blanketing and non-LTE effects in an expanding medium. The distance is estimated by comparing the apparent angular radius of the supernova with expansion velocities under the assumption of homologous expansion. The standardized candle method employs a correlation between the supernova luminosity and the measured expansion velocity to infer the distances. While they are still not competitive with Type Ia supernovae for cosmology, Type II-P SNe offer an independent and promising method for estimating distances with different astrophysics and systematics.

In practice, the extraction of supernova distances, explosion energies, nickel yields and progenitor masses from observations is not straightforward. Most of all, the light from the supernova or the progenitor is extinguished by gas and dust around the star, in its host galaxy and in our Galaxy. The extinction can be estimated from the maps of galactic dust (Schlegel et al., 1998), the spectral lines of the intervening gas imprinted on the supernova spectrum (e.g. Munari & Zwitter, 1997; Poznanski et al., 2011, 2012), by comparing the supernova colors to well-observed template supernovae (Olivares E. et al., 2010), or by modelling of the supernova spectra in the well-defined continuum windows (Dessart & Hillier, 2006; Dessart et al., 2008; Baron et al., 2007). All of these recipes provide potentially biased results for the explosion progenitor masses, explosion energies and nickel mass from the heterogeneous data of different supernovae. Furthermore, Faran et al. (2014) tested several commonly used methods for dust-extinction correction and argued that none of them increase the uniformity of the sample.

To expand on our previous effort to understand the supernova progenitors, the explosion mechanism and its observational signatures (Prieto et al., 2008a, b, c, 2012, 2013; Pejcha & Thompson, 2012, 2014; Pejcha et al., 2012a, b) and to address the uncertainties and inhomogeneities in determining Type II-P supernova parameters, we present a new method to fit the multi-band light curves and expansion velocities of Type II-P supernovae. Our method combines the founding principles of expanding photosphere method with the generalization of the Baade-Wesselink approach in Cepheids (Pejcha & Kochanek, 2012) to decompose the observed multi-band light curves and expansion velocities of Type II-P supernovae into radius and temperature variations. The changes in the photospheric radius are constrained by the expansion velocities under the assumption of homologous expansion and affect all observed photometric bands in the same achromatic manner. We assume that the changes in the spectral energy distribution of supernovae are driven by variations in a single underlying parameter, the temperature. This represents the chromatic part of the light curve, which enters differently for each photometric band. The parameters necessary to project the observed light curves and velocities on radius and temperature changes are obtained in the fitting process. Our method does not require any input from theoretical supernova atmosphere models. In fact, and importantly, our results can constrain these models. Furthermore, we fit each photometric band independently, which assures that supernova observed in any combination of our photometric bands can be successfully fitted and its parameters estimated.

In this paper, we present the model and the first findings based on the limited set of publicly-available data. In Section 2, we present the basic equations of the model, identify the degeneracies and design priors to ensure reasonable fits, describe the available data and the fitting methods, and discuss the relation to previous methods. In Section 3, we present the fits to the light curves along with the range of applicability of the model, discuss the morphology of the light curves, and present the distance estimates to individual supernovae. We also construct spectral energy distributions, bolometric light curves and corrections, and estimate nickel masses of individual supernovae. We compare our results to the theoretical supernova atmosphere models by calculating the dilution factors. In Section 4, we discuss the reddening law and constraints from our model. We also describe possible extensions to our model. In Section 5, we summarize our findings. In the Appendix, we describe our implementation of priors on the model.

2. Model

We model the supernova magnitude in a photometric band at a time

 mi(t)=¯¯¯¯¯¯Mi+μj+RiE(B−V)−2.5Π(t)−2.5Θi(t), (1)

where is the absolute magnitude in band of the object at a distance of  pc, is the distance modulus of the host galaxy , is the ratio of total to selective extinction, is the total reddening, represents changes in the luminosity due to changes in the surface area at constant temperature and other achromatic luminosity changes, and describes the chromatic magnitude changes in band due to temperature changes. In this work, we assign a single value of to each supernova and assume that there is a universal extinction law , which we discuss in Section 4.1. We also emphasize that our is the total reddening that includes the Galaxy and the supernova host galaxy contributions. We do not pre-correct the observations for Galaxy reddening, because this could potentially introduce additional unknown systematical errors in our analysis and we are interested in the total photometric reddening for our supernovae.

