Astronomy & Astrophysics manuscript no. p0603 c© ESO 2018
November 29, 2018
Generalized
ǫ-law
The role of unphysical source terms in resonance line polarization transfer and
its importance as an additional test of NLTE radiative transfer codes.
J. Štěpán1,2 and V. Bommier2
1 Astronomical Institute, Academy of Sciences of the Czech Republic, 251 65 Ondřejov, Czech Republic
e-mail: stepan@asu.cas.cz
2 LERMA, Observatoire de Paris – Meudon, CNRS UMR 8112, 5, Place Jules Janssen, 92195 Meudon Cedex, France
e-mail: [jiri.stepan;V.Bommier]@obspm.fr
Received 4 October 2006 / Accepted 26 March 2007
ABSTRACT
Context. A derivation of a generalized
ǫ-law for nonthermal collisional rates of excitation by charged perturbers is presented.
Aims. Aim of this paper is to find a more general analytical expression for a surface value of the source function which can be used
as an addtional tool for verification of the non-LTE radiative transfer codes.
Methods. Under the impact approximation hypothesis, static, one-dimensional, plane-parallel atmosphere, constant magnetic field of
arbitrary strength and direction, two-level atom model with unpolarized lower level and stimulated emission neglected, we introduce
the unphysical terms into the equations of statistical equilibrium and solve the appropriate non-LTE integral equations.
Results. We derive a new analytical condition for the surface values of the source function components expressed in the basis of
irreducible spherical tensors.
Key words. line: formation – polarization – radiative transfer
1. Introduction
In the series of papers of Landi Degl’Innocenti et al. (1991a,b),
Landi Degl’Innocenti & Bommier (1994) (from now on refer-
enced as Paper I), the general formalism of resonance line po-
larization scattering for a two-level atom has been developed.
The non-LTE problem of the 2nd kind for an arbitrary magnetic
field, three-dimensional geometry of the medium and arbitrary
irradiation by external sources has been discussed. The effect of
inelastic collisions with charged perturbers has been considered
for the particular case of a relative Maxwellian velocity distribu-
tion.
Paper I analysed the analytical properties of the solutions in
the particular case of a one-dimensional, semi-infinite, static at-
mosphere with a constant magnetic field of arbitrary strength
and direction and assuming zero external irradiation of the at-
mosphere. They derived a generalization of the well known
law (e.g. Avrett & Hummer 1965; Mihalas 1970; Hubený 1987)
for the case of polarized radiation and extended the previous re-
sults of Ivanov (1990) who studied scattering in a non-magnetic
regime.
In most cases of practical interest the polarization degree is
rather small. The purpose of this paper is to find a new analytical
solution of the non-LTE problem in unphysical conditions in or-
der to better verify the accuracy of the polarized radiation trans-
fer codes. This is done by introduction of an unphysical source
term in the polarization into the equations of statistical equilib-
rium. Such a generalization can be useful in testing the accuracy
of the radiative transfer codes whose purpose is to deal with the
Send offprint requests to: J. Štěpán
non-thermal collisional processes (for instance in the impact po-
larization studies of solar flares).
Following the approach of the papers quoted above, we adopt
the formalism of density matrix in the representation of irre-
ducible tensorial operators (e.g. Fano 1957). We consider the
lower level with total angular momentum j to be unpolarized.
This level is completely described by the overall population
which is set to 1 for normalization reasons. The upper level with
angular momentum j′ is described by the multipole components
of the statistical tensor ρKQ. Coherences between different levels
j and j′ are neglected but coherences between Zeeman sublevels
of level j′ are in general taken into account. The calculation is
performed in the Wien limit of line frequency whose assump-
tion makes it possible to neglect stimulated emission effect, and
to preserve the linearity of the non-LTE problem.
2. Equations of statistical equilibrium
The suitable coordinate system Σ0 for atomic state description
is the one with the z-axis directed along the magnetic field (see
Figure 1).
Radiative rate contributions to the evolution of statistical op-
erator ρKQ are given by (Landi Degl’Innocenti 1985)


= −iA j′ jΓQρKQ − A j′ jρ
w(K)j′ j (−1)
2 j′ + 1
B j j′J
−Q. (1)
http://arxiv.org/abs/0704.1573v1
2 J. Štěpán and V. Bommier: Generalized
ǫ-law
In this equation A j′ j (B j j′) is the Einstein coefficient of spon-
taneous emission (absorption) from level j′ ( j) to level j ( j′).
