ASTRONOMY AND ASTROPHYSICS. Applications of an improved Monte Carlo code to the synthesis of early-time Supernova spectra. P.A.

Transcription

1 Astron. Astrophys. 363, (2) Applications of an improved Monte Carlo code to the synthesis of early-time Supernova spectra ASTRONOMY AND ASTROPHYSICS P.A. Mazzali Osservatorio Astronomico di Trieste, via Tiepolo 11, Trieste, Italy and Research Centre for the Early Universe, University of Tokyo, Bunkyo-ku, Tokyo , Japan Received 5 November 1999 / Accepted 28 March 2 Abstract. A Monte Carlo code developed originally for stellar winds and subsequently modified to compute synthetic spectra for the photospheric phases of supernovae adopted the simplifying assumption of line formation by coherent scattering in the matter frame. In an improved code (Lucy 1999a), the line absorption of a packet of radiant energy is followed as before by the emission of an identical amount of energy in the matter frame but now the frequency of the packet s photons is chosen so that, statistically, the branching into the available permitted downward transitions is correctly represented. This inclusion of branching is of some importance for supernova spectra, as emphasized in the light curve studies of Pinto & Eastman (2). An improved line list is also adopted in the new code, and the emergent spectrum is computed using the formal integral. Examples of synthetic spectra computed with this new code for supernovae of types Ia and II are presented and the improvements introduced by the incorporation of branching are discussed. Previous results obtained with a pure scattering code are shown not to be invalidated. Possible further code developments to achieve a fully NLTE Monte Carlo code are briefly described. Key words: stars: supernovae: general radiative transfer methods: numerical 1. Introduction The Monte Carlo technique is well suited for the calculation of synthetic spectra from expanding atmospheres. Abbott & Lucy (1985, hereafter AL85) first applied this technique to the winds of hot stars, and were able to obtain mass-loss rates and wind terminal velocities in very good agreement with the results of much more complex codes. While on the one hand the code was adapted to treat a variety of stellar envelopes, like W-R winds (Lucy & Abbott 1993), on the other it was transformed to deal with the ejecta of supernovae (SNe), which in the early photospheric epoch share the condition that most if not all the energy is deposited within a sharply defined radius (the Schuster-Schwarzschild approximation) and that the rate of change of the velocity of the material with radius is large (the Sobolev approximation), and are therefore similar to hot star winds in their radiation transfer properties. Mazzali & Lucy (1993, hereafter ML93), described the original Monte Carlo (MC) SN code and its differences with respect to the stellar wind code. With only minor modifications, this code has been used to analyse the observed spectra of a variety of objects, starting with SN 1987A (Lucy 1987, Fosbury et al. 1987, Mazzali et al. 1992) and including several recent SNe observed in the context of the ESO Key Programme: Supernovae. Some examples are the type Ia SNe 199N (Mazzali et al. 1993), 1991T (Mazzali et al. 1995), 1994D (Patat et al. 1996), 1991bg (Mazzali et al. 1997), and the type IIb SN 1993J (Zhang et al. 1995). In all of these codes, the interaction between photons and spectral lines was treated with the approximation of coherent scattering in the co-moving frame (cmf). This major simplifying assumption has several advantages: it lends itself naturally to a MC treatment because it eliminates photon splitting, and it automatically generates a divergence-free model of the radiation field in the local matter frame, thus incorporating a fundamental constraint even though the temperature stratification appropriate for radiative equilibrium may be only crudely approximated. Additionally, the assumption of coherent scattering describes frequency redistribution rather accurately, since when the constraint of radiative equilibrium is added to the constraints of selection rules the rate of photon emission in a non-resonant line is often obliged to approximate closely the rate of absorption in that line (AL85, Chugai 198), thus restricting the fluorescent degradation expected in diluted radiation fields. Nevertheless, coherent scattering is only a very simplified description of the process of line formation, and the consequences of this choice must be particularly significant in a SN envelope, where line opacity is actually the dominant form of interaction for the photons. Accordingly, a new code has been developed by Lucy (1999a, hereafter L99a), where the coherent scattering approximation is dropped and replaced by a treatment of branching. In this modified code, the photons in the packet emitted following the absorption by a spectral line are assigned a new frequency in such a way that the branching probabilities for radiative decays from excited levels are obeyed, subject only to sampling errors. The divergence-free constraint on the radiation field seen by the matter is preserved by again demanding that the energies of the absorbed and emitted packets of radiant energy are equal in the local co-moving frame.

