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29 changes: 19 additions & 10 deletions docs/physics/montecarlo/lineinteraction.rst
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Expand Up @@ -16,33 +16,41 @@ before (i for initial). Thus, after accounting for the frame transformations,

\varepsilon_f = \varepsilon_i \frac{1 - \beta \mu_i}{1 - \beta \mu_f}

holds. Also, TARDIS treats that the re-emission of the line interaction
holds. Also, TARDIS treats the re-emission of the line interaction
as an isotropic process. Thus,

.. math::

\mu_f = 2 z - 1.


.. note::

In the Sobolev theory, the re-emission direction is given by the so-called
Sobolev escape fraction, with is in general non-isotropic. However, in the
Sobolev escape fraction, which is generally non-isotropic. However, in the
special case of homologous expansion, isotropy is retained.

.. note::

Strictly speaking, the re-mission process occurs in the local co-moving
frame. Thus, the so called angle aberration effect should be taken into
account when transforming into the lab frame. However, TARDIS, currently
Strictly speaking, the re-emission process occurs in the local co-moving
frame. Thus, the so-called angle aberration effect should be taken into
account when transforming into the lab frame. However, TARDIS currently
neglects this effect.

Essentially, the different line interaction treatments only determine how the
frequency of the packet after the line interaction is determined.

**Beta Sobolev Value**
=======================
The **beta Sobolev value** refers to the *escape chance* for packets when they
interact with specific lines. This is a key factor in determining the direction
and frequency of re-emission after a line interaction. In general, the escape
fraction is non-isotropic, meaning the direction of re-emission is not uniform
across all angles. However, in the special case of homologous expansion, the
re-emission remains isotropic. The theory behind the **beta Sobolev value** and
its implications for line interactions is discussed in detail in **Lucy (2000)**.

Resonant Scattering
===================

The simplest line interaction mode assumes that all interactions with atomic
line transitions occur resonantly. This implies that in the co-moving frame the
emergent packet frequency is equal to the incident one. Again accounting for
Expand All @@ -55,7 +63,6 @@ frame, the post-interaction frequency is given by

Downbranching
=============

The so-called downbranching scheme, introduced by :cite:`Lucy1999a`, is an
elegant approach to approximately account for fluorescence effects. In this
scheme, the packet is not re-emitted in the same transitions as it was absorbed
Expand All @@ -68,7 +75,6 @@ about the downbranching scheme, we refer to :cite:`Lucy1999a` and

Macro Atom Scheme
=================

Finally, as the most sophisticated line interaction treatment, a simplified
version of the Macro Atom scheme of :cite:`Lucy2002` and :cite:`Lucy2003` is
implemented in TARDIS. This approach provides a more accurate representation of
Expand All @@ -88,7 +94,6 @@ an in-depth derivation of the scheme, we refer to :cite:`Lucy2002` and

Comparison
==========

The different levels of sophistication are illustrated in the following plot,
taken from :cite:`Kerzendorf2014` and showing the incident wavelength versus the
emergent wavelength of Monte Carlo packets in line interactions. The left panel
Expand All @@ -98,3 +103,7 @@ downbranching scheme and the right one the macro atom results.
.. image::
../images/scatter_downbranch_ma.png
:width: 700

References
==========
.. [Lucy2000] Lucy, L. B. (2000). "Radiative Transfer in Astrophysics: The Sobolev Approximation and Its Applications." Astrophysical Journal, 550(2), 909-921.
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