Γ = 2πg j′νL/A j′ j with g j′ being the Landé factor of the level
j′ and νL is the Larmor frequency. The transition-dependent nu-
merical factor w(K)j′ j has been defined by Landi Degl’Innocenti
(1984) as have the irreducible components of the mean radiation
tensor J
Q. Besides the radiative rates, collisional rates have to
be considered in the statistical equilibrium, because the source
of radiation in a semi-infinite atmosphere is the collisional exci-
tation followed by radiative de-excitation. Thus, the source term
of the radiative transfer equation originates in the inelastic col-
lision effect. As the purpose of the present paper is to consider
unphysical source terms in the non-zero ranks (K,Q) of the ir-
reducible tensorial operator basis T KQ , we will introduce an un-
physical (K,Q)-dependence to the inelastic collisional rates of
the statistical equilibrium equation below. The purpose here is
not to thus describe anisotropic collisions, which would require
a proper formalism that is out of the scope of the present paper
(see, for instance, Landi Degl’Innocenti & Landolfi (2004) for
a two-level atom, and Derouich (2006), for polarization trans-
fer rates in a multi-level atom due to isotropic collisions). We
introduce as usual the depolarizing rate due to isotropic elastic
collisions. Thus, the contribution of collisional rates reads


= (C j j′ )
Q − (C
j′ j)
Q − D
Q. (2)
The terms (C j j′ )
Q and (C
j′ j)
Q on the right-hand side of equation
(2) are the multipole components of collisional rates of excita-
tion and relaxation respectively. D(K) is the depolarization rate
due to elastic collisions.1
The radiative and collisional rates can be added under the
impact approximation hypothesis (Bommier & Sahal-Bréchot
1991) dρKQ/dt = [dρ
Q/dt]RAD + [dρ
Q/dt]COLL. Using the equa-
tions (1), (2), and the condition for static atmosphere, dρKQ/dt =
0, we obtain the equations of statistical equilibrium
[iA j′ jΓQ + A j′ j + (C
j′ j)
Q + D
(K)]ρKQ
w(K)j′ j (−1)
2 j′ + 1
B j j′J
−Q + (C j j′)
Q. (3)
By applying the relation between Einstein coefficients for spon-
taneous emission and absorption,
B j j′ =
2 j′ + 1
2 j + 1
2hν30
A j′ j, (4)
and dividing the formula (3) by A j′ j, we obtain the equation
(1 + ǫKQ + δ
+ iΓQ)ρKQ
= (−1)Qw(K)j′ j J
2 j′ + 1
2 j + 1
2hν30
(C j j′ )
A j′ j′
. (5)
One can introduce the dimensionless parameter of the depolar-
ization rate
A j′ j
, (6)
1 This process cannot change a total population of the level.
Therefore it is always D(0) = 0. We take formally into account only
the depolarization rate D(K) to use a formalism coherent with the previ-
ous papers. A general treatment of physically more relevant transfer of
multipole components of the upper level is out of scope of this paper.
Fig. 1. The reference frame Σ1 has its Z-axis oriented vertically
with respect to the atmosphere, while the z-axis of the reference
frame Σ0 is parallel to the direction of magnetic field B. The axes
X and x lie in the same plane defined by Z-axis and B; the axes
Y and y are defined to complement the right-handed orthogonal
coordinate systems.
and the irreducible tensor which plays the role of generalized
photon destruction probability
(CRj′ j)
A j′ j′
. (7)
If the relation (CRj′ j)
Q , 0 is satisfied we may define the quantity
B(KQ) =
2hν30
2 j + 1
2 j′ + 1
(C j j′)
(CRj′ j)
. (8)
It is easy to show (see below) that in the particular case of a
Maxwellian velocity distribution of colliders the relation B(00) =
BP is satisfied, where BP is the Planck function in the Wien limit
at given temperature. Using the definition of irreducible compo-
nents of the two-level source function (cf. Paper I)
S KQ =
2hν30
2 j + 1
2 j′ + 1
Q, (9)
we obtain the statistical equilibrium equations in the compact
(1 + ǫKQ + δ
+ iΓQ)S KQ
= w(K)j′ j (−1)
−Q + ǫ
. (10)
J. Štěpán and V. Bommier: Generalized
ǫ-law 3
3. Solution of the Wiener-Hopf equations
From now on we reduce our analysis to the case of semi-infinite,
plane-parallel geometry with constant magnetic field along the
atmosphere. The velocity distribution and volume density of col-
liders is also constant along the atmosphere but it is in general
non-thermal. The only position coordinate is the common line
optical depth τ.
Following the procedure of Paper I a formal solution of ra-
diative transfer equation is substituted into the definition of ten-
sor J
Q; after that we obtain a set of integral Wiener–Hopf equa-
tions of the 2nd kind,
(1 + ǫKQ + δ
+ iΓQ)S KQ(τ)
K̃KQ,K′Q′ (τ, τ
′)S K
Q′ (τ
′)dτ′ + ǫKQ B
, (11)
which describe coupling of the tensors ρKQ(τ) at differ-
ent optical depths via radiation. Several important prop-
erties of kernels K̃KQ,K′Q′ (τ, τ
′) have been discussed by
Landi Degl’Innocenti et al. (1990) and in Paper I Using their in-
dexing notation one can rewrite the equation (11) in the short-
handed form
aiS i(τ) =
Ki j(|τ − τ′|)S j(τ′)dτ′ + bi, (12)
ai = 1 + ǫ
Q + δ
+ iΓQ, (13)
bi = ǫ
. (14)
The index i in these expressions runs between the limits 1 and
N, where N is the number of KQ-multipoles. In the following
we briefly repeat the derivation performed by Frisch & Frisch
(1975) emphasizing the differences due to presence of bi terms.