2 76 P.A. Mazzali: Improved Monte Carlo code for early-time Supernova spectra A further improvement to the code s efficiency is obtained by computing the emergent spectrum using the formal integral rather than by simply binning the escaping MC energy packets according to their frequency. This improvement, which was also discussed in L99a, allows an accurate, noise-free synthetic spectrum to be computed using a much smaller number of energy packets, thus speeding up the calculation significantly, which is important if the code is to be a useful diagnostic tool. In this paper we describe how the modifications introduced by L99a have been implemented in the SN MC code, discuss some of the features of this new code, present some applications of it to problems which the pure scattering code could not address, discuss the advantages of the new code over the old one, assess the validity of results obtained with the old code in the light of the new results, and finally indicate areas of possible further refinement. 2. The new SN code: model SN envelope The new Monte Carlo (MC) Supernova (SN) code including photon branching was developed in close analogy to the previous, pure scattering code described in ML93. In this section we describe the code following the line of argument of ML93, emphasizing the differences from the pure scattering code where we have implemented the modifications described by L99a. The assumptions described in ML93 with regard to the density and temperature stratification (homologous expansion and radiative equilibrium, respectively) are retained, and so is the treatment of excitation and ionisation (modified nebular approximation). The envelope is divided into a number of shells, whose thickness increases as a function of radius. Density, abundances, temperature and thus the ionisation and excitation states are uniform within each shell, but velocity is treated as a continuous function, given by v = r/t, where r is the radius and t the time elapsed since the explosion, so that the Sobolev approximation can be applied. The conditions in the envelope are assumed to remain constant during one calculation, i.e. time-dependent effects are neglected. Spectrum formation in a SN at early times is of course a time-dependent process (Pinto & Eastman 2), because photons emitted deep in the ejecta have a diffusion time larger than the characteristic time of the main physical conditions. This effect, together with the different decay times of 56 Ni and 56 Co, is at the basis of the typical SN Ia light curve shape. However, our MC code only uses the momentary luminosity of the SN, which is itself the result of time-dependent photon diffusion, to compute line formation and hence the emerging spectrum, and does not attempt to determine the diffusion time and the photospheric properties ab initio. Since the diffusion time through our reversing layer is small, the assumption of constant physical properties is justified. The treatment of the radiative transfer is conceptually the same as in ML93, based on the Schuster-Schwarzschild approximation. Photons (energy packets) propagating through the envelope can interact with electrons and atoms. The treatment of electron scattering is unchanged. In the treatment of photonatom interaction, however, the assumption of purely scattering lines has now been dropped. Once a photon is absorbed in a line transition l u, all radiative decays from the upper level u of that transition down to lower levels are considered. All downward rates are computed, and each transition is assigned a weight proportional to its effective downward rate (i.e. correcting for photon trapping). One of the downward transitions is then selected randomly, but according to the weights assigned. The energy packet is finally re-emitted with the new selected frequency in a random direction, and the MC procedure continues as described in Sect. 2.3 of ML93. If during the MC experiment many excitations to level u occur, then the ensemble of packets randomly assigned in this way to decay channels u i will clearly constitute an accurate realisation of the emission from level u. Since this method avoids the explicit treatment of photon splitting or the inverse process, photon combination, i.e. the consecutive absorption of two bluer photons followed by the emission of one, redder photon, the code remains relatively simple and compact, and is also fully compatible with the introduction of NLTE. 3. Numerical technique 3.1. Input data This is unchanged from the previous version. Necessary inputs are the time elapsed from the explosion, t e, the emerging luminosity, L, the photospheric velocity v ph, the density stratification and the abundances in the ejecta. Using L makes it quicker to fit observed spectra. L can either refer to the bolometric luminosity or to the luminosity in a specific wavelength interval Line selection Previous versions of the code were based on the list compiled by Abbott (1982), which was specifically designed for hot-star winds, supplemented by lines from the list of Kurucz & Petreymann (1975) for lower ions. In view of the availability of the new massive line list compiled by Kurucz & Bell (1995), which contains several million lines, we have adopted this list as the basic source for the updated version of the code. Weak lines have been eliminated to make the handling of the data more manageable, and a final list comprising some 5, lines has been generated. In one particular case (C II) we have adopted the gf values from the recent compilation of Wiese et al. (1996). The gf values from the two lists are not always in agreement. In particular, the Kurucz & Bell (1995) line list gives significantly larger gf values for some transitions, notably the multiplet near 578 Å. As described in L99a, the data are compacted in two files. In the first file, for each ion a list of levels and of their J-values is stored. The second file contains the information on the line transitions, but for each transition only a reference to the upper and lower energy levels u and l and the gf value are stored. In keeping with the treatment of AL85 and ML93, the list differ-

