| Issue |
A&A
Volume 549, January 2013
|
|
|---|---|---|
| Article Number | C1 | |
| Number of page(s) | 1 | |
| Section | Numerical methods and codes | |
| DOI | https://doi.org/10.1051/0004-6361/200810982e | |
| Published online | 12 December 2012 | |
Time-dependent radiative transfer with PHOENIX (Corrigendum)
1 Hamburger Sternwarte, Gojenbergsweg 112, 21029 Hamburg, Germany
e-mail: yeti@hs.uni-hamburg.de
2 Departamento de Astronomía, Universidad de Guanajuato, Apartado Postal 144, 36000 Guanajuato, Mexico
e-mail: dennis@astro.ugto.mx
3 Homer L. Dodge Department of Physics and Astronomy, University of Oklahoma, 440 W Brooks, Rm 100, Norman, OK 73019-2061 USA
e-mail: baron@ou.edu
Key words: supernovae: general / radiative transfer / methods: numerical / errata, addenda
4. Time-dependent radiative transfer
The equation of radiative transfer is given by:
where λ∞/λ = γ(1 + βμ) is the Doppler factor (Baron et al. 2012). In the original work the transformation of the time-dependent term from frequency space to wavelength space was neglected. Thus, our discussion of Eqs. (24)–(27) should read:
For an implementation of the time dependence in the radiative transfer itself, the spherical symmetric special relativistic radiative transfer equation (SSRTE) for expanding atmospheres (Hauschildt & Baron 1999) is extended so that the additional time-dependent term is given by 
(24)where 
is the velocity in units of the speed of light c and γ = (1 − β2) − 1/2 is the usual Lorentz factor. Here, I is the intensity, μ the cosine of the angle between the radial direction and the propagation vector of the light. Using the notation of Hauschildt & Baron (2004), the comoving frame SSRTE with the additional time-dependent term is given by 
(25)where η is the emissivity and χ the extinction coefficient. The wavelength is represented by λ. The additional time-dependent coefficient is given by 
(26)Along the characteristics the equation has the form 
(27)where ds is a line element along a (curved) characteristic and Il() is the specific intensity along the characteristic at point s ≥ 0 (s = 0 denotes the beginning of the characteristic). For a definition of the other coefficients see Hauschildt & Baron (2004).
For the tests presented in this paper, the error is small, since the velocities considered were less than 10% of the speed of light, but it will affect the results at the 10% level. We have tested that the correction does not materially alter our conclusions.
References
- Baron, E., Hauschildt, P. H., Chen, B., & Knop, S. 2012, A&A, 548, A67 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
- Hauschildt, P. H., & Baron, E. 1999, J. Comp. Appl. Math., 109, 41 [Google Scholar]
- Hauschildt, P. H., & Baron, E. 2004, A&A, 417, 317 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
© ESO, 2012
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