Calculation of the derivative of (12) with respect to τ, split-
ting the integral on the right-hand side into two parts, multiplica-
tion of the equation by S i(τ), summation over index i, and finally
integration with respect to τ leads to the set of equations
S i(τ)
dS i(τ)
S j(0)
Ki j(τ)S i(τ)dτ
dτS i(τ)
dτ′Ki j(|τ′ − τ|)
dS j(τ
. (15)
The left-hand side of (15) is easily evaluated as
S i(∞)2 − S i(0)2
The first term on the right-hand side of (15) is evaluated using
the kernels symmetry Ki j(t) = K ji(t) and the equation (12), so
that we obtain
S i(0) [aiS i(0) − bi] , (17)
while the second term equals
S i(∞)2 − S i(0)2
bi [S i(∞) − S i(0)] . (18)
We put these results into (15) to get
aiS i(0)
biS i(∞). (19)
Calculation of the limit τ → ∞ of both sides of the equation
(12) leads to the set of linear algebraic equations for the compo-
nents of source function tensor in the infinite depth:
a jδi j −
Ki j(t)dt
S j(∞) = bi. (20)
We can solve these equations and write
S(∞) = L−1b, (21)
where S is the formal vector of S i components, b is the formal
vector of bi components, and the elements of matrix L are de-
fined by relation
{L}i j = a jδi j −
Ki j(t)dt. (22)
Establishing a new matrix ℓ = L−1 and substituting (21) into
(19) leads to the generalized form of the
ǫ-law
aiS i(0)
bib jℓi j. (23)
4. Particular solutions
Setting the special conditions for magnetic field and collisional
rates, one recovers the less general but more common and ex-
plicit formulations of the
ǫ-law than the one given by (23).
In the following sections we will verify this result in the limit-
ing conditions assumed in recent papers and we will analyse the
simple examples of non-thermal collisional excitation.
4.1. Maxwellian velocity distribution of colliders
In the case of Maxwellian velocity distribution of colliders, re-
laxation rates of all multipole components ρKQ are the same:
(CRj′ j)
Q = C
j′ j, (24)
where CRj′ j is the usual relaxation rate for collisional deexcitation
from j′ to j. For excitation rates one has
(C j j′)
C j j′√
2 j′ + 1
δK0δQ0, (25)
where the factor (2 j′ + 1)−1/2 has been introduced to make a
connection with the usual collisional rate C j j′ of standard unpo-
larized theory. In this isotropic case, there is no collisional exci-
tation of higher ranks of density matrix. From the assumption of
thermodynamic equilibrium one has
C j j′
CRj′ j
2 j′ + 1
2 j + 1
e−hν0/kBT , (26)
where kB stands for the Boltzmann constant and T for a tem-
perature of the atmosphere. From (24) and (7) it is evident that
ǫKQ = ǫ for all possible K and Q, where ǫ is the common photon
destruction probability. Further
B(KQ) = BPδK0δQ0. (27)
4 J. Štěpán and V. Bommier: Generalized
ǫ-law
Substituting the rates (24) and (25) into the formula (22) and
employing the general identity
−∞ Ki1(t)dt = δi1 (see Paper I)
together with bi = δi1, we recover form (23) the formula (16) of
the previously cited paper:
(1 + ǫ + δ(K) + iΓQ)[S KQ(0)]
= ǫB2P. (28)
Assuming that there is zero magnetic field, i.e. Γ = 0, the
source function tensor reduces due to symmetry reasons to the
two non-vanishing components S 00 and S
0 in the reference frame
Σ1. This reference frame is suitable for descriptions of the atomic
system under these conditions, so that we may identify Σ0 ≡ Σ1,
with X and Y axes oriented arbitrary in the plane parallel to atmo-
spheric surface. Further, assuming that there is no depolarization
of the upper level (δ(K) = 0), we realize from (28):
S 00(0)
S 20(0)
1 + ǫ
ǫ′BP, (29)
which is the same result derived in different notation by Ivanov
(1990). For simplicity the common alternative to the photon de-
struction probability has been introduced: ǫ′ = ǫ/(1 + ǫ).
If depolarization of the upper level is high enough to destroy
atomic level polarization (δ(K) → ∞ for K > 0), or the upper
level is unpolarizable, the common
ǫ-law for scalar radiation
is recovered,
S 00(0) =
1 + ǫ
ǫ′BP. (30)
4.2. Anisotropic alignment (de)excitation
The relation ǫKQ = ǫ is not in general satisfied for all the mul-
tipoles because the relaxation of the ρKQ state depends on the
velocity distribution of colliders. In the following text we will
neglect the effects of magnetic field.