3 P.A. Mazzali: Improved Monte Carlo code for early-time Supernova spectra 77 entiates between excited and metastable states. This is useful because the excitation formula used changes for the two different kinds of levels (AL85, Sect. IIe). This differentiation is achieved by replacing E i with E i in the energy level file and l with l in the line transition file for the metastable states. This organisation of the data greatly reduces the time spent handling atomic data, which is important in a code whose aim remains the fast (and accurate) modelling of observed spectra. From this large line list, a reduced list is selected case by case, according to the problem at hand. The selection is based on the assumed abundances in the ejecta, and on an approximate run of the radiation temperature, as described in ML93, Sect Line selection can be done directly in the first temperature iteration. Although in this first iteration all temperatures are set to a constant value (see Sect. 3.3), this does not introduce major differences in the number of lines selected and in their type, since the optical depth threshold for inclusion in the reduced line list is typically set to a small value (.1 in at least one shell). A typical working line list includes 5 1, lines. The list is organised in two files, one ordered by frequency with ν k+1 >ν k and the other with lines grouped according to ion and then, for each ion, according to the index u of the upper levels of the transitions. Pinto & Eastman (2) found that including weaker lines (i.e. the second 1, or more) has a significant effect upon the diffusion time. This does not affect these results since the effect they found occurs below our inner boundary. The diffusion time from our photosphere is sufficiently small that weak lines do not affect it significantly. Because of the expansion of the SN envelope, the co-moving frequency of a photon in free flight decreases with time. The list organised by frequency serves therefore as before to identify efficiently the next transition with which a photon packet might interact. On the other hand, after a packet has been absorbed by the transition l u, the possible radiative decays are the set of permitted transitions u i (with i < u) originating from the same upper level u. But these transitions are contiguous in the second line list, and so an appropriate random selection of the frequency of the re-emitted packet can be made simply and efficiently Determination of the temperature structure The temperature structure in the SN envelope is determined iteratively with a series of small MC experiments, as described in Sect. 3.3 of ML93 and similarly to the method outlined by Lucy (1999b). Radiative equilibrium is assumed, as described in ML93, Sect. 2.2, using the fact that the mean energy of a photon in a black body radiation field is x = h ν = (1) k B T The frequency moment ν is computed in the MC calculation, and hence an equivalent radiation temperature is obtained. Upon the first iteration, the radiation temperature T R(r) and the electron temperature T e (r) are given the constant values T R = T, T e =.9T R throughout the envelope. T is the effective temperature at the photosphere, which is obtained approximately from the relation L =4πR σt 2 4, where the photospheric radius R = v ph t e. The calculation of a converged temperature structure in the envelope using an approximate radiative equilibrium method leads also to the determination of the temperature of the photospheric black body necessary to produce the required emerging luminosity. This value T B is usually higher than T, since a considerable fraction of the photons emitted at the photosphere is eventually re-absorbed in the photosphere after a series of scattering events in the envelope. This process, known as backwarming, leads to an increased temperature gradient at and immediately above the photosphere. The method outlined above is well suited to handle the photon branching regime, since every energy packet carries the same energy but is characterised by a particular frequency. The computation of the various radiation field moments is therefore unchanged with respect to ML93. Of course, the assumption T e =.9T R is only valid approximately, and the real temperature structure depends on the ionisation regime, so that a fine analysis performed in NLTE and taking into account the physical processes coupling the kinetic temperature to the radiation field may lead to somewhat different results. We can examine the effect of using photon branching on the temperature structure by running the MC SN code with the same input parameters with branching or in the pure scattering approximation. The number of escaping photons is different in the two regimes. In the fast differentially expanding envelope of a SN, where lines efficiently block wide regions of the spectrum, photons can only escape in certain line-free spectral windows. In the pure scattering case, this window is always the next available one to the red of the cmf frequency of the photon as it was emitted at the photosphere. In order to reach these windows a photon may have to scatter a large number of times in the envelope, until differential Doppler shift (to the red) in the cmf eventually takes it into one such window. Since scattering is isotropic, the more times a photon scatters around in the large optical depth near-photospheric regions the more likely it is that it will be re-emitted in a backward direction and reabsorbed in the photosphere. The introduction of branching gives photons the possibility to make large jumps in frequency. Given the typical distribution of energy levels in an ion, such jumps are more likely to be from blue to red than from red to blue, as we shall describe later. During the photospheric phase, a SN emits most of its photons at blue and UV wavelengths, while the line-free windows are typically in the visible part of the spectrum. With branching, a fraction of these blue photons are transformed into visible-wavelength photons deep in the envelope and can therefore escape more easily. Backwarming is thus reduced, and the temperature structure is flatter than in the scattering regime. Also, the rapid transfer of photons from high- to low-optical depth wavelengths reduces the computation time significantly. Since computation time is proportional to diffusion time in our

