Ring
Spectroscopy
Todd
Bradley
January
6,
2010
Inves=ga=on
Summary
• New
data
reduc=on
techniques
• Results
from
spectral
fiEng
• New
inves=ga=ons
pertaining
to
ring
par=cle
phase
func=on
2
New
Data
Reduc=on
Techniques
• Account
for
smeared
field
of
view
(discussed
in
Bradley
cube
generator
presenta=on
on
1/5/2010)
• Throw
out
observa=ons
with
significant
off
axis
light
and
Saturn
shine
• When
averaging
data
within
a
radial
bin,
weight
the
data
to
account
for
differences
in
sampling
3
Sampling
and
weigh=ng
issues
pixels
Idealized
hi
resolu=on
I/F
4
Last
year
5
Now
6
Water
ice
absorp=on
feature
The
magnitude
of
I/F
depends
on:
1) the
ring
par=cle
albedo
2) the
ring
par=cle
phase
func=on
3) the
scaVering
func=on
4) the
regolith
grain
single
scaVering
albedo
5) mul=ple
scaVering
among
grains
7
Spectral
FiEng
• The
shape
and
spectral
loca=on
of
the
water
ice
absorp=on
feature
has
been
used
without
regard
to
the
absolute
value
of
I/F
• Hapke
and
Shkuratov
models
were
used
to
fit
to
the
spectral
shape
of
the
water
ice
absorp=on
feature
• Retrieved
proper=es
were
the
photon
mean
path
length
and
the
asymmetry
parameter
• The
UV
slope
was
computed
to
determine
rela=ve
varia=ons
in
water
ice
abundance
as
a
func=on
of
ring
plane
radius
8
Shkuratov
model
Re
=
average
external
reflectance
coefficient
which
=
average
backwards
reflectance
coefficient
(Rb)
+
average
forward
reflectance
coefficient
(Rf)
Ri
=
average
internal
reflectance
coefficient
Te
=
average
transmission
from
outside
to
inside
Slab
model
of
regolith
grain
Ti
=
average
transmission
from
inside
to
outside
Wm
=
Probability
for
beam
to
emerge
acer
mth
scaVering
Poulet
et
al.,
2002
t =
4kL/#
k
=
imaginary
index
of
refrac=on
9
Shkuratov
model
Use
real
part
of
indices
of
refrac=on
(n)
to
determine
Re,
Rb,
and
Ri.
Empirical
approxima=ons
from
Shkuratov
(1999)
give:
Re
~
(n‐1)2
/
(n
+
1)2
+
0.05
Rb
~
(0.28
n
–
0.20)Re
Ri
~
1.04
–
1/n2
Shkuratov
assumes
W2
=
0
and
Wm
=
1/2
for
m
>
2.
Then
adding
all
the
terms
shown
in
the
last
figure
becomes
a
geometric
series
and
gives:
rb
=
Rb
+
1/2TeTiRi
exp(‐2t)/(1
–
Ri
exp(‐t))
rf
=
Rf
+
Te
Ti
exp(‐t)
+
1/2
Te
Ti
Ri
exp(‐2t)/(1
–
Ri
exp(‐t))
where
rb
+
rr
is
assumed
to
be
the
single
scaVering
albedo
of
a
regolith
par=cle
(Poulet
et
al.,
2002)
10
Shkuratov
Model
ω = rf + rb
rf − rb
g=
rf + rb
Denote
“q”
as
the
volume
frac=on
filled
by
par=cles.
Then:
€
b
=
q
*
rb
f
=
q*rf
+
1
–
q
11
Hapke
Model
2
Se =
2
n
−1
+
k
( )
(n + 1)
2
+k
2
+ 0.05
n,k
=
complex
indices
of
refrac=on.
