2. Photospheric epoch
2.1. The velocity at photosphere
According to general results of hydrodynamical simulations of SNe II-P the radiation cooling of the expanding envelope at the plateau phase proceeds in a specific regime of the cooling recombination wave (Grassberg et al. 1971). As a result the photosphere in SN II-P resides at the well defined jump between the almost completely recombined (ionization degree ) transparent atmosphere and the fully ionized sub-photospheric layers of high opacity. The velocity at the photosphere determined from the observed scattering line profiles during photospheric epoch (plateau) thus gives us a position of the cooling recombination wave and therefore is of vital importance for constraining parameters of the hydrodynamical model.
To measure the photospheric velocity in the Jan. 14 spectrum of SN 1997D we concentrated on the 5700-6700 Å band which contains strong, clearly-cut spectral lines (Fig. 1) of H, Na I, Ba II, Fe II, and Sc II. Most of them are well observed in other SNe II-P. However, due to the low expansion velocity it is possible to distinguish here some spectral features never observed before, e.g. Sc II 6605 Å line (Fig. 1). A Monte Carlo technique used for modelling the spectrum (Fig. 1) suggests an absorbing photosphere and a line scattering atmosphere (Schwarzschild-Schuster model). In total 19 lines are included for this spectral range. The Sobolev optical depth was computed assuming the analytical density distribution in supernova ejecta
which corresponds to a plateau at velocities and a steep slope in the outer layers at . Parameters and are defined by the ejecta mass M, kinetic energy E, and index n. Such a density distribution for closely approximates hydrodynamical models discussed in Sect. 2.4. The case shown in Fig. 1 is characterized by , erg, and , although one may easily fit the spectrum using higher mass and higher energy. Since we do not solve the full problem of radiation transfer in ultraviolet we adopt that metals are singly ionized and find level populations assuming appropriate excitation temperature (4200 K). For the standard abundance we assume here, Sc II 6605 Å line is too strong; therefore its abundance is reduced by a factor of two. It may well be that the odd behavior of Sc II line reflects different excitation conditions for Sc II and other metals rather than abundance pattern. We found that hydrogen excitation has to be cut beyond km s-1 to prevent washing out the 6500 Å peak. Close to the photosphere within the layer km s-1 the net emission in H is comparable to its scattering component. We simulated this emission assuming the line scattering albedo greater than unity.
In spite of its simplicity the model is appropriate for the confident estimate of the photospheric velocity which was found to be km s-1 (Fig. 1) with an uncertainty less than 100 km s-1. The value of km s-1 reported by Turatto et al. (1998) is consistent with the above estimate. Our choice was a compromise between two possibilities: (1) washing out many observed features in the spectrum if km s-1, and (2) producing significant excess in emission components for Na I and Ba II 6142 Å lines if km s-1. The value km s-1, although optimal, still leads to some extra emission in Na I , Ba II 6142 Å and Fe II 6249 Å lines (Fig. 1). Preliminary analysis indicated that this drawback of the model may be overcome, if Rayleigh scattering on neutral hydrogen is taken into account.
2.2. Rayleigh scattering effects
Rayleigh scattering on neutral hydrogen in the optical dominates over Thomson scattering at an extremely low ionization degree which is the case for SN II-P atmospheres at the photospheric epoch. To get an idea of the role of Rayleigh scattering in the spectrum of SN 1997D we adopt the analytical density profile given by Eq. (1) with a power index . Let us first estimate the Rayleigh optical depth using the cross-section by Gavrila (1967) and assuming conditions of the atmosphere of a normal SN II-P (e.g. SN 1987A) for two extreme cases: completely mixed and unmixed envelopes. Assuming for SN 1987A erg, , and a helium/metal core mass (Woosley 1988; Shigeyama & Nomoto 1990; Utrobin 1993), one gets at the wavelength 6142 Å (Ba II line) a Rayleigh optical depth of 0.07 (mixed case) and 0.1 (unmixed case) at the age d and km s-1. At the light maximum, on day 90 and km s-1, the Rayleigh optical depth is 0.04 (mixed case) and 0.07 (unmixed case).
SN 1997D is essentially different in that respect. Adopting the ejecta model by Turatto et al. (1998), viz. total mass , helium/metal core mass , and km s-1 at the end of photospheric epoch d, one finds the Rayleigh optical depth at Å in the range 1.3-1.8, more than one order of magnitude exceeding that in normal SNe II-P at the similar evolution phase. For the model with parameters scaled-down by a factor of four ( and helium/metal core mass ), one obtains . Our study showed that such values cannot be ignored in modelling line profiles.