4 78 P.A. Mazzali: Improved Monte Carlo code for early-time Supernova spectra Scattering Branching Fig. 1. Number of photons crossing a shell s midpoint as a function of ejecta velocity (normalised to v ph ) for a typical SN Ia model near maximum light in the two regimes (pure scattering and branching) Branching Scattering obviously the same in the two models, while the marks represent the equivalent photospheric black body temperatures T B. T R (r) is different in the two regimes. In the model that includes branching there is less backscattering, and more photons escape. Since this effect concerns mostly blue photons, and T R (r) depends on the mean photon frequency (Eq. (1)), T R (r) is lower near the photosphere and higher in the outer, high velocity regions. The resulting temperature structure is therefore flatter when branching is used. In the case of SNe II the behaviour is similar, but the change in T R (r) going from the scattering to the branching regime is smaller because of the lower T for these objects Radiative transfer Although this is the part that is most affected by the introduction of branching, the formal changes are actually quite small. As in the previous version of the code, continuum formation is assumed to take place only below a sharply defined photosphere, i.e. the Schuster-Schwarzschild approximation still holds. The only possible interactions for a photon are electron scattering and line excitation. Therefore, the calculation of the probability of interaction with an electron or a line is unchanged with respect to the previous code. The electron scattering opacity is treated as a continuous function of distance travelled by the packet, the physical properties (electron density in this case) being held constant within each shell, while line opacity takes the form of a series of δ-functions, because a photon can only interact with a spectral line when the relative Doppler shift caused by the expansion of the envelope brings the two in resonance. This was extensively described in Sects. 2.3 and 3.4 of ML93. However, in the case of branching, following an absorption by a spectral line the packet may be re-emitted at a completely different frequency from that at which it was absorbed. One particular downward transition is selected randomly with a MC simulation. The probability of each transition downward from level j is assigned according to the effective downward rate A ji β ji, where A ji is the Einstein coefficient for spontaneous emission, and β ji =(1 exp( τ ji ))/τ ji is the escape probability, which introduces a correction for photon trapping. The Sobolev optical depth τ ij = hν ji (B ij n i B ji n j ) λ jit 4π (2) Fig. 2. Temperature structures for a typical SN Ia model near maximum light in the two regimes (pure scattering and branching). MC calculation this confirms, at least qualitatively, the results of Pinto & Eastman (2). In Fig. 1 we plot the number of photons crossing a shell s midpoint as a function of velocity for a typical SN Ia model near maximum light in the two regimes (pure scattering and branching). In Fig. 2 we show the corresponding temperature structures. The dashed line is the effective temperature T, which is is given here in the form appropriate for a velocity law v = r/t. Einstein B-coefficients have been preferred to the oscillator strengths and statistical weights. If a large enough number of packets is used in the simulation, the distribution of the re-emitted packets in the various lines can be accurately described. One consequence of this somewhat simplified approach is that it is still possible to conserve a packet s energy in the co-moving frame, which simplifies our calculations and is an accurate description of reality as long as enough energy packets are sampled. In our code all packets have been defined to have the same energy. This is simply achieved

5 P.A. Mazzali: Improved Monte Carlo code for early-time Supernova spectra 79 when branching occurs by only changing the frequency of the emitted packet Branching ratios After absorption of a photon, the electron is on level j, whence it can decay to lower levels i. The probability of spontaneous decay to a particular level i is then: p ji = A jiβ ji hν ji i A jiβ ji hν ji. (3) The factor hν ji must be introduced because energy packets are used in the MC experiment. It would not be necessary if photon number were used. Branching can be switched on and off easily in the code. Applying this technique does not actually lead to a significant increase in computation time, and so the code can still be used for the rapid analysis of SN spectra. It is also clear from the description above that this scheme represents the starting point for developing a fully NLTE code, because the various radiative rates can be used to solve a matrix of level populations. A NLTE rate equation solution based on MC estimates of the mean intensities would of course also suffer from MC noise The emergent spectrum In the earlier MC codes (Abbott & Lucy 1985; ML93), the emergent spectrum was derived directly from the packets that escape to infinity. A frequency grid was set up and the rest energy of each escaping packet was added to the bin appropriate for the rest frequency of the packet s photons. Accordingly, at the end of a simulation, a somewhat noisy estimate of the object s luminosity density L ν was obtained. For SNe, simulations with at least 5, packets are necessary with this method if sampling errors are not to hinder comparisons with observed spectra. For this new code, an alternative procedure has been developed that yields high quality synthetic spectra from simulations with as little as 1, packets, thus greatly speeding up the calculation and therefore facilitating the interactive fitting of observed spectra. This technique derives from the formal integral for the emergent intensity and is not restricted to spherical geometry nor to homologous expansion, though these assumptions are made in this application to SNe. The method is thoroughly described in L99a, and we shall not discuss it again here Comparison to observed spectra When fitting a spectrum with our MC code, our input parameters lead to changes in the overall flux (L), in the line velocity (v ph ), and in the overall nature of the spectrum (the temperature, which depends on both L and v ph and also on the details of the explosion model used). Therefore, given an observed spectrum and an assumed epoch and distance, we can try to narrow down the range of the parameters L and v ph based on the quality of the fit. Thus we can also estimate the distance to the SN under study. We judge the quality of a synthetic spectrum by its ability to reproduce the observed spectral features. This is essentially the result of deriving an appropriate temperature structure, and hence the correct runs of ionisation and excitation with radius. For example, if we tested distances that are different by a factor of 2 (i.e. 1.5 mag in distance modulus), we would have to compute two sets of models with L s different by a factor of 4 in order to reproduce the observed flux level. If this change in L is not accompanied by a