To apply Equation (1) in the supernova context, we assume that after the moment of explosion , the optical emission comes predominantly from a photosphere in the optically-thick expanding medium with a relatively well-defined radius and temperature , the “plateau”. We assume that the expanding medium is homologous so that is related to the spectroscopically measured expansion velocity as

 R=v(t−t0), (2)

where we are neglecting the initial radius of the progenitor star. Here, is the zero-point time for each supernova, which might be systematically offset from the true moment of explosion due to missing physics in our model. We assume that the line used to infer expansion velocity is directly related to the photosphere so that Equation (2) is valid. Expansion velocities are modeled as

 v=ω0(t−t0)ω1+ω2, (3)

where , , are parameters. For in km sand in days, the reference magnitude corresponds to an object with  cm (). We parameterize the change in the spectral energy distribution as the supernova evolves using a parameter . Motivated by observational studies of supernovae during this phase, we model the time evolution of as a linear decay

 τthick=α0(t−t0)+α1. (4)

Over time , the plateau smoothly transitions to an optically-thin phase, where the optical luminosity comes from the deposition of energy from the decay of radioactive nuclei. Motivated by observations (e.g. Hamuy et al., 2001), we model this part of the evolution as an achromatic exponential luminosity decay with constant in time. Improvements in the treatment of the exponential decay will be a subject of future work, but in this paper, we focus predominantly on the optically-thick plateau phase.

We combine the photospheric phase and the exponential decay phase as

 Π = log10{(1−w)R2+w10γ0(t−t0−tP)+γ1}, (5) τ = (1−w)τthick+wτthin, (6)

where and parameterize the exponential decay, is the duration of the photospheric phase, and is the weight function

 w=[1+exp(t−t0−tPtw)]−1, (7)

which we choose after Olivares E. et al. (2010) to resemble the typical transition phase in Type II-P supernovae. Equation (7) smoothly connects the photospheric and exponential decay phases, which allows for the convergence of the fitting routine. Equation (7) implies that is defined at the midpoint between the end of the plateau and the beginning of the exponential luminosity decay, similarly to Hamuy (2003). This is somewhat different from other recent studies that measure the duration of the photospheric phase at the end of the plateau (Anderson et al., 2014a; Sanders et al., 2014).

To calculate the magnitude at any given time, we need to know how temperature changes affect individual photometric bands through . We model the temperature dependence of as a low-order polynomial in

 Θi=3∑n=11n!βn,iτn (8)

with a matrix of coefficients . In principle, depends also on metallicity and other parameters, however, since the will be obtained by fitting the data, these additional dependencies will matter only when comparing fits of individual objects. Metallicity variations between supernovae will be projected on other parameters such a , , and , making the metallicity signature much smaller than one would expect (Pejcha & Kochanek, 2012). In our previous work on Cepheids (Pejcha & Kochanek, 2012), we found that the metallicity term is typically few percent of the . We believe that this is also the case for supernovae. We will apply our model to local low-redshift supernovae with good observational coverage. As a result, we do not include K-corrections. Changes to our model in this direction would be an obvious extension.

Finally, we want to address the meaning of the parameter that we have so far left unexplained. The spectral energy distribution changes primarily due to changes in the photospheric temperature, however, in the model presented here, is neither the effective nor the color temperature of the supernova. Instead, it is advantageous to think about as a parameter that describes changes in the spectral energy distribution and that its meaning is limited by the model presented above. We show later in Section 3.7 that is approximately linearly proportional to the logarithm of the effective temperature. Alternatively, we could have forced to be identical to the effective temperature by integrating the spectral energy distribution. However, this approach would mean that the definition of changes when a new band is added.

2.1. Degeneracies and priors

The parameters of the model are obtained by minimizing the master constraint

 H=∑all data(mobs−mσ)2+∑all data(vobs−v(1−w)−1σ)2+S, (9)

where and are observed magnitudes and expansion velocities with uncertainties , includes contributions from all priors, and the sums are over all supernovae, photometric bands, and measurements. We modify the uncertainties of the expansion velocities by a factor of to lower the fitting weight of velocity measurements during the time when the supernova ejecta is becoming transparent. We collectively denote the first two terms in Equation (9) as .

The model in Equation (9) is hierarchical in the sense that it includes parameters pertaining only to a single supernova, (, , , , , , , , , , , ), only to the host galaxies (), and globally to all data (, , ). Although we fit Equation (9) to a large amount of data, some of the model parameters cannot be constrained completely independently. We address these issues by fixing some parameters and adding priors. Equation (1) implies that the model magnitude remains unchanged if we move all objects by while simultaneously changing . We fix this degeneracy by fixing the distance modulus to M95, the host galaxy of SN2012aw, to  mag. This number is very close to the weighted average of  mag from the NASA Extragalactic Database (NED), which includes numerous distance estimates using Cepheids (e.g. Kochanek, 1997; Kelson et al., 2000; Freedman et al., 2001; Saha et al., 2006). A more statistically appropriate method would be to fix the distance scale by introducing a number of distance priors on individual galaxies with distance estimates with Cepheids or Type Ia supernovae or some other method. We do not take this road in order to make our model more transparent.

Another degeneracy comes from defining the reddening zeropoint, applying is equivalent to . We remove this degeneracy by fixing the total reddening of SN2012A to  mag (Tomasella et al., 2013). This reddening estimate is based on high-resolution spectroscopy of Na I D lines. The small value of reddening to this supernova guarantees that any potential absolute error in will be relatively small when propagated to other supernovae.

