Time-dependent radiative transfer with PHOENIX (Corrigendum)

D. Jack; P. H. Hauschildt; E. Baron

doi:10.1051/0004-6361/200810982e

Home

All issues

Volume 549 (January 2013)

A&A, 549 (2013) C1

Full HTML

Free Access

Erratum

This article is an erratum for:
[https://doi.org/10.1051/0004-6361/200810982]

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

A&A 549, C1 (2013)

Time-dependent radiative transfer with PHOENIX (Corrigendum)

D. Jack¹^,2, P. H. Hauschildt¹ and E. Baron¹^,3

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

Erratum for: A&A 502, 1043-1049 (2009), DOI: 10.1051/0004-6361/200810982

4. Time-dependent radiative transfer

The equation of radiative transfer is given by: ${\frac{λ}{λ_{\infty}} \frac{1}{c} \frac{\partial I_{λ}}{∂t} |}_{λ} + {\frac{\partial I_{λ}}{∂s} |}_{λ} + \frac{d λ}{d s} \frac{\partial I_{λ}}{∂λ} = - (χ_{λ} + \frac{5}{λ} \frac{d λ}{d s}) I_{λ} + η_{λ},$ $\begin{equation*} \left. \frac{\lambda}{\lambda_\infty}\frac{1}{c}\frac{\partial I_\lambda}{\partial t}\right|_\lambda + \left.\frac{\partial I_\lambda}{\partial s}\right|_\lambda +\frac{{\rm d}\lambda}{{\rm d}s}\frac{\partial I_\lambda}{\partial \lambda}=-\left(\chi_\lambda +\frac{5}{\lambda}\frac{{\rm d}\lambda}{{\rm d}s}\right)I_\lambda+\eta_\lambda, \end{equation*}$ 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 ${[γ (1 + βμ)]}^{-1} \frac{1}{c} \frac{∂I}{∂t}$ $\begin{equation} \left[\gamma\left(1+\beta\mu\right)\right]^{-1}\frac{1}{c}\frac{\partial I}{\partial t} \end{equation}$ (24)where $β = \frac{v}{c}$ $\hbox{$\beta=\frac{v}{c}$}$ 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 $a_{t} \frac{∂I}{∂t} + a_{r} \frac{∂I}{∂r} + a_{μ} \frac{∂I}{∂μ} + a_{λ} \frac{∂λI}{∂λ} + 4 a_{λ} I = η - χI$ $\begin{equation} a_{t}\frac{\partial I}{\partial t} +a_{r}\frac{\partial I}{\partial r}+a_{\mu}\frac{\partial I}{\partial \mu}+a_{\lambda}\frac{\partial \lambda I}{\partial \lambda} +4a_{\lambda}I=\eta-\chi I \end{equation}$ (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 $a_{t} = {[γ (1 + βμ)]}^{-1} \frac{1}{c} \cdot$ $\begin{equation} a_{t}=\left[\gamma\left(1+\beta\mu\right)\right]^{-1}\frac{1}{c}\cdot \end{equation}$ (26)Along the characteristics the equation has the form $\frac{d I_{l}}{d s} + a_{t} \frac{∂I}{∂t} + a_{l} \frac{∂λI}{∂λ} = η_{l} - (χ_{l} + 4 a_{l}) I_{l}$ $\begin{equation} \frac{{\rm d}I_{l}}{{\rm d}s}+a_{t}\frac{\partial I}{\partial t}+a_{l}\frac{\partial \lambda I}{\partial \lambda} =\eta_{l}-(\chi_{l}+4a_{l})I_{l} \end{equation}$ (27)where ds is a line element along a (curved) characteristic and I_l() 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

Current usage metrics show cumulative count of Article Views (full-text article views including HTML views, PDF and ePub downloads, according to the available data) and Abstracts Views on Vision4Press platform.

Data correspond to usage on the plateform after 2015. The current usage metrics is available 48-96 hours after online publication and is updated daily on week days.

Initial download of the metrics may take a while.

[1] Baron, E., Hauschildt, P. H., Chen, B., & Knop, S. 2012, A&A, 548, A67 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]

[2] Hauschildt, P. H., & Baron, E. 1999, J. Comp. Appl. Math., 109, 41 [Google Scholar]

[3] Hauschildt, P. H., & Baron, E. 2004, A&A, 417, 317 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]