simultaneous change in v ph, the two sets of models would have temperatures different by 4 1/ This would lead to significant differences in the synthetic spectra. On the other hand, the two models can be forced to have an almost identical temperature if the increase in L by a factor of 4 is accompanied by an increase of v ph by a factor 4 1/2 =2.In this case the temperature does not change, but the photosphere moves significantly in radius/velocity space, and thus the line features have different redshifts: a larger v ph means bluer synthetic lines. This is particularly noticeable for those lines that are reasonably well isolated in wavelength space. Agreement with the observed line velocity indicates which distance should be favoured. A second order effect is also that for a large v ph the ejecta mass above the photosphere, with which the energy packets released at the photosphere can interact, is smaller. This leads to reduced backwarming and to a flatter temperature structure. If v ph is very large, a given explosion model may not even have sufficient mass above the photosphere to yield the observed line strenghts. Thus, explosion models with different properties (ejected mass, kinetic energy) can be discriminated on the basis of how they reproduce the spectra. A further uncertainty in the determination of the distance using our MC code is introduced by the use of black body radiation as the lower boundary at long wavelengths. This is discussed further in Sect For some objects and at some epochs this is not a correct assumption, and it leads to an overestimate of the luminosity, and hence of the distance, but fortunately this error is small, of the order of 5% at most. However, the overall aspect of the emerging spectrum depends on the line opacity, which is treated accurately in our code. If we do not perform the temperature iteration, using for the lower boundary a black body or a delta function at a wavelength smaller than that of the region where line opacity is very strong, e.g. 2 Å, yields synthetic spectra that are indistinguishable. We use a black body because this greatly simplifies the temperature iteration procedure. 4. Applications Having described the features of the code, we now concentrate on a few examples of applications to real SN model spectra. We chose cases where the introduction of branching could be expected to make a clear impact on the quality of the synthetic spectra, and possibly on the physical parameters required to reproduce the observations. The branching regime has one main effect on the propagation of the packets: it allows jumps from the blue to the red region of the spectrum, where the optical depth due to line transitions is smaller and whence photons can more

6 71 P.A. Mazzali: Improved Monte Carlo code for early-time Supernova spectra Fig. 3. Ionization structures of Si and Fe for a typical SN Ia model near maximum light in the two regimes (pure scattering and branching). easily escape. As shown in the previous section, this affects the fraction of escaping photon, but also the temperature structure in the envelope and, of course, the emerging spectrum. The examples we give below are for those cases where the emergent spectrum is most affected. We must note that in general the effect of branching on the optical spectrum turns out to be fairly subtle, which has the comforting implication that the results obtained with the previous scattering code are not invalidated. The effect of branching on the properties of the envelope and on the synthetic spectra can be assessed by comparing models computed with branching and with scattering. Since in the branching regime the intensity of the radiation field is not reduced as greatly with increasing radius in the ejecta as in the pure scattering regime, and the temperature structure is flatter (see Fig. 2), the ionisation and excitation conditions change less rapidly. Pinto & Eastman (2) obtained a similar result for the same reasons. In Fig. 3 we compare the run of ionisation with radius obtained using branching for Si and Fe in the envelope of a typical SN Ia near maximum with the results obtained for pure scattering. When branching is introduced the higher temperature in the outer ejecta leads to a larger extension of the more highly ionized species. This means that the lines that are strong near the photosphere (usually lines arising from lower levels of higher excitation energy than lines which are important further out) will weaken less rapidly with radius when branching is introduced, leaving a stronger signature on the synthetic spectrum The UV spectrum of SNe Ia Although branching is mostly from blue to red, some blue and UV photons are also generated by decays down to low lying Fig. 4. Branching matrix for a typical SN Ia model near maximum. levels in the outer part of the envelope, where the supporting radiation field is weaker. This turns out to be a very important process, because these blue photons can escape more easily than those generated deep in the envelope, which encounter a very large UV line opacity. Therefore, while the pure scattering model had difficulty reproducing the UV flux of SNe Ia near maximum, with the synthetic spectrum being usually far too weak (see, e.g., Pauldrach et al. 1996), in the branching model UV photons are produced far above the photosphere, whence they can escape and give rise to a synthetic UV spectrum which compares well with the observations. In Fig. 4 we show the branching matrix obtained from the SN Ia calculation. The diagonal line represents events for which the absorption and emission wavelengths were the same, i.e. scattering events. For clarity, all points in the diagram have the same size, independently of the number of times a particular jump actually happened in the MC experiment. Scattering events actually dominate in number over branching ones, as shown also by Pinto & Eastman (2), who showed a similar matrix obtained from an explicit NLTE calculation. Nevertheless, the diagram shows clearly how absorption in the UV (especially at λ 3 A ) is often followed by emission in the visible (λ 4 A ). These events are marked by points above the diagonal. Points below the diagonal represent reversely fluorescent events, for which the wavelength of emission is shorter than that of absorption. Although reversely fluorescent events are much rarer than jumps to the red, they are very important, because it is these events that give rise to the observed UV spectrum of SNe Ia at early times. In fact, the UV photons produced at the photosphere face too large an optical depth to be able to escape without first being redshifted to optical wavelengths. In Fig. 5 we compare the combined optical-uv spectrum of the SN Ia 1992A observed about 5 days after maximum with