Let us assume an example of a relative velocity distribution
of particles that is axially symmetric with the axis of symmetry
parallel to the vertical of the atmosphere (so that it is as in the
former case Σ0 ≡ Σ1) and that the collisional interaction can be
fully described by only the first two even multipole components
of this distribution. Thanks to these assumptions the only non-
vanishing excitation collisional rates are (C j j′ )
0 and (C j j′ )
0, the
relaxation rates (CRj′ j)
0 and (C
j′ j)
0 and for the same reasons the
only non-zero source function components are S 00 and S
An explicit evaluation of the integrals of kernels∫ ∞
−∞ K̃KQ,K′Q′ (τ, τ
′)dτ′ under these conditions shows that
the only non-zero ones are given by (A5) and (A12) of
Landi Degl’Innocenti et al. (1991b). In our notation they read
K̃00,00(τ, τ
′)dτ′ = 1, (31)
K̃20,20(τ, τ
′)dτ′ =
W2, (32)
with W2 = (w
j′ j)
2. Substituting these results into (23) we see that
(1 + ǫ00 )(S
+ (1 + ǫ20 )(S
(00))2 +
(ǫ20 B
(20))2
1 + ǫ20 −
10 W2
. (33)
To check the validity of polarized radiative transfer codes,
it is advantageous if one can verify that the transfer of higher
ranks of the radiation tensor is accurate enough. In the realistic
scattering polarization models the polarization degree does not
exceed a few percent so that |S 00(0)| ≫ |S
Q(0)|. By setting ar-
bitrary (even unphysical) collisional rates it is possible to verify
transfer codes in conditions with |S 00| ≪ |S
To privilege transfer in higher ranks of the radiation tensor
one can artificially suppress the excitation rate (C j j′)
0. In the
extremal case one can set (C j j′)
0 → 0. The easiest way to do
this is the formal interchange of the role of excitation rates of
population and alignment, i.e. (C j j′ )
0 ↔ (C j j′)
0 of the original
Maxwellian velocity distribution:
(C j j′)
0 = 0, (C j j′ )
C j j′√
2 j′ + 1
(no collisional excitation to upper level population) and the re-
laxation rates set to the Maxwellian ones
(CRj′ j)
0 = (C
j′ j)
0 = C
j′ j. (35)
In this case we have
B(00) = 0, B(20) = BP, (36)
and again
0 = ǫ
0 = ǫ. (37)
Substituting this into (33) we find out the
ǫ-law in the form
[S 00(0)]
2 + [S 20(0)]
1 − 710 W2(1 − ǫ
BP. (38)
The particular collisional rates (34) are in fact arbitrary and
have been chosen to obtain a formula similar to the one of the
Maxwellian distribution case.
This relation is useful to test polarized radiative transfer
codes, because in this unphysical case S 20(0) is the largest term,
unlike the physical case where the largest term is S 00(0) and S
is only a few percent of it. By applying Eq. (38) the test is much
more sensitive to the polarization, and the polarization is better
tested. We have thus successfully tested a multilevel non-LTE
radiative transfer code that we are developing, but this code and
its results are the subjects of a forthcoming paper.
5. Conclusions
We have derived a more general formulation of the so-called√
ǫ-law of radiation transfer. This analytical condition couples
the value of source function tensor of a two-level atom with
other physical properties of the atmosphere. The simplest re-
sult obtained in conditions of a non-magnetic, isothermal, plane-
parallel, semi-infinite atmosphere with thermal velocity distri-
bution of particles and unpolarized atomic levels (e.g. Mihalas
1970) has been generalized by Ivanov (1990) to account for scat-
tering of polarized radiation and polarized upper atomic level.
Further generalizations done in Paper I, which account for a
magnetic field of arbitrary strength and direction, has been ex-
tended in the present paper to account for non-thermal colli-
sional interactions. It was done by introducing the tensor of the
photon destruction probability ǫKQ and by defining the function
B(KQ).
The resulting formula (23) reduces to the cases mentioned
above if the physical conditions become more symmetric. On
J. Štěpán and V. Bommier: Generalized
ǫ-law 5
the other hand, situations with a high degree of perturbers ve-
locity distribution anisotropy and especially ones with unphysi-
cal collisional rates result in a wide range of models which can
be calculated both numerically and analytically. Thus they offer
new possibilities for verification of the non-LTE radiation trans-
fer codes.