We fix the degeneracy between and by fixing for SN2012aw and for all supernovae. In principle, it would be sufficient to fix only for a single supernova, but we found that this causes artificial features in the model for supernovae with incomplete data around the transition. For the default calculation we fix to the reddening law of Cardelli et al. (1989) with . In principle, our model allows for independent determination of the reddening law on a filter-by-filter basis, which we study in Section 4.1.

We impose hard limits on the values of several parameters pertaining to individual supernovae. We constrain to be at least  days before the first observation of every supernova to prevent singularities when calculating . We require that so that the photometric radius is increasing with time (Eq. [2]). We also require that to prevent negative expansion velocities and to get positive reddenings. If we removed the constraint for the two supernovae with fitted , the resulting would be negative only by few thousands of a magnitude.

Finally, we apply a number of priors on individual supernova parameters to aid fitting in cases when there are gaps in photometric coverage or small number or complete lack of expansion velocities. We build these priors based on well-observed supernovae and we take into account any potential correlations between the parameters. We describe the implementation of the priors in the Appendix.

2.2. Data and fitting method

In order to constrain the parameters of our model (Eqs. [19]) we require as large a sample of measurements as possible. Our final sample includes supernovae in galaxies (there are three supernovae in NGC6946 in our sample). The measurement database consists of roughly photometric measurements distributed among photometric bands with central wavelengths between and m, and roughly expansion velocities measured using the Fe II line at  Å. This line is commonly used in the expanding photosphere method (e.g. Schmidt et al., 1994b) so that Equation (2) is valid. Adding velocities measured on other lines would not necessarily increase the time coverage, because they are often measured from the same spectra. The bulk of the photometric measurements are in , but we also include the Sloan bands, six Swift bands , , , and , the passband of the ROTSE telescope (SN2006bp; Quimby et al., 2007), band (SN1999em; Hamuy et al., 2001), and band (SN2009N; Takáts et al., 2014). The references we used for obtaining the data are given in Table 1, the photometric bands are summarized in Table 4, and the supernovae with corresponding galaxies are listed in Table 2. The master constraint of Equation (9) is minimized over parameters for each supernova minus the three fixed parameters discussed in Section 2.1, galaxy distances, and global parameters, which gives an overall total of parameters.

An improvement over our previous work in Pejcha & Kochanek (2012) is that we use the Levenberg-Marquardt technique in cmpfit2 (Moré, 1978; Markwardt, 2009) to solve the least-squares problem of Equation (9). cmpfit can limit each parameter to a specified range, which we use for some parameters (Sec. 2.1). The implementation of priors in cmpfit is described in the Appendix. The non-linear least-squares fitting technique requires partial derivatives of the model with respect to all parameters. In principle, these derivatives can be obtained analytically, but we found that the numerical derivatives calculated in cmpfit are more robust.

The complexity of our model precludes obtaining all parameter values from scratch. We started with a small subset of well-observed supernovae with some individual parameters fixed, first obtaining estimates of and . We then gradually increased the complexity of the model by introducing more photometric bands and the non-linear parts of the model ( and ), alternatively holding either the supernova individual parameters or global parameters fixed. However, in the end, we perform a global fit of all parameters together. This ensures that uncertainties of all parameters are fully coupled and we include covariances between the global parameters of the model and individual supernova parameters. Our default calculation has for degrees of freedom.

In this paper, we will discuss also quantities derived from the parameters such as the bolometric luminosities and nickel masses. Since our fit simultaneously adjusts global and individual parameters, collectively denoted as , we have the full covariance matrix of the problem including covariances between global and individual parameters. The uncertainty of a derived quantity is then

 σ2f=(∂f∂a)TC(∂f∂a)=∑i,jCij(∂f∂ai)(∂f∂aj). (10)

The partial derivatives are calculated numerically.

2.3. Relation to previous methods

Traditionally, the expanding photosphere method requires construction of synthetic magnitudes from theoretical models of supernova spectral evolution. The requisite dilution factors are tabulated for a specific combination of filters (, , , , etc.), where each typically yields a different distance (e.g. Fig. 12 of Hamuy et al., 2001). Furthermore, this method cannot be used for an arbitrary combination of filters and adding new filters is not straightforward. These shortcomings can be alleviated by modelling of the supernova spectra (Dessart & Hillier, 2006; Dessart et al., 2008; Baron et al., 2007, e.g.). In the model presented here, we use the assumption of homologous expansion to construct the evolution of the photospheric radius. We use information from many different supernovae to learn how the photometric color evolves and what are the typical parameters in supernovae and their covariances. Our method naturally works for any combination of filters in our set and adding new filters is relatively easy, requiring a single supernova well-observed in the new band and one of the “old” bands.

