## 2. Photospheric epoch## 2.1. The velocity at photosphereAccording 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
In spite of its simplicity the model is appropriate for the
confident estimate of the photospheric velocity which was found to be
km s ## 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 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 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
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
## 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 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
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 An extended grid of hydrodynamical models of SN 1997D led us
to the conclusion that requirements imposed by the
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 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 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 |