7 P.A. Mazzali: Improved Monte Carlo code for early-time Supernova spectra old MC new MC SN 1992A t = 25 d log L = Fig. 5. The combined UV-optical spectrum of the type Ia SN 1992A (Kirshner et al. 1993, bold line) is compared with two synthetic spectra obtained with the new code: the thin continuous line is the spectrum computed using photon branching, the dotted line is the spectrum computed using the pure scattering approximation. two synthetic spectra, obtained with the new MC code in the scattering and branching regime, respectively. In Fig. 6 a blowup of the UV spectrum is shown. In both figures, the dashed line shows the model computed for pure scattering, but using the new line list, while the thin continuous line is the spectrum computed using branching. The synthetic UV spectrum produced when branching is included is almost entirely due to photons generated via reverse fluorescence sufficiently far above the photoshere that the optical depth they encounter at UV wavelengths is small. The ions principally responsible for transforming UV photons into optical photons are Fe ii, Feiii, Co ii, Niii, Crii and Ti ii, which are the main contributors to the UV opacity. Reverse fluorescence is due mostly to the same ions, but with a significant contribution from Si ii, which strongly absorbs in the λλ6347, 6371 doublet. Although not all the observed features are perfectly reproduced, the overall agreement is quite good. This is particularly important for the estimate of the luminosity. The parameters used for the fit were not very different from those used in the model shown in Pauldrach et al. (1996) (log(l/l )=9.4, v ph = 725 km s 1 ). The abundances are also similar, but C had to be reduced to avoid the spurious formation of the 65 Å line, which is not present in the observed spectrum. This suggests that C must be confined to the outermost layers of the ejecta SN 1992A t = 25 d branching scattering Fig. 6. Same as Fig. 5, but showing only the UV part of the spectra. The escaping photon fraction, the temperature structure and some ionisation runs for the two models were shown in Figs. 1, 2 and 3. The similarity of the two synthetic spectra in the optical

8 712 P.A. Mazzali: Improved Monte Carlo code for early-time Supernova spectra SN 1991bg t = 18 d old MC (scatt.) new MC, scatt. new MC, branch Fig. 7. The optical spectrum of the peculiar type Ia SN 1991bg (Turatto et al. 1996, bold line) is compared with three synthetic spectra: two of these were computed with the new MC code, one (dashed line) using the pure scattering approximation, the other (thin continuous line) with branching. The third synthetic spectrum (dash-dotted line) is that shown in Fig. 3 of Mazzali et al. (1997), and was computed with the old scattering code. region stems essentially from the fact that line opacity is almost entirely to be found shortwards of a limiting wavelength of about 65 Å, as discussed earlier. Thus, the validity of the results obtained with the previous scattering code in the optical part of the spectrum is not affected. Also, since only a small fraction of the total luminosity is in fact emitted at UV wavelengths, the error on the previous estimates for L was also small. In the red part of the optical region, both synthetic spectra lie higher than the observed one (Fig. 5). This poor fit is due to our assumption of black body radiation at the lower boundary. The lack of spectral lines in the red means that the effective opacity is much lower in that region than in the B and V part of the spectrum, and hence our assumed photospheric radius is larger than its real value. Pinto & Eastman (2) indeed show that the thermalization depth to the centre is insufficient to produce a blackbody photosphere. The error introduced is similar to that which was made in the UV when branching was not used, but now it has the opposite sign: for the spectrum shown in Fig. 5 L is overestimated by about 1% at most, and so the distance may be overestimated by about 5%, which is not a very large error considering the range of values presently quoted for, e.g. the Hubble constant. Fortunately, this problem does not manifest itself in every calculation: for cooler objects, such as SN 1991bg or SN 1987A (see the next two sections), the red flux is reproduced correctly The spectrum of the peculiar SN Ia 1991bg We have shown that branching occurs mostly from bluer to redder wavelengths, and that the optical spectrum of normal SNe Ia is not greatly affected by this process. Nevertheless, we are aware of at least one case of a SN Ia where the synthetic optical spectrum obtained with a pure scattering model shows an unexplained emission feature which we already suggested could perhaps be eliminated if branching was used. Models for the peculiar SN Ia 1991bg near maximum (Mazzali et al. 1997) produced a spurious peak in one of the bluest line-free windows, near 45 Å. We attributed that feature to the neglect of branching, or to the absence in the previous code of some source of opacity. We can now test the effect of branching, and we also have available a much bigger line database, so we computed the synthetic spectrum again, in both the branching and scattering approximations. In Fig. 7 we show the maximum-light spectrum of SN 1991bg and three synthetic spectra: two of these were computed with the new MC code, one (dashed line) using the pure scattering approximation, the other (thin continuous line) with branching. The third synthetic spectrum (dash-dotted line) is that shown in Fig. 3 of Mazzali et al. (1997), which was computed with the old scattering code. All three models have the same parameters, those used in Mazzali et al. (1997). When