Similarly, the standardized candle method requires velocity and bolometric measurements at  days, which are used as an input to a pre-calibrated correlation (Hamuy & Pinto, 2002), and this often requires extrapolation to infer quantities at a desired time (e.g. Fig. 3 in Olivares E. et al., 2010). We can understand why the standardized candle method works within our model. Evaluating Equation (1) at the moment of , , gives for absolute magnitude

 Mi=¯¯¯¯¯¯Mi−5logv−5log(tτ=0−t0), (11)

where we assume . For our supernovae, typically occurs between and  days after with a peak at  days, making it similar to the usual standardized candle method. Since the chromatic function at , the velocity is a measure of total radius of the supernova if is constant for all supernovae. The fact that is not constant for all supernovae is responsible for factors in front of in the empirical fits (Hamuy & Pinto, 2002; Nugent et al., 2006; Poznanski et al., 2010; Olivares E. et al., 2010).

Equation (1) offers another approach to the standardized candle method. Assume that magnitudes and velocity measurements of a supernova are obtained on a single arbitrary epoch , which is different for each supernova. Equation (1) can then be approximated3 to provide a fitting formula for the supernova absolute magnitude

 Mi=¯¯¯¯¯¯Mi+RiE(B−V)−Kvlogv−Ktlog(t−t0)−2.5β1,iτ. (12)

Here, , , , , and are fitting coefficients that have a specific meaning within our model (e.g. ), but their values can be obtained empirically from the data. Equation (12) reduces to the common standardized candle method by assuming a constant and the same color and hence for all supernovae (Nugent et al., 2006; Poznanski et al., 2009, 2010; Olivares E. et al., 2010). The model in Equation (12) requires observations in at least three bands per supernova to constrain and , and a minimum of supernovae to constrain the parameters , , , , and if each supernova is observed in exactly three bands. The requirements on the total number of supernovae can be decreased by assuming a fixed reddening law (e.g. Cardelli et al., 1989) and fitting only for . Alternatively, with four bands per supernova, for each supernova can be determined. Number of requisite bands and supernovae can be further decreased by using external values of .

To summarize, Equation (12) differs from the common standardized candle method by not requiring the supernova measurements at any specific post-explosion time and by not assuming the same color for all supernovae. This has practical advantages, because it does not require extrapolation of expansion velocities. This process is uncertain, because each supernova does have a different value of as we show in Section 3.1. Furthermore, uncertainties in determining can be explicitly factored in the fit providing more realistic uncertainties. More degrees of freedom of the model should also provide tighter Hubble diagram. Our sample includes predominantly local objects and is thus not suitable for applying Equation (12) to construct the Hubble diagram. Applying Equation (12) to a sample of more distant supernovae will the subject of future work.

3. Results

3.1. Fits of supernovae

In Figure 1 we show the resulting fits to the light curves and expansion velocities for several well-observed supernovae. The fits of the remaining objects are available in the on-line version of the journal. The behavior of the model in supernovae with less data than what is shown in Figure 1 can be ascertained later in Figure 9 or by the fits of ASAS13co in Holoien et al. (2014). Fit parameters for individual supernovae are given in Tables 2 and 3. We see that our model reproduces well the key features of Type II-P supernova light curves and can fit them simultaneously in many bands. The model reproduces the light curve just after the explosion when the optical flux rises due to photospheric expansion and cooling so that peak of the spectral energy distribution moves into the optical. At the same time, the supernova color evolves from blue to red, which is captured by the rapidly decreasing flux in the band and in the Swift near-UV bands. Another key feature is the plateau, where our model reproduces the observed variety between supernovae, ranging from the rather flat plateau of SN1999em to the gradually declining one in SN2012A. Furthermore, our model naturally reproduces the flattening of the decline in the and bands just before the plateau ends. Our model predicts that this flattening should manifest itself as a distinct bump in the near-UV Swift bands. The only supernova with Swift data covering this part of the light curve is SN2012aw and we indeed see indications of a marked flattening and possibly upturn of brightness in . However, the data are missing just during the predicted bump and the supernova was already very faint. This part of the near-UV evolution could be better constrained with more data, which could allow higher order determination of the temperature coefficients (Eq. [8]).

Our model does reasonably well in reproducing the transition from the optically-thick plateau to the optically-thin exponential decay part of the light curve. We fit very sharp transitions such as in SN2005cs as well as more gradual transitions such as in SN2004et. In objects without observations during the transition phase like SN2012aw, the model typically favors very long . This does not affect the quantities derived from the model, because has little correlation with other parameters as we show below. We also see that our assumption of constant and hence constant color during the exponential decay is reasonably good in objects like SN2012A and SN2012aw. In some objects like SN2004et or SN2005cs we observe a distinct color evolution during this phase. However, in this paper we focus predominantly on the supernova plateau, which exhibits the highest luminosities and allows for distance determination. Improving the description of the exponential decay phase is a subject for future work.

