Transfer equation in general
curvilinear coordinates
Juris Freimanis
Ventspils International Radio Astronomy Centre, Ventspils
University College;
Institute of Mathematics and Information Technology, Liepaja
University
e-mail: jurisf@venta.lv
Expanding the Universe
Conference devoted to the 200th anniversary of Tartu Observatory
Tartu, Estonia, April 27 – 29, 2011
Introduction
Research in the theory of radiative transfer has a long tradition in Tartu
Observatory (A.Sapar, T.Viik, R.Rõõm, A.Heinlo, I.Vurm and others):
- studies of Ambartzumian / Chandrasekhar and Hopf functions, resolvent functions;
- algorithms for the calculation of Voigt and Holtsmark functions;
- multilayer atmospheres, Viik’s principle of invariance (Viik 1982);
- theoretical studies of Rayleigh and molecular scattering;
- scattering within spectral lines with redistribution of radiation over frequencies;
- studies of the connection between RTE, classical and quantum electrodynamics
(Sapar 1968);
- NLTE phenomena in stellar atmospheres;
- theory of Compton scattering,
- etc.
The differential operator of RTE
Stationary integro-differential radiative transfer equation in a polydisperse
medium consists of the following basic terms:
- the differential operator, sometimes called “the streaming operator”,
- the extinction term,
- the integral term describing scattering,
- the term describing the primary sources of radiation.
The first of those will be the object of study in this contribution. Just it
depends on the chosen coordinate system.
In inhomogeneous and / or anisotropic medium the linear polarization plane of
electromagnetic radiation physically rotates around the direction of propagation:
1) S.M.Rytov. “Dokl. AN SSSR”,1938, vol. 18, pp. 263 – 268 (in Russian);
2) L.D.Landau, E.M.Lifshitz. Electrodynamics of Continuous Media. Moscow,
“Nauka”, 1982, paragraph 85. (In Russian; English translation in 1984.)
3) L.A.Apresyan, Yu.A.Kravtsov. Radiation Transfer. Statistical and Wave
Aspects. Amsterdam, Gordon and Breach Publishers, 1996.
In particular, this happens if, due to refraction within inhomogeneous
medium, the light ray moves along curve with nonzero torsion.
Polarized radiative transfer equation in curved spacetime (general
relativity) – accretion disks around neutron stars and black holes:
1) C. Fanton, M. Calvani, F. de Felice, A. Čadež. “Publications of the Astronomical
Society of Japan”, 1997, vol. 49, pp. 159 – 169.
2) C.F. Gammie, J.C. McKinney, G. Tóth. “The Astrophysical Journal”, 2003, vol.
589, pp. 444 – 457.
3) A. Broderick, R. Blandford. “Monthly Notices of the Royal Astronomical
Society”, 2003, vol. 342, pp. 1280 – 1290; ibid., 2004, vol. 349, pp. 994 – 1008.
4) C. Pitrou. “Classical and Quantum Gravity”, 2009, 26, Issue 6, pp. 065006.
Broderick and Blandford (2004): due to general relativistic rotation of
the polarization basis vectors e||m and e^m along the path of propagation, the
linear polarization angle apparently rotates in vacuum:
dQ
d
 2U
,
d
d
dU
d
 2Q
,
d
d
1
where d /d is the angular rotation speed of polarization basis vectors.
Some radiative transfer problems in approximately flat spacetime
invites the researcher to use curvilinear coordinate systems:
J.L.Ortiz et al. Observation of light echoes around very young stars.
“Astronomy & Astrophysics”, 2010, vol. 519, A7, 8 pages
Imaging-polarimetric maps of IRAS
04395+3601 in the (a) total intensity
I, (b) polarized intensity Ip, (c)
polarization strength P, and (d)
polarization position angle.
From: T. Ueta et al. Hubble Space
Telescope NICMOS imaging
polarimetry of protoplanetary
nebulae. II. Macromorphology of the
dust shell structure via polarized
light. “The Astronomical Journal”,
2007, vol. 133, pp. 1345 – 1360.
Near-infrared images of IRAS 17150-3224: (a) F160W, (b) F222M, (c) flux-ratio map of F160W/F606W, and (d) flux-ratio map of
F222M/F160W. – From: K.Y.L. Su et al. High-resolution near-infrared imaging and polarimetry of four protoplanetary nebulae. “The
Astronomical Journal”, 2003, vol. 126, pp. 848 – 862.
(d): 40% > p > 30%
(e): 30% > p > 10%
(f): p < 10%