4
Si = 1−
2
n ( n + 1)
Qs = Se + (1− Se )
Θ=e
L
=
mean
path
length
(1− Si )Θ
1− SiΘ
−4 πkL / λ
12
€
Hapke
model
Assume
Qs
=
single
scaVering
albedo
(ῶl)
Aλ (scaled) = ϖ λ (P(g,α ) + H µ H µ o −1)
Hx ≈
1+ 2x
, x = µ, µo
1+ 2γx
µ = cos(emission angle), µo = cos(incidence angle), γ = 1− ϖ λ
13
Model
fits
to
the
data
• Fit
the
Hapke
and
Shkuratov
models
to
the
spectral
shape
of
the
absorp=on
feature
• Least
squares
fit
over
photon
mean
path
length
(L)
and
the
asymmetry
parameter
(g)
14
Uncertainty
in
the
g
value
15
Uncertainty
in
the
L
value
16
Retrieved
L
value
• L
values
from
Shkuratov
are
consistently
higher
than
the
Hapke
L
values
by
a
few
percent
• They
both
give
the
same
overall
trend
17
Retrieved
g
value
• Nega=ve
values
imply
backscaVering
• Retrieved
g
values
follow
a
similar
trend
for
the
C
and
B
rings
• The
disagreement
could
be
due
to
the
way
the
asymmetry
parameter
is
treated
in
each
model
18
UV
color
ra=o
19
Physical
picture
of
ring
par=cles
Ring
par=cles
range
in
size
from
mm
to
meters
Ring
par=cles
are
covered
by
icy
regolith
grains
Ring
par=cles
exhibit
clumping
Volume
filling
frac=on
is
unknown
20
Interac=on
of
photon
with
a
ring
par=cle
Incident
photon
Emission
of
photon
from
ring
par=cle
Regolith
ice
grain
(model
as
single
scaVering)
Ring
par=cle
composed
of
many
grains
(mul=ple
scaVering
between
grains)
21
Macroscopic
proper=es
Microscopic
proper=es
Ring
par=cles
have
their
own
phase
func=on
Each
regolith
grain
has
its
own
phase
func=on
Ring
par=cles
have
their
own
albedo
(Bond
albedo)
Each
regolith
grain
has
its
own
single
scaVering
albedo
Mul=ple
scaVering
may
or
may
not
be
significant
depending
on
loca=on
and
geometry
Mul=ple
scaVering
must
be
taken
into
account
for
regolith
grains
22
Jus=fica=on
for
ring
par=cle
single
scaVering
assump=on
• ScaVering
asymmetry
implies
the
rings
are
moderate
to
strongly
backscaVering
• Theore=cal
calcula=ons
show
that
as
the
incidence
angle
increases
single
scaVering
dominates
(Cuzzi
et
al,
1984)
23
Equa=on
for
reflectance
from
Saturn’s
rings
for
single
scaVering
assump=on
I /F = A * P(α ) * O(τ , µ, µo )
A
is
the
ring
par=cle
single
scaVering
albedo
(also
referred
to
as
the
Bond
albedo)
P
is
the
ring
par=cle
phase
func=on
O
Is
the
scaVering
func=on
24
Ring
par=cle
phase
func=on
I /F = A * P(α ) * O(τ , µ, µo )
• Constraining
the
phase
func=on
allows
for
the
absolute
determina=on
of
A
(provided
O
is
known)
• Gives
informa=on
pertaining
to
the
direc=onality
of
scaVering
• Says
something
about
the
morphology
of
ring
par=cles
25
µo 1
−τ / µ o −τ / µ
If O =
1− exp
(
)
µo + µ 4
I /F
then
= A*P
O
€
This
is
probably
valid
for
the
C
ring,
Cassini
Division,
and
maybe
the
A
ring
€
Using
measured
phase
curve,
solve
for
A
and
P
26
Phase
func=ons
Power
Law
P = Cn * (π − α )
Minnaert
n
C
is
a
normaliza=on
constant
n
is
the
power
law
index
α
is
the
phase
angle
€
k
−Sα
P = (µoµ) exp
k
is
the
Minnaert
parameter
S
is
the
steepness
parameter
α
is
the
phase
angle
μo
=
cosine
of
incidence
angle
μ
=
cosine
of
emission
angle
Note:
Minnaert
phase
func=on
is
not
yet
normalized
27
Phase
curve
in
outer
C
ring
• I/F
is
averaged
from
175‐190
nm
• Minnaert
func=on
is
a
beVer
fit
• Implies
that
a
“smooth”
phase
func=on
is
inadequate
• Dependence
on
incidence
and
emission
angle
implies
“roughness”
of
ring
par=cles
28
Minnaert
results
• Least
squares
fit
performed
at
discrete
ring
plane
radii
for
the
steepness
and
Minnaert
parameter
• Larger
values
of
the
steepness
parameter
imply
a
more
backscaVering
phase
func=on
29
ScaVer
plot
of
steepness
parameter
vs.
retrieved
g
values
• Correla=on
coefficient
=
0.71
for
Shkuratov
and
0.64
for
Hapke
• As
g
value
becomes
more
backscaVering,
steepness
parameter
becomes
less
back
scaVering
• It
could
be
that
the
shape
of
the
absorp=on
feature
is
affected
by
the
ring
par=cle
phase
func=on
30
Future
work
• Model
scaVering
func=on
for
B
ring
• Normalize
Minnaert
phase
func=on
• Determine
Minneart
phase
func=on
on
short
wavelength
side
of
absorp=on
feature
• Model
absolute
value
of
ring
par=cle
albedo
using
Shkuratov
and
Hapke
models
• May
have
wavelength
dependent
ring
par=cle
phase
func=on
that
will
require
modifica=ons
of
model
values
31