In Figure 1, we also show fits to the expansion velocities, which show excellent agreement with the data. In the case of SN2004et, we observe a systematic shift between the data and the model. The reason is that the distance to SN2004et host galaxy NGC6946 is constrained also by the data of SN2002hh and SN1980K, which results in a small systematic offset in expansion velocities of SN2004et. If SN2004et was fit independently, then there would be no systematic shift between the observed expansion velocities and the model. In several supernovae like SN1999em, the fit deviates from the observations close to . This probably indicates that these velocities are biased because the supernova is becoming transparent. We emphasize that in our model, the radius changes in supernovae with incomplete velocity coverage are constrained not only by velocities but also by the achromatic part of the photometry.

Now we investigate the range of validity of our model. Both the total value of and the values of individual objects given in Table 2 indicate that to . Normally, this would indicate that the model is not fitting the data well and it should be modified. However, our model is phenomenological and we would not expect it to provide a perfect match to data of all supernovae. The agreement between the data and the model could be improved by adding degrees of freedom in the model, for example, by using higher-order temperature terms in Equation (8) or by assuming non-linear evolution of in Equation (4). However, we prefer to keep the model as simple as possible, even at the cost of higher . One way to rectify this situation is to multiply the measurement uncertainties by and repeat the fit, which ensures that . At several places of the paper, we apply this procedure to obtain perhaps more realistic uncertainties of the fit parameters and we explicitly mention when we do so. Uncertainties in Tables 2 and 3 are not rescaled.

Another aspect of the phenomenological description of supernova light curves and expansion velocities is the possibility to arrive to misleading results if the model is pushed too far. For example, assume that the “true” evolution of in Equation (4) is nonlinear. If there are no observations of the initial part of the light curve, the two coefficients in Equation (4) will be different than if the full light curve was fitted. As a result, in the initial part of the light curve will be wrong, which can have dramatic consequences, because of the temperature sensitivity of the blue bands (Fig. 4). Therefore, we will present bolometric light curves and other quantities only for epochs after the first observation of each supernova.

Despite this caveat, if is well determined by observations in a subset of bands, we can robustly predict the flux in the remaining bands if the coefficients are well-constrained at this by observations of some other supernova. In Figure 2 we present the distribution of assigned to all our photometric measurements by the best-fit model. We see that there are plenty of observations for suggesting that there are no global “holes” in our coverage of . The peak at is due to the fact that we fix during the exponential decay part of the light curve. We will show in Section 3.4 that the near-UV Swift bands are extremely important to constrain the total flux from supernovae in the early part of the light curve. We show in Figure 2 with dotted blue line the distribution of the three bluest Swift bands. We see that we can reliably predict near-UV magnitudes only for . There is a lack of near-UV measurements at low as well, but this is not important for bolometric fluxes due to the negligible contribution of the near-UV to the flux in later epochs. However, we find that the coefficients are only determined reliably for for the Swift bands.

In Figure 3 we show the mutual dependencies of the individual supernova parameters from Tables 2 and 3. The uncertainties come from a modified fit with . We see that the plateau duration correlates with almost all parameters except for several outliers with  d, which typically have poor coverage of the early phases of the light curve. We also see that and (Eq. [3]) correlate in the sense that velocity decays faster in supernovae with initially higher velocity. Both of these parameters correlate with , which is the slope of the temperature parameter decrease (Eq. [4]), in the sense that in supernovae with initially higher expansion velocity the temperature parameter decays more slowly. Our values of are typically smaller than the mean obtained by Nugent et al. (2006) and Faran et al. (2014). The reason is that we have also a constant velocity offset in our model, which typically makes more negative. We have a different value of for each of our supernova and is determined not only by velocity measurements, but also by the photometry.

In Table 4, we present values of the global parameters of the model. We see that the parameters are very well constrained by the data, with typical uncertainties of a few per cent. The only exception is (Eq. [8]) for the ROTSE photometric band, where the value is relatively small when compared to or band with similar central wavelength and it is comparable to its uncertainty. The reason is that we have ROTSE photometry only for SN2006bp and there is no photometry in other bands during and after the transition, which means that high-order temperature term is poorly constrained in relation to other bands.

In Figure 4 we investigate the behavior of the temperature polynomial . We see that there is a clear trend with the central wavelength of the filter in the sense that bluer filters have steeper . The only exception is band, which is steeper than and for and the Swift  and bands, which are only shown where data exist to constrain their values. This reflects the time evolution of the spectral energy distribution of Type II-P supernovae, potentially indicating evolution of strong emission or absorption lines. We apply our results on the global parameters to determine supernova spectral energy distributions (Sec. 3.4), bolometric light curves (Sec. 3.5), bolometric corrections (Sec. 3.7), and compare to theoretical supernova spectrophotometric models through the dilution factors (Sec. 3.8).