9 P.A. Mazzali: Improved Monte Carlo code for early-time Supernova spectra 713 the new code (which introduces the new atomic data) is used, even in the pure scattering approximation the peak at 45 Å is weakened, and a second region of flux excess is now present at about 53 Å, indicating a shift to the red of the emergent flux caused only by using the new line list. It is also possible to note that the complex feature between 45 and 52 Åis reproduced much better with the new line list. The two facts are indeed related. The lines of Fe ii λλ4924, 518 and 5169, which were very strong with the old code, have smaller gf values in the new line list. Also, the new list includes a Cr ii multiplet (λλ4824, 4848, 4864) which was not present in the old one. The consequent strong Cr ii absorption near 475 Å makes the entire absorption trough between 46 and 52 Å broader. Many photons that could escape near 45 Å when the old line list was used are now scattered further to the red, into the high line optical depth region between 47 and 52 Å, where they are scattered again. Thus, these photons can only escape in the next low line optical depth window, near 53 Å, giving rise to the second peak in the synthetic spectrum. Of course this is not correct either, as shown by the excess flux in that region of the synthetic spectrum. Once again, the possible reasons for the incorrect behaviour are either that branching is important or that some other element that we have not considered gives rise to further opacity near 53 Å. Elsewhere, the synthetic spectrum is similar to that obtained with the old code. There is even less flux in the near-uv compared to the old model, and this is due mostly to the presence of strong lines of Cr ii which were not present in the old line list, but also to some new lines of Co ii and Ti ii. Let us now look at the model with branching (thin continuous line): the extra emission at 45 Å has almost disappeared, which is just what we expected as a result of introducing branching. The emission near 55 Å is also greatly reduced. Fe ii and Ti ii are the ions mostly responsible for removing photons from the two large optical depth regions (4 45 Å and Å) through branching, so that these photons do not have to escape in the two narrow low optical depth regions located immediately to the red (45 46 Å and Å). Some packets have been recycled into the UV (the near-uv flux is higher in this model) and some into the red. The entire spectrum shortwards of 52 Å is well reproduced now. Several strong lines are weaker now (e.g. Si ii 6347 and 5978, O i 7772, Mg ii 9218), while others have strengthened, like the Ca ii IR triplet, which shares the same upper level with the very optically thick H&K doublet, and therefore is fed photons that result in net emission. Note that all three models had the same input parameters, so in this case using branching does not lead to new parameters for a best fit, but just to a better best fit. We have thus shown that the spurious emission in the scattering models results from the presence of broad absorption troughs: when a photon is trapped in one such trough it can only escape in the next line-free window to the red. Therefore, the broader the trough the larger the number of photons escaping in the next window. This gives rise to the peaks seen in the scattering models. The introduction of branching addresses the physical problem of transferring radiation directly from the blue to the red, and in so doing overcomes the unwanted production of the peaks The spectrum of SN 1987A The final example of the effect of introducing photon branching is that of the best observed SN II, 1987A. In previous work (Mazzali et al. 1992, Mazzali & Chugai 1995) we had used the MC model in the pure scattering regime to derive the abundance of Ba in SN 1987A. In the second of those papers we remarked that the pure scattering approximation was inadequate to reproduce the observed strength of Hα. We developed a simple model for Hα, and showed that on day 8 the Hα emission is due mostly to recombination, although the emission component computed with that model was too strong (Mazzali & Chugai 1995, Fig. 6). In the day 8 spectrum the emission component was much stronger than the absorption one. On day 13, however, the emission and absorption components have similar equivalent widths, but still the scattering model yields an Hα profile which is too weak, both in absorption and in emission (Mazzali & Chugai 1995, Fig. 2). Therefore, Hα emission following recombination should not be very important at this epoch, and the process responsible for the extra line strength at that epoch must also be absorptive. We can now use the new MC code with branching to take into account emission of an Hα photon following absorption in other lines, and therefore we might hope to be able to account for more of the missing Hα emission strength. Of course, recombination emission is not included, so we do not expect that even this model can reproduce the observations perfectly. We have recomputed the model for day 13. Initially, we used the same input luminosity and photospheric velocity as in Mazzali & Chugai (1995) (log L =41.31 erg s 1, v ph = 6 km s 1 ). In Fig. 8 we compare the synthetic spectrum published by Mazzali & Chugai (1995) with one computed with the new code, using the same input parameters and the pure scattering approximation. The new code yields a definitely better spectrum. The temperature is somewhat higher, so that the continuum in the visible is reproduced more accurately, and the various line features are much improved. The Balmer lines, in particular Hβ, are now stronger, and the entire blue spectrum, from 4 to 6 Å, compares very well with the observed one. This is essentially the effect of using a better line list. The strength of Hα, however, does not change significantly. When we switch branching on, we find a noticeable drop in the temperature, so that the flux in the blue falls below the observations. In fact, a better fit for the model with branching is found if a lower v ph is adopted. In Fig. 9 we show a synthetic spectrum computed using branching for v ph = 525 km s 1. This has a temperature structure similar to the pure scattering model with v ph = 6 km s 1, and it has a dilution factor W =.47 at the photosphere, which is more realistic than the value W =.42 of the branching model with v ph = 6 km s 1. The observed spectrum is reproduced very well. Agreement in line velocity does not change significantly when the lower v ph is adopted. Encouragingly, the lower v ph value is very close to that

