4. The Monte Carlo code
4.2. Numerical technique
4.2.5. Emergent spectrum
whereonlythese packets contribute to the summation which are actually inter-
acting with this particular line in volume elementV. The statistical accuracy can
be increased by modifying the estimator: ˙ E`u= ∆0 t 1 V(1−e −τ`u )X 0 (4.23) The term (1−e−τ`u
) denotes the probability that a photon redshifted to frequency
νu` is absorbed by the line, whereτ`uis the Sobolev optical depth. In this case
the summation extends overallpackets inVwhich encounter the frequencyνu`
at some time during the MC experiment. Therefore, ˙E`u>0 even if no packet is
absorbed or emitted by this line at any time in the calculation.
4.2.5. Emergent spectrum
The last step is to calculate the emergent spectrum. This is done by solving the formal integral for the emergent intensity but using line and continuum source functions derived from the MC experiment.
Noise-free SN spectra can be computed by incorporating Sobolev theory into
a formal integral calculation of the luminosity densityLν(e.g. Lucy 1991). For
this purpose the source functions or equivalently the level populations have to be known. In fact, if we treat the level populations in NLTE, the formal integral solution is preferable to the MC approach. One could even think of eliminating the MC calculation completely and instead of simply computing the emergent spectrum from the formal integral using the source function obtained from the modified nebular approximation described in Section 4.1.3. However,
4.2 Numerical technique 47
the resulting spectrum differs significantly from the corresponding MC spectrum
and it is further from the real spectrum than the MC experiment.
To be more precise, we know that for some ions a normal level u can be
overpopulated. In the above hypothetical formal integral calculation the error
increases emissivities for all transitionsu→`resulting in emission bumps in the
emergent spectrum. On the other hand, in the MC procedure emissionsu→`
only occur if leveluis excited by an absorption from the ground-state or other
low-lying (mostly metastable) levels. The populations of the levels are typically much more reliably known.
Line source function
The general expression for the line source function after the level populations are determined in NLTE is (Mihalas 1978):
Su` = B Au`nu
`un`−Bu`nu (4.24)
Since our calculation is not carried out in NLTE, we use first the line’s effective emissivity in the Sobolev approximation:
4πju`=Au`nuhνu`βu` (4.25)
where the escape probabilityβu`is defined in Equation 4.18. The line emissivity
in a SN envelope follows immediately from Equations 4.24 and 4.25, together with the Sobolev optical depthτu` (Eq. 4.19):
4πju` = λ4π u`t
(1−e−τu`)S
u` (4.26)
The line source function can then be computed via the following steps:
1. The total rate per unit volume at which energy is absorbed in exciting level
uusing Equation 4.23 is given by:
˙ Eu= X i<u ˙ Eiu (4.27)
2. A fractionqu` of this absorbed energy escapes via the branchu→ `, and
the effective line emissivity becomes:
4πju` =qu`E˙u (4.28)
3. Finally, the line source function Su` is obtained by substituting this value
of 4πju`into Equation 4.26.
Continuum source function
In order to calculate the formal integral line transitions and electron scattering must be considered. Hence the corresponding source function for electron scat- tering, which is the mean intensity in the co-moving frame, must be derived. The co-moving mean intensity of the incident radiation in the far blue wing of
the transition ` → u is Jb
`u. Thus, β`uJ b
`u is given as the mean intensity of the
partially attenuated incident radiation averaged over the line profile. The rate at which energy is removed from the radiation field is:
˙
E`u=(B`un`−Bu`nu)β`uJb`uhν`u (4.29)
If this formula is combined with Equations 4.24 and 4.25 and since the values of ˙
E`uare given by Equation 4.23,Jb`ucan be computed from:
˙ E`u= 4π λu` (1−e−τ`u)Jb `u (4.30)
Given the velocity law in the SN envelopev=r/t, Lucy (1971) showed that J`ru, which is the corresponding quantity in the extreme red wing of the transition, can be derived from Sobolev theory via the formula:
J`ru=J`bue−τ`u+S
u`(1−e
−τ`u
) (4.31)
Therefore, the mean intensities in the blue and red wings of every line in the line list can be derived. They have non-zero values if at least one packet came into resonance with the line at any time during the calculation, even though it is not
4.2 Numerical technique 49
Formal integral
A line of sight intersects the SN envelope with impact parameterp. IfIν(p) is
the limiting specific intensity at rest frequencyνon that line of sight, then the
luminosity density is given by:
Lν=8π2
Z ∞
0
Iν(p)pdp (4.32)
The beam intensity can be incremented either by line interaction or by electron scattering between two consecutive resonances. Both contributions have to be
known in order to computeIν. According to Equation 4.31, the line formation
increment at resonance frequencyνkis given by:
Irk=Ikbe−τk+S
k(1−e
−τk
) (4.33)
with the same notation and superscripts as before.
Electron scattering contributes between thekand (k+1)th resonance with:
Ibk+1 =Irk+ ∆τe(Jk,k+1−Irk) (4.34)
where∆τe(1) is the electron scattering optical depth along that segment and
Jk,k+1denotes the average co-moving mean intensity along the same way. Jk,k+1
is approximated by: Jk,k+1 = 1 2(J r k+J b k+1) (4.35)
with the quantities in parentheses derived from Equation 4.31.
The following initial conditions are required to solve Equations 4.33 and 4.34 recursively: Ib
1ifp>RandIbm =Bν(TB) ifp<R, wheremis the first transition in
the line list with its resonance point sitting above the lower boundaryr=R.
That procedure results in a much more realistic and noise-free spectrum than we would obtain by simply binning the escaped photons according to their frequency unless we used a very large number of packets. Before that spectrum can be compared with observations, a suitable reddening law has to be applied and we must correct for the attenuation due to the distance of the SN.