Moreover, to treat Rayleigh scattering in an adequate way one has to abandon the assumption of a fully absorbing photosphere and instead include a diffuse reflection of photons from the photosphere. We describe the diffuse reflection by a plane albedo which is a function of cosine µ of incident angle and thermalization parameter . Here is the absorption coefficient and is the scattering coefficient. In the approximation of the isotropic scattering the plane albedo reads
where the function is defined by the integral equation (cf. Sobolev 1975)
which was solved numerically to create a table of .
In the absence of Rayleigh scattering, non-zero albedo for slightly (by 4%) increases the intensity of the emission component compared to the purely absorbing photosphere (Fig. 2). The difference obviously becomes larger for a smaller thermalization parameter. Rayleigh scattering significantly decreases the emission component due to backscatter and subsequent absorption of photons by the photosphere in the case of and . Another effect of Rayleigh scattering is washing out of the absorption trough by continuum photons drifted from blue to red; this effect is especially pronounced for weak lines and is of minor importance for strong lines. This modelling shows how the emission excess in Na I and Ba II lines (Fig. 1) may be suppressed.
Apart from Rayleigh scattering and diffuse reflection by the photosphere we made two other essential modifications to our Monte Carlo model of line formation. First, we took electron scattering into account. The electron density distribution is recovered from the H line profile using a two-level plus continuum approximation. Second, we calculated the population of three lowest levels of Ba II using the observed flux in the spectrum on Jan. 14. This approximation is fairly good in analyzing the blue side of the absorption trough of the Ba II 6142 Å line. We adopted the standard barium abundance (Grevesse & Sauval 1998) and the Ba II fractional ionization . The latter seems to be a good approximation for the outer layers of SN 1987A at the stage when strong Ba II lines are present (Mazzali et al. 1992).
With the modified Monte Carlo model the synthetic spectrum is calculated for two relevant cases: a high-mass model with parameters and (Fig. 3a) and a low-mass model with parameters and (Fig. 3b). Note, that both models have the same photospheric velocity km s-1 and the same ratio , where is the kinetic energy in units of erg and M in . Complete mixing, which implies a minimum Rayleigh optical depth, gives and 0.33 for high and low-mass models, respectively. The thermalization parameter in sub-photospheric layers is 0.35 and 0.24 for high and low-mass models, respectively. In the high-mass model Rayleigh scattering suppresses emission components of Na I , Ba II 6142 Å, and Ba II/Fe II peak at 6500 Å down to an unacceptably low level (Fig. 3a). The low-mass case fits the observations fairly well (Fig. 3b). Computations of spectra for different values of Rayleigh optical depth led us to conclude that the tolerated upper limit is 0.6. Yet the Rayleigh optical depth cannot be lower than , otherwise the emission in the Na I and Ba II 6142 Å lines becomes too strong. We find an optimal value is with an uncertainty of about 0.15 for the Jan. 14 spectrum.
2.3. Diagnostics of ejecta mass and kinetic energy
The observational limitations upon Rayleigh optical depth in the atmosphere of SN 1997D may be combined with the restriction on the density in the outer layers imposed by the blue absorption edge of Ba II 6142 Å in order to get the first guess about ejecta mass and kinetic energy. The idea may be illustrated using a toy model, in which the supernova envelope is represented by a homogeneous sphere with the boundary velocity . Given the photospheric velocity and Rayleigh scattering optical depth one finds the product , whereas the blue edge of the Ba II 6142 Å absorption gives the outer velocity . At moment t one gets then ejecta mass and kinetic energy .
For the density profile given by Eq. (1) with a power index and a certain ejecta mass one can find the corresponding value of the kinetic energy consistent with the blue edge of the Ba II 6142 Å absorption in the SN 1997D spectrum on Jan. 14. Again, we adopted a standard barium abundance with the Ba II ion as the dominant ionization state. Variation of the model mass under the condition that the Ba II 6142 Å absorption is reproduced results in the corresponding variation of the kinetic energy. The magnitude of this variation is determined by an uncertainty in the description of continuum around 4500 Å responsible for the Ba II excitation and by an error in fixing the position of the blue absorption wing ( km s-1) for the Ba II 6142 Å line. These uncertainties result in a region of allowed parameters ("barium" strip) in the mass-kinetic energy () plane (Fig. 4). The lower and upper limits of Rayleigh optical depth, 0.3 and 0.6, respectively, produce another strip of allowed parameters ("Rayleigh" strip) in this plane. The overlap of "barium" and "Rayleigh" strips gives a tetragonal region where the ejecta mass and kinetic energy of SN 1997D are confined. One sees that optimal values of ejecta mass should reside around , while kinetic energy should be close to erg.