References
Avrett, E. H. & Hummer, D. G. 1965, MNRAS, 130, 295
Bommier, V. & Sahal-Bréchot, S. 1991, Ann. Phys. Fr., 16, 555
Derouich, M. 2006, A&A, 449, 1
Fano, U. 1957, Rev. Mod. Phys., 29, 74
Frisch, U. & Frisch, H. 1975, MNRAS, 173, 167
Hubený, I. 1987, A&A, 185, 332
Ivanov, V. V. 1990, Soviet Astronomy, 34, 621
Landi Degl’Innocenti, E. 1984, Sol. Phys., 91, 1
Landi Degl’Innocenti, E. 1985, Sol. Phys., 102, 1
Landi Degl’Innocenti, E. & Bommier, V. 1994, A&A, 284, 865
Landi Degl’Innocenti, E., Bommier, V., & Sahal-Bréchot, S. 1990, A&A, 235,
Landi Degl’Innocenti, E., Bommier, V., & Sahal-Bréchot, S. 1991a, A&A, 244,
Landi Degl’Innocenti, E., Bommier, V., & Sahal-Bréchot, S. 1991b, A&A, 244,
Landi Degl’Innocenti, E. & Landolfi, M. 2004, Polarization in Spectral Lines
(Kluwer acad. Publ.)
Mihalas, D. 1970, Stellar Atmospheres (W. H. Freeman and Company)
	Introduction
	Equations of statistical equilibrium
	Solution of the Wiener-Hopf equations
	Particular solutions
	Maxwellian velocity distribution of colliders
	Anisotropic alignment (de)excitation
	Conclusions

  Context. A derivation of a generalized sqrt(epsilon)-law for nonthermal
collisional rates of excitation by charged perturbers is presented. Aims. Aim
of this paper is to find a more general analytical expression for a surface
value of the source function which can be used as an addtional tool for
verification of the non-LTE radiative transfer codes. Methods. Under the impact
approximation hypothesis, static, one-dimensional, plane-parallel atmosphere,
constant magnetic field of arbitrary strength and direction, two-level atom
model with unpolarized lower level and stimulated emission neglected, we
introduce the unphysical terms into the equations of statistical equilibrium
and solve the appropriate non-LTE integral equations. Results. We derive a new
analytical condition for the surface values of the source function components
expressed in the basis of irreducible spherical tensors.

Introduction
In the series of papers of Landi Degl’Innocenti et al. (1991a,b),
Landi Degl’Innocenti & Bommier (1994) (from now on refer-
enced as Paper I), the general formalism of resonance line po-
larization scattering for a two-level atom has been developed.
The non-LTE problem of the 2nd kind for an arbitrary magnetic
field, three-dimensional geometry of the medium and arbitrary
irradiation by external sources has been discussed. The effect of
inelastic collisions with charged perturbers has been considered
for the particular case of a relative Maxwellian velocity distribu-
tion.
Paper I analysed the analytical properties of the solutions in
the particular case of a one-dimensional, semi-infinite, static at-
mosphere with a constant magnetic field of arbitrary strength
and direction and assuming zero external irradiation of the at-
mosphere. They derived a generalization of the well known
law (e.g. Avrett & Hummer 1965; Mihalas 1970; Hubený 1987)
for the case of polarized radiation and extended the previous re-
sults of Ivanov (1990) who studied scattering in a non-magnetic
regime.
In most cases of practical interest the polarization degree is
rather small. The purpose of this paper is to find a new analytical
solution of the non-LTE problem in unphysical conditions in or-
der to better verify the accuracy of the polarized radiation trans-
fer codes. This is done by introduction of an unphysical source
term in the polarization into the equations of statistical equilib-
rium. Such a generalization can be useful in testing the accuracy
of the radiative transfer codes whose purpose is to deal with the
Send offprint requests to: J. Štěpán
non-thermal collisional processes (for instance in the impact po-
larization studies of solar flares).
Following the approach of the papers quoted above, we adopt
the formalism of density matrix in the representation of irre-
ducible tensorial operators (e.g. Fano 1957). We consider the
lower level with total angular momentum j to be unpolarized.
This level is completely described by the overall population
which is set to 1 for normalization reasons. The upper level with
angular momentum j′ is described by the multipole components
of the statistical tensor ρKQ. Coherences between different levels
j and j′ are neglected but coherences between Zeeman sublevels
of level j′ are in general taken into account. The calculation is
performed in the Wien limit of line frequency whose assump-
tion makes it possible to neglect stimulated emission effect, and
to preserve the linearity of the non-LTE problem.
2. Equations of statistical equilibrium
The suitable coordinate system Σ0 for atomic state description
is the one with the z-axis directed along the magnetic field (see
Figure 1).
Radiative rate contributions to the evolution of statistical op-
erator ρKQ are given by (Landi Degl’Innocenti 1985)


= −iA j′ jΓQρKQ − A j′ jρ
w(K)j′ j (−1)
2 j′ + 1
B j j′J
−Q. (1)
http://arxiv.org/abs/0704.1573v1
2 J. Štěpán and V. Bommier: Generalized
ǫ-law
In this equation A j′ j (B j j′) is the Einstein coefficient of spon-
taneous emission (absorption) from level j′ ( j) to level j ( j′).