Now we investigate whether the components of the model have the intended meaning. In Figure 5, we show the time evolution of the components of Equation (1), specifically the expansion velocity , temperature parameter , and the achromatic and chromatic parts of the light curves, and . As expected, and decrease with time in a similar manner in all supernovae. The achromatic component of the light curve proportional to the radius increases in time because until , when the supernova becomes transparent. After time , drops, and the subsequent evolution is dominated by the radioactive decay. The chromatic component of the light curve, , which we evaluate in the band for the purposes of Figure 5, decreases similarly to , but with additional “wiggles” to account for color changes during the plateau. The evolution of and is qualitatively similar to the evolution of the color temperature and apparent angular radius presented by Hamuy et al. (2001, Fig. 6). One would not expect a detailed quantitative match, especially in case of , because it is not identical to either the color or effective temperature. This indicates that the individual components of our model indeed have the intended meaning presented in Section 2 and that they closely match the results obtained in the traditional expanding photosphere method. We discuss the relation between and effective temperature in Section 3.7.

With the physics of the model verified, we now ask whether the parameters describing the data are robustly obtained by the fitting process. In Figure 6, we show the correlation matrix of the supernova-specific parameters obtained by calculating the median of each element over all supernovae in our sample. Using the median should mitigate the effect of supernovae lacking some of the observations, which leads to excessively high correlation between some parameters unrelated to the underlying model. For example, it is impossible to constrain all of , , and for a supernova with less than three velocity measurements. From Figure 6 we see that the time of explosion and the plateau duration are anticorrelated mutually and correlated with . This can be understood by realizing that supernovae generally lack coverage early on and thus uncertainty in implies uncertainty in the duration of the plateau and the temperature at the moment of explosion . The transition width is little correlated with other parameters, as was found also by Anderson et al. (2014a). The velocity parameters , , and are mutually relatively highly correlated. The distance modulus is highly correlated with the explosion time and . This is similar to the usual expanding photosphere method, where the distance and explosion time are obtained from fitting the apparent angular radius versus expansion velocity: uncertainty in the distance directly translates to uncertainty in the explosion time. These correlations imply that to precisely determine the distance of a Type II-P supernova, we require photometric observations constraining the explosion time and velocity measurements spanning long-enough interval so that the exponent is well-constrained. The highest correlation of with other parameters is , which indicates that we robustly determine the reddening with little influence from other supernova-specific parameters.

Our method works so well for determination of not only because we have multi-band photometry for many of our objects. If a supernova has enough velocity measurements or the distance is well-known, the radius is constrained and changing results not only in changes of the color, but also in changes of the supernova magnitude. In other works, making the supernova redder to mimick the reddening would also make it fainter, with constant radius. Keeping the global parameters fixed, we fitted the individual supernova parameters of SN2004et and SN2009N based on a dataset that includes only velocity and and measurements. We found that in this case was smaller than what we obtain in Table 2. This opens a path to reliable determination of supernova reddenings. Note that our determination of is potentially biased, because the model will try to absorb differences between supernovae not parameterized in the model (such as the metallicity) into changes of and other parameters, as we have shown for Cepheids in Pejcha & Kochanek (2012).

In Figure 7, we compare our estimates of to results of Rodríguez et al. (2014), which are based on color standardization of Type II supernovae using their C3() method. The overlap of our sample and the sample of Rodríguez et al. (2014) is 13 objects and we see relatively good agreement, especially at high . At lower reddenings, the discrepancies are greater. We also show regions of perfect correlation between the C3() results and spectrum-fitting estimates determined by Rodríguez et al. (2014) for all of their supernovae and with outliers removed. All of the supernovae fall within these regions, except one supernova, where Rodríguez et al. (2014) obtained significantly negative . This suggests that our reddening estimates are definitely not worse than previous or contemporary results. We emphasize that our results make use of all available colors, while Rodríguez et al. (2014) use only pass bands.

3.2. Morphology of light curves and the differences between plateau and linear supernovae

Recently, attention has been devoted to understand the morphology of Type II supernova light curves with properties ranging from flat plateaus to steep declines (Anderson et al., 2014a; Faran et al., 2014; Sanders et al., 2014). In particular, Arcavi et al. (2012) suggested the possibility that there are no supernovae with properties in between Type II-P and Type II-Linear supernovae. Conversely, Anderson et al. (2014a) and Sanders et al. (2014) found a continuum of light curves of Type II supernovae. Since our model can disentangle the observed data into radius and temperature variations, we decided to investigate these issues within our model.

In Figure 5 we show three Type II-P supernovae with the different shapes of the plateau: the relatively steeply declining SN2009bw, the flat plateau of SN2005cs with a noticeable bump just before the transition, and the intermediate case of SN2004et. We see that the overall behavior of velocity and chromatic part of the light curve are very similar for these supernovae. In fact, this is true for all supernovae with enough data, as we show in Figure 8. The small differences in the velocity evolution, however, translate into different steepness for the achromatic part of the light curve . In SN2004et and SN2009bw, there is initially a fast rise in the radius, which then gradually slows down. As a result, the decrease of brightness due to dominates and we observe a declining plateau. In SN2005cs, the radius is increasing fast4 even before the transition, which almost exactly compensates for the decrease of brightness due to the decreasing temperature. As a result, we observe a flat plateau with a bump. Since the temperature evolution is so similar, the morphological differences in the Type II-P supernova light curves are thus due to different photospheric velocity evolution.