Polarimetric results for IRAS 17150-3224, displayed in different ranges of the percentage polarization: (a) 70% >
p > 60%, (b) 60% > p > 50%, (c) 50% > p > 40%, (d) 40% > p > 30%, (e) 30% > p > 10%, (f) p < 10%. – From:
K.Y.L. Su et al. High-resolution near-infrared imaging and polarimetry of four protoplanetary nebulae. “The
Astronomical Journal”, 2003, vol. 126, pp. 848 – 862.
H2 S(1) line emission (red) and
nearby 2.15 m continuum intensity
(green) in CRL 2688. A logarithmic
stretch has been used for both
images. – From: R.Sahai et al. The
Structure of the Prototype Bipolar
Protoplanetary Nebula CRL 2688
(Egg Nebula): Broadband,
Polarimetric, and H2 Line Imaging
with NICMOS on the Hubble Space
Telescope. “The Astrophysical
Journal”, 1998, vol. 492, L163 –
L167.
(a) NICMOS and (b) ground-based (K-band) polarimetric images of CRL 2688. The tick marks on both axes show offsets in
arcseconds from the center. Orange lines indicate polarization and position angle q , with a maximum length of p = 85%.
Each inverse gray-scale image is the logarithm of the total intensity. Vectors indicating the percentage polarization (length)
and position angles are overplotted in all panels and represent the average value in 4 × 4 and 2 × 2 pixel bins in NICMOS and
COB, respectively. – From: R.Sahai et al. The Structure of the Prototype Bipolar Protoplanetary Nebula CRL 2688 (Egg Nebula):
Broadband, Polarimetric, and H2 Line Imaging with NICMOS on the Hubble Space Telescope. “The Astrophysical Journal”, 1998,
vol. 492, L163 – L167.
Below the main results from: J.Freimanis. On vector radiative transfer
equation in curvilinear coordinate systems. – “JQSRT”, 2011, in press, will be
delivered.
The basic assumptions
It is desirable that the coordinate system at least partially reflects the symmetry
of radiative transfer problem. Let as assume that:
1) The 4-dimensional spacetime is a pseudo-Euclidean (i.e. Minkowski) one.
Consequently, the 3-dimensional space is Euclidean. It is equipped with the
coordinate system {xi}, i = 1, 2, 3, and the corresponding metric tensor gik:
ds2  g ik dxi dxk ,
  det g ik  0,
i
Rklm
 0.
The coordinate system is not assumed to be orthogonal. Orthogonal
coordinate systems will be reviewed as a special (but very convienent) case.
2) The host medium is homogeneous and isotropic, and it is piecewise
homogeneously filled with polydisperse scatterers. Consequently, plane
electromagnetic waves propagate along straight lines within each homogeneous part
of the space. The real part of the effective refractive index is almost isotropic, and
there is no significant birefringence in the medium.
A little reminder
Accordingly to G.A.Korn and T.M.Korn (1968) one can define the
covariant basis vectors ei ,
ei 
r
,
i
x
i  1, 2, 3,
as well as the contravariant basis vectors ei:
eie j   ij ,
having the properties
eie j  g ij ,
eie j  g ij ,
g ijg jk  ik .
Arbitrary vector E(r) can be defined either by its contravariant
components Ei(r) or by its covariant components Ei(r):
Er   Ei r  ei r   Ei r  ei r  .
The direction of propagation of radiation
The direction of propagation of radiation can be characterized by the unit
vector W:
dxi
W 
,
ds
i
where the derivative is taken along the (straight) line of propagation.
It is most convenient to characterize the direction of propagation by the
spherical angles J, j). Usually polar axis J = 0 and zero azimuthal plane j = 0 are
tied to the basis vectors of the chosen spatial coordinate system in the given point.
Let us designate each of the possible different tensor index values (1, 2, 3)
with one of the letters a, b, c, where a  b  c  a. Henceforth, let us define that the
vector and tensor indices designated by a, b, c are fixed, i.e. the tensor expression of
kind Aaibk Baibk involves summation over indices (i, k) but it does not involve
summation over indices (a, b).
Option A:
ec
J
W
S [ec ea] / |[ec ea]| = Se abc(gaa eb – gab ea) / (gaa g cc ) 1/2
ea
j
1, if e1 , e 2 , e3  is theright - handed trihedron,
S 
- 1, if e1 , e 2 , e3  is theleft - handed trihedron.
In this case the contravariant components of W are as follows:
W 
a
sin J cos j
g aa
W  Se abc
b

