I s
Q s
U s
V s
0
B
B
B
B
B
@
1
C
C
C
C
C
A
¼
1
k N
2 r 2
op
S 11 S 12
0
0
S 12 S 22
0
0
0
0
S 33 S 34
0
0 ÀS 34 S 44
0
B
B
B
@
1
C
C
C
A
I i
Q i
U i
V i
0
B
B
B
B
B
B
@
1
C
C
C
C
C
C
A
ð4:17Þ
Furthermore, if the incident light is unpolarized (as in the case of solar illumination) then the Stokes vector of the scattered light reduces to
I s ¼ S 11 I i =k N
2 r
2
op
Q s ¼ S 12 Q i =k N
2 r
2
op
U s ¼ 0
V s ¼ 0
ð4:18Þ
We can now define the scattering angles, θ and ϕ as in Fig. 4.3. The angle θ
(frequently referred to as the scattering angle) can be recognized as π–α (where α is
the phase angle in planetary photometry) while the angle ϕ is a rotation about the
vector defining the direction of the incident light.
The linear polarization of the scattered light for unpolarized incident light also has
a simple form being
Ƥ L θ, ϕ
ð
Þ ¼ À
S 12 θ, ϕ
ð
Þ
S 11 θ, ϕ
ð
Þ
ð4:19Þ
where Ƥ L is a function of the scattering angles in a polar coordinate system with θ
being the scattering angle (e.g. Frattin et al. 2019). If Ƥ L is positive, the scattered
light is, to some degree, polarized perpendicular to the scattering plane. A negative
value for Ƥ L implies that the scattered light is partially polarized parallel to the
scattering plane.
Within a cometary coma, dependencies on ϕ, are often ignored as it is assumed
that the number of particles involved in scattering combined with their rotation
would lead to randomization and smoothing out of any dependencies. However, this
Fig. 4.3 Definition of the scattering angle for radiation incident upon a scattering particle
4.2 Scattering of Light by Dust
287
Q s
U s
V s
0
B
B
B
B
B
@
1
C
C
C
C
C
A
¼
1
k N
2 r 2
op
S 11 S 12
0
0
S 12 S 22
0
0
0
0
S 33 S 34
0
0 ÀS 34 S 44
0
B
B
B
@
1
C
C
C
A
I i
Q i
U i
V i
0
B
B
B
B
B
B
@
1
C
C
C
C
C
C
A
ð4:17Þ
Furthermore, if the incident light is unpolarized (as in the case of solar illumination) then the Stokes vector of the scattered light reduces to
I s ¼ S 11 I i =k N
2 r
2
op
Q s ¼ S 12 Q i =k N
2 r
2
op
U s ¼ 0
V s ¼ 0
ð4:18Þ
We can now define the scattering angles, θ and ϕ as in Fig. 4.3. The angle θ
(frequently referred to as the scattering angle) can be recognized as π–α (where α is
the phase angle in planetary photometry) while the angle ϕ is a rotation about the
vector defining the direction of the incident light.
The linear polarization of the scattered light for unpolarized incident light also has
a simple form being
Ƥ L θ, ϕ
ð
Þ ¼ À
S 12 θ, ϕ
ð
Þ
S 11 θ, ϕ
ð
Þ
ð4:19Þ
where Ƥ L is a function of the scattering angles in a polar coordinate system with θ
being the scattering angle (e.g. Frattin et al. 2019). If Ƥ L is positive, the scattered
light is, to some degree, polarized perpendicular to the scattering plane. A negative
value for Ƥ L implies that the scattered light is partially polarized parallel to the
scattering plane.
Within a cometary coma, dependencies on ϕ, are often ignored as it is assumed
that the number of particles involved in scattering combined with their rotation
would lead to randomization and smoothing out of any dependencies. However, this
Fig. 4.3 Definition of the scattering angle for radiation incident upon a scattering particle
4.2 Scattering of Light by Dust
287