A further understanding of the light curve morphology can be obtained by applying our model to Type II-Linear supernovae. The only Type II-Linear supernova we found in the literature with enough data for a successful fit is SN1980K (Barbon et al., 1982; Buta, 1982). SN1980K is convenient for our purposes, because it exploded in NGC6946 with a distance very well constrained by SN2004et (Fig. 11 and Table 5) and we do not have to worry about uncertainties in the overall radius scaling. In Figure 9 we show the light curve and expansion velocity fits, and Figure 5 shows the decomposed changes in radius and temperature. We obtained a reasonable fit to the data. The estimated optically-thick phase duration of  d is noticeably shorter than for normal plateau supernovae. However, the transition width is fairly long,  d, and as a result the temperature parameter reaches values typical for exponential decay () about  days after the explosion, which is more similar to ordinary Type II-P supernovae. Still, after  d, the temperature falls faster than in normal plateau supernovae. This is consistent with the finding of Anderson et al. (2014a) that faster declining supernova have shorter duration of the optically-thick phase. The radius of SN1980K stays almost constant during the optically-thick phase, but this is mostly because of the large , which makes the supernova nearly transparent for most of the observed time. As a result, the steeply declining light curve is mainly due to temperature changes of the photosphere. The successful fit and the resulting parameters indicate that Type II-Linear supernovae can be modelled along with normal Type II-P explosions and that Type II-Linear brightness variations are driven primarily by changes in the photospheric temperature.

To put the discussion of the light curve morphology on more quantitative grounds, we address the recent finding of Anderson et al. (2014a) that the -band maximum magnitude and the plateau rate of magnitude decline are correlated. The maximum magnitude in our model is not easy to obtain analytically, but we instead consider the absolute magnitude at , . The epoch corresponding to is (Eq. [4]) and the color temperature at this point is  K using the relations between color index and temperature of Hamuy et al. (2001) as implemented below in Section 3.8. The effective temperature at is  K as discussed in Section 3.7. The light curve slope at is

 (13)

where the approximation comes from the assumption of .

We show the absolute magnitude as a function of light curve slope in Figure 10 for , , and bands. The uncertainties in both quantities were obtained by applying Equation (10) to Equation (13) with the full covariance matrix of the fit. We see that we reproduce the band correlation of Anderson et al. (2014a) and that this correlation exists also for the band, but not for the band. In general, we find that the correlation does not exist for bands with m, in other words, the correlation exists only when the first term in Equation (13) dominates, which occurs when is small. Why does this correlation exist? Early after explosion, in Equation (3) can be neglected, which gives , , and . In other words, both quantities are roughly proportional to leading to a correlation that is not perfect because of the other terms in both expressions.

Clearly, the parameter , measuring the curvature of the velocity decrease, plays an important role in determining the morphology of Type II-P supernovae. To gain insight about the physical property of the exploding medium controlling , we consider an homologously expanding adiabatic medium with density profile , where the outer boundary is expanding linearly in time. We assume that the medium is dominated by radiation and that the temperature is constant as a function of radius giving (Arnett, 1980). Assuming a general opacity law , the photospheric radius in such medium is determined by

 τ=∫RoutRκρdr=23. (14)

Using our expressions for density, opacity, and temperature, and assuming that , we obtain the rate of photospheric expansion

 dlnRdlnt=(k−3)(n+1)+sk(n+1)−1. (15)

Equations (23) give the rate of photospheric expansion from observed quantities as

 dlnRdlnt=1+ω0ω1t1+ω1ω0t1+ω1+ω2t, (16)

which reduces to if is negligible. The opacity law can be either Thompson opacity (), Kramers opacity (, ) or H opacity (, ), where the latter is perhaps most appropriate at with effective temperature of about  K. The density exponent implied by Equations (1516) for H opacity is , , and for SN2005cs, SN2004et, and SN2009bw, respectively, at  days. This is in relatively good agreement with the density exponents in the theoretical models of Dessart & Hillier (2008, 2011). As can be seen in Figure 5, steeper density profiles correspond to flatter plateaus.

Since the temperature evolution and thus the chromatic part of the light curve are very similar for all supernovae (Fig. 8), this implies that for a fixed opacity law, the shape of the supernova plateau is controlled by the homologous density profile of the ejecta, with steeper density profiles resulting in flatter light curves. The supernova ejecta have also temperature structure that evolves in time so that further insight into this issue requires much more realistic models than the simplistic estimate presented here. The relation of the light curve morphology to the mass of hydrogen has also been proposed by Anderson et al. (2014a, b). Nickel mixing in the ejecta can have also effect on the morphology of the optically-thick phase (Bersten et al., 2011; Kasen & Woosley, 2009).