Se abc g ab
g aag 
cc
sin J sin j 
g ac
g
g aa
g bc
sin J sin j 
cosJ ,
cc
cc
g 
g
cc
cos J ,
W c  g cc cos J.
Option B:
ec
J
W
S [ec ea] / |[ec ea]| = Seabc (g aa eb – g ab ea) (/(g aa gcc)) 1/2
j
ea
Now the contravariant components of W are as follows:
W a  g aa sin J cos j ,
W 
b
Wc 
g ab
g
aa
g ac
g
aa
sin J cosj  Se abc
sin J cos j 
g cc
sin J sin j ,
aa
g 
Se abc g bc
g g cc
aa
sin J sin j 
cos J
g cc
.
Orthogonal coordinate systems:
g ij  H i   ij ,
2
g  H i   ,
ij
2
ij
3
   H i  .
2
i 1
In this case both options A and B are equivalent, and they lead to one and
the same expressions
sin J cosj
W 
,
Ha
a
W  Se abc
b
sin J sin j
,
Hb
Wc 
cosJ
.
Hc
Dyadics as tensors
Dyadics used as the basic mathematical objects in M.I.Mishchenko,
L.D.Travis and A.A.Lacis (2006) are essentially tensors (Morse & Feshbach 1953;
Arfken 1970), obeying all the corresponding transformation laws in case of spatial
coordinate transformation. They can be objects of absolute (covariant) differentiation
etc.
Two-point dyadics of type E(r)  E*(r’) are split tensors with contravariant
components Ei(r)E*k(r’), covariant components Ei(r)E*k(r’), and mixed components
Ei(r)E*k(r’) and Ei(r)E*k(r’) accordingly to Goląb (1974).
Two-point dyadics implying parallel displacement of vectors and tensors
such as the identity dyadic I(r,r’) and the differential dyadic (r r ) × I(r,r’) in case
r  r’ are uniquely defined if and only if the three-dimensional space is Euclidean, with
zero Riemann curvature tensor.
Propagation of the ladder specific coherency tensor in
a polydisperse medium
Rewriting dyadics defined in Mishchenko, Travis and Lacis (2006) as tensors, we
have:
the ladder coherency tensor
CikL  r   E i r E *k r 
R ,
,
and the ladder specific coherency tensor S(L)ik(r,W) as its directional component having the
properties
CikL  r    S ik L  r, Ω  dΩ,
W i S ik L  r, Ω   S ik L  r, Ω  W k  0.
4
Due to transversality of S(L)ik(r,W) it is convenient to project it onto two orthogonal
polarization basis unit vectors (e||, e^):
ec  Ω
e^  S c
e Ω
for Option A,
e^  S
e||  S e^  Ω.
ec  Ω
ec  Ω
for Option B,
The projections rik(r,W) of the ladder specific coherency tensor onto polarization
basis vectors are the elements of the 2x2 specific coherency matrix:
r11 r, Ω 
1 e1 i
e|| r, Ω e||k r, Ω S L ik r, Ω,
2 m0
r12 r, Ω 
1 e1 i
e|| r, Ω e^k r, Ω S  L ik r, Ω,
2 m0
r 21 r, Ω 
1 e1 i
e^ r, Ω e||k r, Ω S  L ik r, Ω,
2 m0
r 22 r, Ω 
1 e1 i
e^ r, Ω e^k r, Ω S  L ik r, Ω.
2 m0
The process of radiative transfer is a true physical interaction between the
scatterers, the primary sources, and the electromagnetic field. Consequently, radiative
transfer equation contains the absolute derivative of the ladder specific coherency tensor:
DS mnL r, Ω 
Ds
Ω  const
mn



