5. Emergent spectrum
In this section, a procedure for calculating the emergent spectrum is developed that is greatly superior to the crude estimate provided by the frequency distribution of escaping packets. This procedure uses the formal integral for the emergent intensity but with line and continuum source functions derived from the Monte Carlo experiment.
5.1. Basic idea
Without recourse to Monte Carlo techniques, noise-free SNe spectra can be computed by incorporating Sobolev theory into a formal integral calculation of the luminosity density (e.g. Lucy 1991). However, this requires that the source functions be known or, equivalently, the level populations. Accordingly, in the context of a NLTE treatment of level populations, the formal integral approach is indeed preferable to a Monte Carlo simulation. However, in the context of approximate treatments of level populations and line formation, the relevance of the formal integral is not at all evident. Nevertheless, the prospect of thereby eliminating or reducing sampling errors makes this approach worth exploring.
One obvious possibility would be to eliminate the Monte Carlo calculation entirely and simply compute the emergent spectrum from the formal integral with source functions evaluated using our approximate formulae (Sect. 2) for the level populations. However, the resulting emergent spectrum would differ systematically from the corresponding Monte Carlo spectrum. Moreover, apart from sampling errors, the Monte Carlo spectrum will in general be closer to the truth than the formal integral spectrum calculated in this way.
To justify this latter remark, suppose that, for some ion, our approximate formulae seriously overpopulate a normal level u. In the above hypothetical formal integral calculation, this error directly translates into enhanced emissivities for all lines , with corresponding spurious emission bumps in the emergent spectrum. On the other hand, in the Monte Carlo simulation, emission in the transitions only arises following absorptions that excite level u, and these occur at a rate governed by the typically much more reliably estimated populations of the ground state and other low-lying (especially metastable) levels.
With the origin of this expected superiority of the Monte Carlo spectrum thus understood, it is now of interest to construct a formal integral procedure that incorporates this highly desirable insensitivity to the population of upper levels. To do this, the line source functions implicit in the Monte Carlo simulation must be extracted. This in turn can be done by noting that, in effect, the Monte Carlo simulation derives line emissivities for transitions by applying approximate branching ratios to the rate at which transitions absorb energy.
5.2. Line source function
Expressed in terms of level population and Einstein coefficients, the general expression for the line source function is
and this is the form useful in a formal integral calculation after a NLTE determination of the level populations.
Because a NLTE calculation is not carried out, it is useful first to express the line source function in terms of the line's effective emissivity, which, in the Sobolev approximation, is
is the escape probability and
is the Sobolev optical depth. Note that is given here in the form appropriate for velocity law and that Einstein B-coefficients have been preferred to the oscillator strength and statistical weights.
From Eqs. (14)-(17), it immediately follows that
a formula valid for line formation in a SN envelope treated in the Sobolev approximation.
Eq. (18) now allows the line source function to be computed from data accumulated during the Monte Carlo simulation. The steps are as follows:
1) From the values of given by Eq. (11), we immediately have an estimate of the total rate per unit volume at which energy is absorbed in exciting level u,
Moreover, from the discussion in Sect. 4.2, we expect this estimate to be superior to that obtained simply by tallying absorbed packets.
2) For the adopted model of line formation, the fraction of this absorbed energy escapes via the branch . Accordingly, the effective line emissivity is estimated to be
Note that the Sobolev optical depths and escape probabilities needed to evaluate these branching probabilities from Eqs. (8) and (15) are derived using our approximate formulae for excitation and ionization (Sect. 2).
3) Finally, substitution of this value of into Eq. (18) gives the desired line source function .
From the discussion in Sect. 4.2 of the superior performance expected of the estimator for given in Eq. (11), it follows that the above steps for extracting from a Monte Carlo simulation should yield data from which a high quality emergent spectrum can be derived. In particular, thousands of weak lines in a typical line list, the majority of which neither absorb nor emit a packet during a simulation, will be assigned non-zero values of by this procedure.
5.3. Continuum source function
In addition to bound-bound transitions, our model of a SN's reversing layer also includes electron scattering. This process must therefore also be included in the formal integral calculation, and so the corresponding source function, the mean intensity in the co-moving frame, must be derived. Fortunately, this quantity is readily evaluated at the many thousands of frequencies in a typical line list by applying Sobolev theory to the estimates of given by Eq. (12).
If denotes the co-moving mean intensity of the incident radiation in the far blue wing of the transition , then the mean intensity of the partially attenuated incident radiation averaged over the line profile is, in the Sobolev approximation, . Accordingly, the rate at which this transition absorbs energy from the incident radiation field is
When combined with Eqs. (14) and (15), this implies that
a formula that allows to be derived from the values of given by Eq. (12). Note that an implicit assumption here is that there are no population inversions so that stimulated emissions can be treated as negative absorptions and remains non-negative. The excitation formulae of Sect. 2 are consistent with this assumption.
From the values of so derived, the corresponding quantity in the extreme red wing, , can also be derived with a further application of Sobolev theory. Applying the analysis in Lucy (1971) to velocity law , we readily find that
Accordingly, with and already derived from , this equation gives us the further quantity . Thus, for each spherical shell of the discretized SN envelope, this analysis gives us mean intensities in the blue and red wings of every line in the line list, with non-zero values whenever at least one packet came into resonance with the line during the simulation. In Sect. 5.5, these discrete values of are used to approximate the electron scattering source function.
5.4. Monte Carlo estimator for
From Eqs. (12) and (22), we immediately derive
as a Monte Carlo estimator for the mean intensity in a line's extreme blue wing, where, as for Eq. (12), the summation is over packets that resonate with the line during the simulation.
In the original version of this investigation, this formula was derived from an energy density argument (cf. Lucy 1999) and was the starting point for the source function calculations developed in this Section. Here preference has been given to the estimator as starting point because the origin of superior performance is then clearer. Nevertheless, it is worth emphasizing that this estimator for provides the radiative coefficients for a NLTE treatment of excitation carried out in the context of a Monte Carlo simulation. Moreover, in accordance with the discussion of Sect. 4.2, these coefficients are non-zero provided only that at least one packet comes into resonance during the simulation: it is not actually necessary that any excitations occur.
5.5. Formal integral
If denotes the limiting specific intensity at rest frequency of a beam that intersects the SN envelope with impact parameter p, then
is the luminosity density in the rest frame.
To calculate , we must evaluate (i) the increments in intensity due to line formation at the points where the beam resonates with lines and (ii) the increments due to electron scattering along the segments between consecutive resonances. If denotes the line frequencies in line list A, then the line formation increment at the point of resonance with is given by Sobolev theory as
where the superscripts r and b denote the far red and blue wings as before, and is derived as described in Sect. 5.2.
Now, in the absence of continuum processes, we would have as in Lucy (1991), and so we could proceed recursively through the line list to calculate the limiting intensity. But here electron scattering is included, and we estimate its contribution over the segment between the k and th resonances as
where is the electron scattering optical depth along that segment, and denotes the average co-moving mean intensity along the same segment. For this latter quantity, we adopt the approximation
with the quantities on the right-hand-side calculated as described in Sect. 5.3.
The initial conditions required for the recursive application of Eqs. (26) and (27) are: if and if , where m denotes the first transition in line list A for which the point of resonance is above but not occulted by the lower boundary .
© European Southern Observatory (ESO) 1999
Online publication: April 12, 1999