discrete rays transfer equation stepsize

3D optically thin NLTE radiative transfer

by Doris Folini and Rolf Walder

Page contents

Transfer part spacer Rate equation part spacer Visualization
History spacer References spacer Versions


D3NEBEL is part of the A-MAZE code package which provides codes for astrophysical simulations.

D3NEBEL solves the NLTE radiative transfer problem for a prescribed 3D density and velocity distribution with a central source of radiation. The code assumes that the transfer part can be treated by a set of independent 1D rays. From a physical point of view, this assumption implies that scattering is neglected as a source of radiation. The code further assumes that the rate equations can be treated using optically thin (nebular) conditions. Also, the atomic physics implemented is based on the assumption that hydrogen is strongly ionized. A temperature distribution may be prescribed as well or it can be calculated consistently.

The code makes use of the 1D code D1NEBEL, which itself has a long history. Some IDL macros to visualize the results accompany D3NEBEL. Some idea of what we have used for D3NEBEL so far you find here.

Transfer part

3D through a set of 1D rays: D3NEBEL assumes nebular conditions. In the 3D case this includes the assumption that the ionizing radiation stems mainly from the central radiation source and propagates radially away from this source. Scattering and emission of ionizing radiation can be neglected or treated 'on the spot'. This allows to split the 3D problem into a set of non-interacting 1D problems, namely a set of 1D rays, each ray originating at the central radiation source. Along each of these rays a 1D radiative transfer problem is solved. The number of these rays in latitudinal and longitudinal direction can be nearly freely chosen.

discrete rays

Proceeding along 1D rays: Along each ray, the spatial step size is adjusted automatically to prevent too large changes in the ionization state of the matter between two points. In practice, this means that a spatial step is made along the ray and the radiation field at this new point is computed. The computation of the radiation field includes geometrical dilution, absorption by matter along the ray, and 'on the spot' or 'diffuse' emission along the ray. Based on this radiation field the rate equation part is solved.

transfer equation

Is the step ok? If the ionization structure is not too different from that at the previous point on the ray the step is accepted. If it differs more than a certain limit, the step is rejected and a new attempt is made with half the step size.


A flux diagram for the solution procedure along a 1D ray you can find here. The diagram is taken from the PhD thesis of Manfred Vogel, 1990, ETH Zürich, Diss. Nr. 9089, and is in german.

Rate equation part

Getting level populations: D3NEBEL is capable of treating nebular condition only, meaning that excited levels are assumed to be not populated. The rate equations are solved solely for the ionization state of the matter. The basic proceeding, somewhat oversimplified, is as follows. The electron temperature is determined, based on the ionization structure of the previous point and the radiation field obtained from the transfer part. Then the ionization state of H and He is deduced, then the ionization of the metals. If necessary, iteration is applied between these parts to reach consistency. The flux diagram from the thesis of Manfred Vogel, 1990, ETH Zürich, Diss. Nr. 9089, illustrates this procedure.

Doppler shifting: Once a spatial step has been accepted, collisional excitations and various recombination processes are computed to obtain the line strengths of optically thin emission lines at this point. Finally, this line emission is appropriately Doppler shifted for each observer and stored in the frequency array each of the observers carries with him. Another spatial step may now be taken.


Software on the basis of IDL has been developed for the visualization of the computed spectra as seen by an observer. Also, the ionization structure along an individual 1D ray can be rendered in the frame of IDL. Some rudimentary matlab tools exist for visualizing the ionization structure in the equatorial plane.


  • 1981: D1NEBEL developed by Hans Rudolf Schild in the frame of his PhD.
  • 1990: Additional physics added to D1NEBEL by Manfred Vogel in the frame of his PhD.
  • 1997: D3NEBEL developed by Michael Psarros and Doris Folini on the basis of D1NEBEL.


D. Folini
Computational approaches to multidimensional radiative transfer and the physics of radiative colliding flows
PhD Thesis, ETH No. 12606, 1998
(Available as a 3.3 MB gzipped ps-file)

M. Psarros
Berechnung optisch dünner Linienprofile in 3 Dimensionen
ETH Semesterarbeit, in german
(Available as a 0.4 MB gzipped ps-file)

H. Nussbaumer and M. Vogel
A new approach to symbiotic stars
A & A, 182, p 51-62, 1987

H. Nussbaumer and H. Schild
A model for V 1016 CYG based on the ultraviolet spectrum
A & A, 101, p 118-131, 1981

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Doris Folini and Rolf Walder
Last Update: October 14, 2002