S
 L  r , Ω 
j
m ln
n
ml
W
  jl S  L  r, Ω    jl S  L  r, Ω 
j


x
Ω




 i k mj Ω S jnL r, Ω   i k n j Ω  S mjL  r, Ω 

jl
mn







 n0  Z mn
Ω
,
Ω
S
r
,
Ω
d
Ω

B
jl
L 
0  r , Ω  ,
4
with Christoffel symbols of the second kind (Goląb (1974)) and the propagation tensor kmj(W):
g kn g mn 
1  g
  g ik  km


,
2  x n x m x k 
i
mn
k mj Ω   k jm 
2 n0 m
A j Ω, Ω  ,

k
and with the understanding that
d S mnL r, Ω
ds
 Wj
W
mn

S

L  r , Ω 
j
Ω  const
x j
S mnL r,J , j 
x j
J const
j  const
Ω  const
mn
mn
dJ S  L  r,J , j  dj S  L  r,J , j 


,
ds
J
ds
j

D Wi  Wi
  j  ijk W k  W j  0,
Ds  x

or
d Wi
  ijk W j W k .
ds
The last equation gives the derivatives of directional angles along the line of propagation:
Option A:
c
i
j
dJ ij W W
cotJ  ln g cc i


W,
i
cc
ds
2
x
g sin J
ijc Wi W j cosJ  ln g cc i
d cosJ 


W,
i
cc
ds
2
x
g
aji W i W j
dj cosJ cotj c i j

ij W W 
cc
2
ds
g aa sin J sin j
g sin J
cot2 J cotj  ln g cc i cotj  ln g aa i

W 
W.
i
i
2
x
2
x
Option B:
cji W i W j cot J  ln g cc i
dJ


W,
i
ds
2
x
g cc sin J
i
j
d cos J  cj W i W
cos J  ln g cc i


W,
i
ds
2
x
g cc
ija W i W j
dj
cosJ cotj i
j



W
W
cj
i
ds
g cc sin 2 J
g aa sin J sin j
cot2 J cotj  ln g cc i cotj  ln g aa i

W 
W.
i
i
2
x
2
x
Orthogonal coordinate systems:
 ln H b
dJ
sin J   ln H a

2
2

cos
j

sin
j


ds
H c  x c
x c

 cosj  ln H c
sin j  ln H c 
 ,
 cosJ 
 Se abc
a
b
x
H b x 
 Ha
Ha
dj cosJ sin 2j 

ln
ds
2H c
x c H b
cos2 J 
cosj  ln H c sin j  ln H c 
 Se abc



b
a
sin J 
Hb
x
H a x 

cosj  ln H a sin j  ln H b 

 .
 sin J  Se abc

b
a
Hb
x
H a x 

Projecting the transfer equation for the ladder specific coherency tensor onto
polarization basic vectors (e||, e^) and making up the Stokes vector from the usual linear
combinations of the elements of the specific coherency matrix,
I r, J , j 
r11  r 22
r11  r 22
Qr,J , j 
r,J , j  ,
Ir, J , j  

 r12  r 21
U r,J , j 
i r 21  r12 
V r,J , j 
one obtains radiative transfer equation for the Stokes vector in arbitrary curvilinear coordinate
system:
Wj
Ir, J , j  dJ I r, J , j  dj I r, J , j  d



U1 Ir,J , j 
j
x
ds
J
ds
j
ds
 k  I r, J , j   n0 K J , j  Ir,J , j 
 n0  ZJ , j ;J , j  I r, J , j sin J dJ dj   B 0  r, J , j ,
4
where the last term in the left hand side contains the speed of rotation d/ds of polarization
basis vectors around the direction of propagation W:
De||i
d
De^i
 e^ i
 e||i
,
ds
Ds
Ds
and matrix U1 is the polarization reference plane rotation matrix:
0
0
U1 
0
0
0 0
0 2
2 0
0 0
0
0
.
0
0
The rotation speed of polarization reference plane d/ds is given by the
following expressions:
d
1
ikc c
j
 cc
S
e