The suggested diagnostics, unlikely useful for ordinary SNe II-P, proved efficient for constraining parameters of SN 1997D. A warning should be kept in mind that a cosmic barium abundance was assumed here. This may in general not be the case since SN 1987A demonstrates that barium overabundance in SNe II-P may be as large as a factor of two relative to the cosmic value (Mazzali et al. 1992). If barium abundance in SN 1997D ejecta is twice the cosmic value, then the "barium" strip in the plane has to be shifted down by a factor towards lower values of kinetic energy. It is remarkable that this diagnostic does not depend on the supernova distance. However there is a weak dependence on reddening via the colour temperature determined from 4500 Å/6140 Å flux ratio which affects the Ba II excitation. Unaccounted reddening leads to the overestimation of the mass obtained from the Ba II line.
2.4. Light curve
The light curve of SN II-P during the plateau phase is determined by the ejecta mass M, kinetic energy E, pre-SN radius , the structure of the outer layers, the 56Ni mass and its distribution, and the chemical composition of the envelope (Grassberg et al. 1971; Utrobin 1989, 1993). The 56Ni mass in SN 1997D is reliably measured by the light curve tail. The structure of outer layers of the pre-SN may normally be recovered from the initial phase of the light curve, which was unfortunately missed in the case of SN 1997D. Therefore we used a standard pre-SN density structure with the polytrope index of three, though models with other density structure were also tried. The abundance of the deeper part of the envelope, e.g., the transition region between the H-rich envelope and metal/helium core affects the final stage of photospheric regime and may therefore be probed by the light curve at the end of the plateau phase. In general, the parameters M, E, and then may be found from the plateau phase duration, luminosity at plateau phase (e.g., in V band), and velocity at the photospheric level. In a situation when the plateau phase duration is unknown the optimal Rayleigh optical depth in the atmosphere () provides the missing constraint. The description of the radiation hydrodynamics code used for supernova study may be found elsewhere (Utrobin 1993, 1996).
An extended grid of hydrodynamical models of SN 1997D led us to the conclusion that requirements imposed by the V light curve, velocity at the photosphere, and Rayleigh optical depth are consistent with those estimated above from the M-E diagram. The optimal hydrodynamical model is characterized by the following parameters: the ejecta mass , kinetic energy erg, and pre-SN radius . To prevent the emergence of a luminosity spike at the end of plateau phase and to explain the narrow peak of the H emission in the nebular spectrum on day , we suggest mixing between the helium layer and the H-rich envelope (Fig. 5). The adopted helium/metal core mass before mixing is . With 0.002 of radioactive 56Ni this choice of parameters results in a V light curve, which fits the observational data (Turatto et al. 1998) and is consistent with the observational upper limits by Evans at early epochs (Fig. 6). The velocity at the photosphere in this model is 830 km s-1 in agreement with that found from the spectrum synthesis. Remarkably we obtained the same ratio as Turatto et al. (1998). In our model the first spectrum on Jan. 14 corresponds to the epoch of 46 days after the explosion in good agreement with the 50 days found by Turatto et al. (1998).
The diffusion approximation used in the hydrodynamical model breaks down at the transition from the plateau to the radioactive tail about d. To reproduce the tail, we translated the bolometric luminosity computed in the hydrodynamical model into the V band luminosity using the two assumptions about the spectrum of escaping radiation at the tail phase. The first one admits that the spectrum is black-body with the constant effective temperature calculated at d. This gives somewhat higher V luminosity compared to observations at the tail stage (Fig. 6). An alternative approach assumes that the spectrum of escaping radiation during the tail phase is the same as in the observed nebular spectrum at d (Turatto et al. 1998). The latter assumption is more realistic and provides a good fit to observations (Fig. 6). This agreement justifies the adopted 56Ni mass of 0.002 originally obtained by Turatto et al. (1998).
The envelope structure computed in the hydrodynamical model was then used to recalculate the synthetic spectrum in a way similar to that described in Sect. 2.2. The model spectrum agrees well with the observed spectrum on Jan. 14 (Fig. 7). Of particular importance is an excellent fit for the emission component of the Na I 5889, 5896 Å doublet which is free of blending, thus being a reliable probe for Rayleigh optical depth in the atmosphere.
Possible variations of parameters of the optimal hydrodynamical model are determined by errors in the photospheric velocity ( km s-1) and the Rayleigh optical depth (), and by the absolute magnitude uncertainty. The latter results primarily from an error in the distance to SN 1997D for which we adopt a 20% value. It translates into the uncertainty of in V absolute magnitude. On the basis of a set of calculated hydrodynamical models in the vicinity of the optimal model we construct the Jacobian matrix for the transformation of the empirical errors into the uncertainties of the model parameters. We find thus the ejected mass to be , the kinetic energy erg, and the radius of pre-SN .
A dust extinction in the host galaxy (NGC 1536) cannot be ruled out. It is unlikely, however, significant since the galaxy is nearly face-on. With some dust extinction (if any) parameters of the optimal hydrodynamical model should be changed accordingly. For instance, the dust extinction of mag results in the increase of mass by , kinetic energy by , and pre-SN radius by .
© European Southern Observatory (ESO) 2000
Online publication: February 9, 2000