Γ = 2πg j′νL/A j′ j with g j′ being the Landé factor of the level
j′ and νL is the Larmor frequency. The transition-dependent nu-
merical factor w(K)j′ j has been defined by Landi Degl’Innocenti
(1984) as have the irreducible components of the mean radiation
tensor J
Q. Besides the radiative rates, collisional rates have to
be considered in the statistical equilibrium, because the source
of radiation in a semi-infinite atmosphere is the collisional exci-
tation followed by radiative de-excitation. Thus, the source term
of the radiative transfer equation originates in the inelastic col-
lision effect. As the purpose of the present paper is to consider
unphysical source terms in the non-zero ranks (K,Q) of the ir-
reducible tensorial operator basis T KQ , we will introduce an un-
physical (K,Q)-dependence to the inelastic collisional rates of
the statistical equilibrium equation below. The purpose here is
not to thus describe anisotropic collisions, which would require
a proper formalism that is out of the scope of the present paper
(see, for instance, Landi Degl’Innocenti & Landolfi (2004) for
a two-level atom, and Derouich (2006), for polarization trans-
fer rates in a multi-level atom due to isotropic collisions). We
introduce as usual the depolarizing rate due to isotropic elastic
collisions. Thus, the contribution of collisional rates reads


= (C j j′ )
Q − (C
j′ j)
Q − D
Q. (2)
The terms (C j j′ )
Q and (C
j′ j)
Q on the right-hand side of equation
(2) are the multipole components of collisional rates of excita-
tion and relaxation respectively. D(K) is the depolarization rate
due to elastic collisions.1
The radiative and collisional rates can be added under the
impact approximation hypothesis (Bommier & Sahal-Bréchot
1991) dρKQ/dt = [dρ
Q/dt]RAD + [dρ
Q/dt]COLL. Using the equa-
tions (1), (2), and the condition for static atmosphere, dρKQ/dt =
0, we obtain the equations of statistical equilibrium
[iA j′ jΓQ + A j′ j + (C
j′ j)
Q + D
(K)]ρKQ
w(K)j′ j (−1)
2 j′ + 1
B j j′J
−Q + (C j j′)
Q. (3)
By applying the relation between Einstein coefficients for spon-
taneous emission and absorption,
B j j′ =
2 j′ + 1
2 j + 1
2hν30
A j′ j, (4)
and dividing the formula (3) by A j′ j, we obtain the equation
(1 + ǫKQ + δ
+ iΓQ)ρKQ
= (−1)Qw(K)j′ j J
2 j′ + 1
2 j + 1
2hν30
(C j j′ )
A j′ j′
. (5)
One can introduce the dimensionless parameter of the depolar-
ization rate
A j′ j
, (6)
1 This process cannot change a total population of the level.
Therefore it is always D(0) = 0. We take formally into account only
the depolarization rate D(K) to use a formalism coherent with the previ-
ous papers. A general treatment of physically more relevant transfer of
multipole components of the upper level is out of scope of this paper.
Fig. 1. The reference frame Σ1 has its Z-axis oriented vertically
with respect to the atmosphere, while the z-axis of the reference
frame Σ0 is parallel to the direction of magnetic field B. The axes
X and x lie in the same plane defined by Z-axis and B; the axes
Y and y are defined to complement the right-handed orthogonal
coordinate systems.
and the irreducible tensor which plays the role of generalized
photon destruction probability
(CRj′ j)
A j′ j′
. (7)
If the relation (CRj′ j)
Q , 0 is satisfied we may define the quantity
B(KQ) =
2hν30
2 j + 1
2 j′ + 1
(C j j′)
(CRj′ j)
. (8)
It is easy to show (see below) that in the particular case of a
Maxwellian velocity distribution of colliders the relation B(00) =
BP is satisfied, where BP is the Planck function in the Wien limit
at given temperature. Using the definition of irreducible compo-
nents of the two-level source function (cf. Paper I)
S KQ =
2hν30
2 j + 1
2 j′ + 1
Q, (9)
we obtain the statistical equilibrium equations in the compact
(1 + ǫKQ + δ
+ iΓQ)S KQ
= w(K)j′ j (−1)
−Q + ǫ
. (10)
J. Štěpán and V. Bommier: Generalized
ǫ-law 3
3. Solution of the Wiener-Hopf equations
From now on we reduce our analysis to the case of semi-infinite,
plane-parallel geometry with constant magnetic field along the
atmosphere. The velocity distribution and volume density of col-
liders is also constant along the atmosphere but it is in general
non-thermal. The only position coordinate is the common line
optical depth τ.