3.3. Distances

In Table 5, we present the distance estimates to individual galaxies in our sample. We compare our results to the entries in the NASA Extragalactic Database (NED), focusing on the distance measurements from Type II supernovae, Cepheids, and Type Ia supernovae in Figure 11. Overall, we see relatively good agreement in the sense that our results fall within the range of previous distance determinations. The overall distance scale can be moved by changing the distance to NGC3351 (host of SN2012aw), which we assume to be  mag. We point out that in principle, our method allows determinations of distances to supernovae with no velocity measurement by applying priors on , , and . However, in this case the uncertainties are rather large ( mag) and there is a great potential for systematic offsets.

We now specifically discuss several galaxies where there is substantial disagreement with previous results. The photometric and spectroscopic observations of SN2006my in NGC3239 start shortly before the transition to optically-thin phase. As a result, is not well constrained and thus is potentially substantially biased as we discussed in Figure 6. This particular aspect can be improved by applying constraints on the age of the supernova from the appearance of the spectrum while treating appropriately the uncertainty associated with this prior information. Including such constraints is beyond the scope of this paper.

Our distance for NGC3389 (host of SN2009md) is noticeably smaller than the one measured from the Type Ia supernova SN1967C (Parodi et al., 2000), but agrees reasonably well with previous estimates from SN2009md (Fraser et al., 2011; Bose & Kumar, 2014). SN2009md has good spectroscopic and multi-band photometric coverage so the disagreement comes perhaps from the inaccurate photographic photometry of SN1967C. Our distance for NGC918 (host of SN2009js) is significantly larger than that based on the recent Type Ia supernova SN2011ek in the same galaxy (Maguire et al., 2012). SN2009js is relatively well-observed in , but has only one velocity measurement so that the distance estimate is more susceptible to systematic errors. The agreement is almost within the uncertainty, if the rescaled fit with is used. There are no previous distance estimates to SN2009js in NED. Our distance to NGC7793 is based on SN2008bk, for which we do not have any velocity measurements and only -band photometry during the plateau and thus the relatively good agreement with NED distances is rather surprising. Our distance estimate almost agrees with Cepheid distance of Pietrzyński et al. (2010). The distance to NGC5377 is based on SN1992H, which has relatively good velocity measurements, but rather sparse photometry. The distance to MCG-01-04-039 is based on SN1992am with photometry starting about  days after . Our lack of K-corrections probably plays some role, but most of the disagreement can probably be attributed to the poor early coverage. Using the fit with yields higher distance uncertainty and thus better agreement with previous measurements. Photometry of SN2001dc in NGC5777 does not cover the first  days and the first velocity measurement is  days after , which explains the disagreement. SN2009dd in NGC4088 has well-determined , but not enough velocity measurements with sufficient precision. The prior supernova distance determination of NGC4088 was with a different Type II SN1991G (Poznanski et al., 2009).

To summarize, Type II-P supernovae can yield solid distances provided there is multi-band photometry, good expansion velocity coverage and that the explosion time can be well constrained from the observations.

3.4. Spectral energy distribution

Within our model, the monochromatic luminosity at the central wavelength of band at a distance of 10 pc () is

 Li=10ΠFi, (17)

where is a monochromatic flux at a specific radius defined as

 Fi=F0,i10−0.4(¯¯¯¯¯Mi−2.5Θi), (18)

where the flux zero points are taken from the Asiago Database of Photometric Systems (Moro & Munari, 2000) and from Poole et al. (2008) for the Swift bands and are given in Table 5. The usual process of dereddening and correcting for the supernova distance is achieved in our model simply by not including terms and in Equation (1).

In Figure 12, we show absolute spectral energy distributions (SEDs) of SN2004et and SN2012aw at several epochs calculated using Equations (1718). We see that the SEDs get progressively redder as the supernova evolution proceeds. Early in the supernova evolution, the flux is dominated by the near-UV emission. In the first two epochs, the fluxes in the three bluest Swift bands are relatively flat and do not show any sign of turning down at shorter wavelengths indicating that the peak of the emission is located at even bluer wavelengths ( K). It is uncertain how to extrapolate the flux to shorter wavelengths. At about  days, the SEDs of both supernovae have  K and are similar to an F0 supergiant. The near-UV emission is already unimportant at this point. At later epochs, the SED achieves a constant shape () very different from a normal star and only the overall normalization changes in time ( K). At all epochs, the SEDs at m are very similar to the Rayleigh-Jeans tail of the black body, as indicated by the dashed line in Figure 12. The relation between and effective temperature will be discussed in Section 16.

3.5. Bolometric light curves

The bolometric luminosity is

 Lbol=10ΠFbol, (19)

where is the bolometric flux at radius

 Fbol=∫Fidλ. (20)

We perform the integral in Equation (20) using the trapezoidal rule assuming monochromatic fluxes at central wavelengths given in Table 4. We correct