W
W
jk
i
ds g
 sin 2 J
for OptionA,
d

i
k
j

S
e

W
W
ikc jc
ds g cc sin 2 J
for OptionB,
and for orthogonal coordinate systems

Ha
d sin 2j 
cosj  ln H c sin j  ln H c 
 .

ln
 cotJ  Se abc

c
b
a
ds
2 H c x
Hb
H b x
H a x 

Examples
1. Elliptic conical coordinate system
Its standard definition (Morse and Feshbach 1953; Korn & Korn 1968)
uses three coordinates (u, v, w) defining mutually orthogonal families of spheres,
elliptic cones around z axis, and elliptic cones around x axis. These coordinates are
8-fold degenerated, and they have several other disadvantages.
Alternative parameterization of essentially the same coordinate system:
x
r sin  cos
1   sin 
2
2
,
sin 2    2
yr
sin ,
1   2 sin 2 
z
r cos
1   sin 
2
2
,
where the arithmetic values of all square roots are taken. Limitations on the
parameter  and coordinates (r, , ):
0    1,
r  0,
arcsin       arcsin  ,
0    2 .
e 
The “conical” viewpoint:
polar angle J is measured from
the normal to the conical surfaces.
r


 e ,

 thepolaraxis (J  0),
er 
r 
  zero azimuthhalf - plane j  0.
r 
The differential operator of radiative transfer equation:
sin
Ir , , ;J , j  cosJ
Lˆ Ir,J , j   sin J cosj

r
r

1  



   2 1   2 sin 2  Ir , , ;J , j 
sin 2    2 1  cos2  sin 2 

2


Ir , , ;J , j 

r sin 2    2 1  cos2  sin 2 
2
sin 2  sin J sin j


cosJ cosj   0 , ,J , j sin J sin j Ir , , ;J , j 
r
J
1  sin j
 Ir , , ;J , j 
 
  0 , ,J , j  cosJ cosj 
r  sin J
j

cotJ sin j   0 , ,J , j  cosj

U1Ir , , ;J , j ,
r

where
 0 , ,J , j  


3
2


sin 2 sin    1   sin  sin j   2 1   2 sin 2 cotJ
2
2

2
2

2 sin 2    2 1  cos2  sin 2 

3
2
.
2. Oblate spheroidal coordinate system
x   cosh sin  cos,
y   cosh sin  sin ,
z   sinh cos,
with limitations
  0,
  0,
0   ,
0    2 .
The differential operator of radiative transfer equation:
Lˆ Ir,J , j  

I , , ;J , j 
sin J cosj
I , , ;J , j 



a sinh2   cos2 
a sinh2   cos2 
cosJ
sin J sin j I , , ;J , j 
a cosh sin 

2 tanh sin

2

 sin 2 j  sinh 2 sin J  sin 2 cosJ cosj I , , ;J , j 
3
J
2
2
2a sinh   cos  2


sin 2  I , , ;J , j 

2
2
tanh

sin

cos
J
cos
j

cosh

cot

sin
J



3
2
sin
J
j
2
2


a sinh   cos  2
sin  sin j tanh sin  cosj  cos cotJ 

U1I , , ;J , j .
3
a sinh2   cos2  2

sin j




Conclusions
1) General expressions for the differential operator of vector radiative transfer
equation in arbitrary curvilinear coordinate system have been obtained.
2) It has been shown that the plane of linear polarization seemingly rotates in the
vacuum if the polarization reference frame is tied to the basis vectors of general
curvilinear coordinate system.
3) Feasibility of the general method has been demonstrated by derivation of RTE in
elliptic conical and oblate spheroidal coordinate systems.
This study was financed from the basic budget of Ventspils International
Radio Astronomy Centre, as well as from co-sponsorship by Ventspils City Council
and from Latvian Council of Science grant No. 11.1856. Some financing was provided
by Institute of Mathematical Sciences and Information Technologies of Liepaja
University as well. The participation of the author at this conference is financed from
ERDF’s grant “SATTEH” being implemented in Ventspils University College. The
author expresses his gratitude to all these entities.
Thank you for your attention!