Following the procedure of Paper I a formal solution of ra-
diative transfer equation is substituted into the definition of ten-
sor J
Q; after that we obtain a set of integral Wiener–Hopf equa-
tions of the 2nd kind,
(1 + ǫKQ + δ
+ iΓQ)S KQ(τ)
K̃KQ,K′Q′ (τ, τ
′)S K
Q′ (τ
′)dτ′ + ǫKQ B
, (11)
which describe coupling of the tensors ρKQ(τ) at differ-
ent optical depths via radiation. Several important prop-
erties of kernels K̃KQ,K′Q′ (τ, τ
′) have been discussed by
Landi Degl’Innocenti et al. (1990) and in Paper I Using their in-
dexing notation one can rewrite the equation (11) in the short-
handed form
aiS i(τ) =
Ki j(|τ − τ′|)S j(τ′)dτ′ + bi, (12)
ai = 1 + ǫ
Q + δ
+ iΓQ, (13)
bi = ǫ
. (14)
The index i in these expressions runs between the limits 1 and
N, where N is the number of KQ-multipoles. In the following
we briefly repeat the derivation performed by Frisch & Frisch
(1975) emphasizing the differences due to presence of bi terms.
Calculation of the derivative of (12) with respect to τ, split-
ting the integral on the right-hand side into two parts, multiplica-
tion of the equation by S i(τ), summation over index i, and finally
integration with respect to τ leads to the set of equations
S i(τ)
dS i(τ)
S j(0)
Ki j(τ)S i(τ)dτ
dτS i(τ)
dτ′Ki j(|τ′ − τ|)
dS j(τ
. (15)
The left-hand side of (15) is easily evaluated as
S i(∞)2 − S i(0)2
The first term on the right-hand side of (15) is evaluated using
the kernels symmetry Ki j(t) = K ji(t) and the equation (12), so
that we obtain
S i(0) [aiS i(0) − bi] , (17)
while the second term equals
S i(∞)2 − S i(0)2
bi [S i(∞) − S i(0)] . (18)
We put these results into (15) to get
aiS i(0)
biS i(∞). (19)
Calculation of the limit τ → ∞ of both sides of the equation
(12) leads to the set of linear algebraic equations for the compo-
nents of source function tensor in the infinite depth:
a jδi j −
Ki j(t)dt
S j(∞) = bi. (20)
We can solve these equations and write
S(∞) = L−1b, (21)
where S is the formal vector of S i components, b is the formal
vector of bi components, and the elements of matrix L are de-
fined by relation
{L}i j = a jδi j −
Ki j(t)dt. (22)
Establishing a new matrix ℓ = L−1 and substituting (21) into
(19) leads to the generalized form of the
ǫ-law
aiS i(0)
bib jℓi j. (23)
4. Particular solutions
Setting the special conditions for magnetic field and collisional
rates, one recovers the less general but more common and ex-
plicit formulations of the
ǫ-law than the one given by (23).
In the following sections we will verify this result in the limit-
ing conditions assumed in recent papers and we will analyse the
simple examples of non-thermal collisional excitation.
4.1. Maxwellian velocity distribution of colliders
In the case of Maxwellian velocity distribution of colliders, re-
laxation rates of all multipole components ρKQ are the same:
(CRj′ j)
Q = C
j′ j, (24)
where CRj′ j is the usual relaxation rate for collisional deexcitation
from j′ to j. For excitation rates one has
(C j j′)
C j j′√
2 j′ + 1
δK0δQ0, (25)
where the factor (2 j′ + 1)−1/2 has been introduced to make a
connection with the usual collisional rate C j j′ of standard unpo-
larized theory. In this isotropic case, there is no collisional exci-
tation of higher ranks of density matrix. From the assumption of
thermodynamic equilibrium one has
C j j′
CRj′ j
2 j′ + 1
2 j + 1
e−hν0/kBT , (26)
where kB stands for the Boltzmann constant and T for a tem-
perature of the atmosphere. From (24) and (7) it is evident that
ǫKQ = ǫ for all possible K and Q, where ǫ is the common photon
destruction probability. Further
B(KQ) = BPδK0δQ0. (27)
4 J. Štěpán and V. Bommier: Generalized
ǫ-law
Substituting the rates (24) and (25) into the formula (22) and
employing the general identity
−∞ Ki1(t)dt = δi1 (see Paper I)
together with bi = δi1, we recover form (23) the formula (16) of
the previously cited paper:
(1 + ǫ + δ(K) + iΓQ)[S KQ(0)]
= ǫB2P. (28)
Assuming that there is zero magnetic field, i.e. Γ = 0, the
source function tensor reduces due to symmetry reasons to the
two non-vanishing components S 00 and S
0 in the reference frame
Σ1. This reference frame is suitable for descriptions of the atomic
system under these conditions, so that we may identify Σ0 ≡ Σ1,
with X and Y axes oriented arbitrary in the plane parallel to atmo-
spheric surface. Further, assuming that there is no depolarization
of the upper level (δ(K) = 0), we realize from (28):
S 00(0)
S 20(0)
1 + ǫ
ǫ′BP, (29)
which is the same result derived in different notation by Ivanov
(1990). For simplicity the common alternative to the photon de-
struction probability has been introduced: ǫ′ = ǫ/(1 + ǫ).