10 714 P.A. Mazzali: Improved Monte Carlo code for early-time Supernova spectra 1.5 SN 1987A t = d log L = old MC, scatt. new MC, scatt Fig. 8. The optical spectrum of the type II SN 1987A (Hanushik & Thimm 199, bold line) is compared with two synthetic spectra obtained with the scattering approximation: the thin continuous line is the spectrum computedwith the new code, which includes the improved line list; the dotted line is the spectrum computed with the old code, which was also shown Fig. 2 of Mazzali & Chugai (1995). obtained at this epoch from light curve calculations (Blinnikov et al. 2). For comparison, we also show in Fig. 9 a model for the same low v ph, but computed with scattering. Clearly, the scattering model is too hot. In fact, we find that although the two models have the same effective temperature, T = 545 K, the model with scattering has an equivalent black body temperature T B = 6 K, while that with branching has T B = 58 K. This difference is due to the increased backwarming in the pure scattering model, as discussed in Sect Therefore, we adopt the branching model with the lower v ph value as our best model. The reprocessing of UV photons into optical and IR photons near the photosphere is much less relevant in the case of a SN II than it was for a SN Ia, because a SN II has a much cooler photosphere and thus it does not emit a strong UV continuum. Branching therefore acts mainly within the optical region of the spectrum, and ions such as Fe i,feii and Ti ii are the most effective sources of line opacity. Also, material is located mostly at much lower velocities in a SN II than in a SN Ia, and so line blanketing is not as strong and photons can find low optical depth windows in the optical spectrum. This reduces significantly the reverse fluorescence effect. Finally, even though the new synehetic spectrum fits the observed one very well, both the Hα absorption and emission are still too weak. What could still be missing in Hα on day 13? Since the observed emission and absorption components have similar equivalent widths, the process responsible for the extra line strength must involve both absorption and emission in Hα, which rules out net recombination. Accordingly, we tested the behaviour of the model profile when the Hα optical depth is increased arbitrarily. We found that if the optical depth is increased by a factor 1, throughout the envelope, the observed profile is reproduced almost perfectly. Both the nature of the effect and the value of the deviation for the departure coefficients necessary to fit the observed profile are reminiscent of the effect of non-thermal excitation and ionisation, which can keep the excitation/ionisation degree higher than what can be estimated from thermal processes alone. This was actually an early suggestion made by Chugai (1991), who interpreted the unusual H ionisation regime with a model of a cool and deeply recombined atmosphere with clumpy 56 Ni mixed up to velocities of about 5 km s 1. However, it is not clear whether non-thermal effects are really significant as early as day 13 (Lucy 1991). Lucy (1987) suggested possible alternatives. Branching was investigated in this paper, but cascades following absorption in higher order lines could also be important. To investigate this effect, a NLTE treatment should be adopted. NLTE computations by Eastman & Kirshner (1989) gave good results for the Hα profiles in the early-time spectra of SN 1987A. Introducing a NLTE treatment of H in the SN MC code would actually not be very difficult.

11 P.A. Mazzali: Improved Monte Carlo code for early-time Supernova spectra SN 1987A t = d log L = new MC, scatt. new MC, branch Fig. 9. The optical spectrum of SN 1987A (Hanushik & Thimm 199, bold line) is compared with our best fit synthetic spectrum, obtained with the new code using photon branching (thin continuous line). The photospheric velocity adopted in this model is significantly smaller than the value given in Mazzali & Chugai (1995). The dashed line is the spectrum computed with the new code for the same input parameters but in the pure scattering approximation. 5. Summary We have presented and discussed a new MC code for the synthesis of SN spectra during the photospheric epoch. The code has been improved over its previous version with the inclusion of photon branching, which allows photon redistribution to be followed more accurately and efficiently than in the pure scattering approximation. The introduction of photon branching and the use of a new extensive line list lead to significant improvements in the ability of the synthetic spectra to reproduce the observations. The implementation of a formal integral approach to computing the emergent spectrum also increases the code s speed significantly. For example, a good spectrum can be obtained with about 1 4 energy packets, while at least 1 5 packets had to be used to suppress MC noise in the direct accumulation approach adopted previously. We have found that an average gain of a factor 3 in time on the same machine is obtained with this method. The time necessary to compute a spectrum obviously depends on the computer and on the model parameters (abundances, number of lines selected etc.), but times of the order of only several minutes are routinely achieved when using latest generation unix workstations. We have shown several examples where photon branching is instrumental in getting improved results: the UV spectrum of SNe Ia near maximum is mostly the result of reverse fluorescence, a process which can only be described with branching; the maximum light spectrum of a cool, peculiar SN Ia like SN 1991bg shows spurious emission lines if scattering is used, but these features disappear when branching is introduced because the photons are processed away from the optical region; and finally the synthetic spectrum of SN 1987A improves greatly when branching is used along with a smaller value of v ph. Part of the visible flux is in fact due to photons absorbed in the near-uv by metal lines, and a the smaller v ph is necessary to make up for the reduced backwarming when branching is used and is consistent with results from light curve modelling. All the modifications introduced have preserved the code s original nature of being a fast and reliable tool to obtain almost on-line analysis of SN spectra. Obviously, the code s compactness limits its accuracy somewhat, so that the code should be regarded as a means to obtain a coarse analysis of the spectra. More refined techniques should be used if very accurate results on specific issues are sought after. The new code including branching has already been employed successfully to compute synthetic spectra for the dim SN II 1997D (Turatto et al. 1998) and for the two type Ic SNe 1998bw (Iwamoto et al. 1998) and 1997ef (Iwamoto et al. 2). These latter SNe are two rather peculiar objects, their spectra near maximum being rather featureless and displaying only broad features which were at first interpreted as emission lines. Using the new code we were able to show that the spectra were indeed of a photospheric nature, and that very extensive line blocking caused by the high ejecta velocity led to the blending