If depolarization of the upper level is high enough to destroy
atomic level polarization (δ(K) → ∞ for K > 0), or the upper
level is unpolarizable, the common
ǫ-law for scalar radiation
is recovered,
S 00(0) =
1 + ǫ
ǫ′BP. (30)
4.2. Anisotropic alignment (de)excitation
The relation ǫKQ = ǫ is not in general satisfied for all the mul-
tipoles because the relaxation of the ρKQ state depends on the
velocity distribution of colliders. In the following text we will
neglect the effects of magnetic field.
Let us assume an example of a relative velocity distribution
of particles that is axially symmetric with the axis of symmetry
parallel to the vertical of the atmosphere (so that it is as in the
former case Σ0 ≡ Σ1) and that the collisional interaction can be
fully described by only the first two even multipole components
of this distribution. Thanks to these assumptions the only non-
vanishing excitation collisional rates are (C j j′ )
0 and (C j j′ )
0, the
relaxation rates (CRj′ j)
0 and (C
j′ j)
0 and for the same reasons the
only non-zero source function components are S 00 and S
An explicit evaluation of the integrals of kernels∫ ∞
−∞ K̃KQ,K′Q′ (τ, τ
′)dτ′ under these conditions shows that
the only non-zero ones are given by (A5) and (A12) of
Landi Degl’Innocenti et al. (1991b). In our notation they read
K̃00,00(τ, τ
′)dτ′ = 1, (31)
K̃20,20(τ, τ
′)dτ′ =
W2, (32)
with W2 = (w
j′ j)
2. Substituting these results into (23) we see that
(1 + ǫ00 )(S
+ (1 + ǫ20 )(S
(00))2 +
(ǫ20 B
(20))2
1 + ǫ20 −
10 W2
. (33)
To check the validity of polarized radiative transfer codes,
it is advantageous if one can verify that the transfer of higher
ranks of the radiation tensor is accurate enough. In the realistic
scattering polarization models the polarization degree does not
exceed a few percent so that |S 00(0)| ≫ |S
Q(0)|. By setting ar-
bitrary (even unphysical) collisional rates it is possible to verify
transfer codes in conditions with |S 00| ≪ |S
To privilege transfer in higher ranks of the radiation tensor
one can artificially suppress the excitation rate (C j j′)
0. In the
extremal case one can set (C j j′)
0 → 0. The easiest way to do
this is the formal interchange of the role of excitation rates of
population and alignment, i.e. (C j j′ )
0 ↔ (C j j′)
0 of the original
Maxwellian velocity distribution:
(C j j′)
0 = 0, (C j j′ )
C j j′√
2 j′ + 1
(no collisional excitation to upper level population) and the re-
laxation rates set to the Maxwellian ones
(CRj′ j)
0 = (C
j′ j)
0 = C
j′ j. (35)
In this case we have
B(00) = 0, B(20) = BP, (36)
and again
0 = ǫ
0 = ǫ. (37)
Substituting this into (33) we find out the
ǫ-law in the form
[S 00(0)]
2 + [S 20(0)]
1 − 710 W2(1 − ǫ
BP. (38)
The particular collisional rates (34) are in fact arbitrary and
have been chosen to obtain a formula similar to the one of the
Maxwellian distribution case.
This relation is useful to test polarized radiative transfer
codes, because in this unphysical case S 20(0) is the largest term,
unlike the physical case where the largest term is S 00(0) and S
is only a few percent of it. By applying Eq. (38) the test is much
more sensitive to the polarization, and the polarization is better
tested. We have thus successfully tested a multilevel non-LTE
radiative transfer code that we are developing, but this code and
its results are the subjects of a forthcoming paper.
5. Conclusions
We have derived a more general formulation of the so-called√
ǫ-law of radiation transfer. This analytical condition couples
the value of source function tensor of a two-level atom with
other physical properties of the atmosphere. The simplest re-
sult obtained in conditions of a non-magnetic, isothermal, plane-
parallel, semi-infinite atmosphere with thermal velocity distri-
bution of particles and unpolarized atomic levels (e.g. Mihalas
1970) has been generalized by Ivanov (1990) to account for scat-
tering of polarized radiation and polarized upper atomic level.
Further generalizations done in Paper I, which account for a
magnetic field of arbitrary strength and direction, has been ex-
tended in the present paper to account for non-thermal colli-
sional interactions. It was done by introducing the tensor of the
photon destruction probability ǫKQ and by defining the function
B(KQ).
The resulting formula (23) reduces to the cases mentioned
above if the physical conditions become more symmetric. On
J. Štěpán and V. Bommier: Generalized
ǫ-law 5
the other hand, situations with a high degree of perturbers ve-
locity distribution anisotropy and especially ones with unphysi-
cal collisional rates result in a wide range of models which can
be calculated both numerically and analytically. Thus they offer
new possibilities for verification of the non-LTE radiation trans-
fer codes.
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	Introduction
	Equations of statistical equilibrium
	Solution of the Wiener-Hopf equations
	Particular solutions
	Maxwellian velocity distribution of colliders
	Anisotropic alignment (de)excitation
	Conclusions
