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Microwave observations of the disk and rings of Saturn at low inclinations

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M ICROW AVE O B SER V A TIO N S OF T H E D ISK A N D R IN G S OF
S A T U R N AT L O W IN C L IN A T IO N S
by
David Erwin Dunn
A thesis subm itted in partial fulfillment of the
requirements for the Doctor of Philosophy
degree in Physics in the
G raduate College of The
University of Iowa
May 1999
Thesis supervisor: Professor Lawrence Molnar
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UMI Number: 9933368
Copyright 1999 by
Dunn, David Erwin
All rights reserved.
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Copyright by
DAVID ERWIN DUNN
1999
All Rights Reserved
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Graduate College
The University of Iowa
Iowa City, Iowa
CERTIFICATE OF APPROVAL
PH.D. THESIS
This is to certify th at the Ph.D. thesis of
David Erwin Dunn
has been approved by the Examining Com m ittee
for the thesis requirement for the Doctor of
Philosophy degree in Physics at the
May 1999 graduation.
Thesis committee:
'&&&&*%
Thesis supervisor
Member
1*>/\y»
Member
Member
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^
To all my friends and family whom without life would be far
less joyful
ii
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ACKNOW LEDGM ENTS
My journey through this thesis could not be accomplished without the help of
my advisor Larry Molnar. His insights into the physics of my work and his patience
with me is rare and invaluable. He was always willing to assist me when I needed it.
I hope he finds his academic niche in Calvin.
I’d like to thank all of the friends and colleagues which I have had the pleasure
of knowing have help me survive here in my nearly ten years in Iowa. First, Fd like to
thank my two good friends Coleen Maddy and Nonalee Gardner who are wonderful
people. Life without them would have been a whole lot emptier. Second, Grant Denn,
who came to Iowa at the same time, has provided me with the thesis templates for
which I’m eternally grateful. Next, Mike and Becky LeDocq are wonderful friends now
living in Lacrosse, WI; Mike was colleague of mine at Iowa for nine years. Continuing
on with my roommate Paul Foth whose toleration of my slobbiness at the apartment
is unparalleled. I close by thanking Jim Thompson who got me started in LINUX,
which was a great aid in completing my thesis.
W ithout my family I would not be the person I am today. Mom and Dad, whom
I dedicate in part this thesis, are the most wonderful parents a son could hope for.
I only wish I called and wrote more often than I did. My brother, who shares an
interest in astronomy, is a good and successful man and has a wonderful wife Cathy
and two sons, Ryan and Will.
I like to acknowledge Prof. Fix for his help in Larry’s absence. His comments on
radiative transfer and the use of his office for the weekly teleconferences were essential
iii
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essential to the thesis process.
Another person I wish to recognize is K athie Staley. Her therapy group helped
me through some rough times in my life, and she’s just a very caring and wonderful
person.
Before I close, I’d like to dedicate part of my thesis to Karen Phelps who died
of Pancreatic Cancer less than a year ago. She was my strength whenever school or
life became too burdonsome for me. I miss her tremendously.
iv
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TABLE OF C O N T E N T S
Page
LIST OF T A B L E S ........................................................................................................
viii
LIST OF F I G U R E S ....................................................................................................
be
CHAPTER
I
IN T R O D U C T IO N .......................................................................................
1.1
1.2
1.3
1
Historical O v e rv ie w .............................................................................
M o tiv a tio n .............................................................................................
Outline of T h e sis...................................................................................
1
2
6
II OBSERVATIONS AND DATA R E D U C T IO N ..........................................
7
2.1
2.2
I n tr o d u c tio n ..........................................................................................
Interferometry and Aperture s y n th e s is ..............................................
2.2.1 P r e f a c e .........................................................................................
2.2.2 V is ib ilitie s ..................................................................................
2.2.3 Intensity derived from v isib ilitie s...........................................
2.2.4 Discrete sampling and u , v c o v e r a g e .....................................
2.2.5 The brightness te m p e ra tu re .....................................................
The Very Large A r r a y ..........................................................................
2.3.1 C a lib ra tio n ..................................................................................
Making observations of Saturn a t the V L A ....................................
2.4.1 Issues of planetary im a g in g .....................................................
2.4.2 Choosing the correct a r r a y s .....................................................
Synthesis Imaging and M o d e lin g ......................................................
2.5.1 Creating the image ..................................................................
2.5.2 Hybrid Mapping/Modeling P r o c e d u r e .................................
7
8
8
8
9
9
10
10
13
15
15
16
18
18
19
III FIRST E P O C H ..............................................................................................
22
2.3
2.4
2.5
3.1
3.2
3.3
3.4
I n tr o d u c tio n .........................................................................................
O b se rv a tio n s..................................................................
Analysis ................................................................................................
3.3.1 Hybrid Mapping/Modeling P r o c e d u r e .................................
3.3.2 Application to Equinox D a t a ..................................................
3.3.3 R e s u l ts ........................................................................................
Discussion .............................................................................................
3.4.1 Atmospheric D y n a m ic s ...........................................................
3.4.2 R i n g s ............................................................................................
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22
24
26
26
27
30
46
46
48
IV SECOND E P O C H ..........................................................................................
4.1
4.2
4.3
4.4
V
I n tr o d u c tio n ..........................................................................................
O b serv a tio n s..........................................................................................
Analysis ................................................................................................
Issues with our second epoch d a t a ...................................................
4.4.1 R e s u lts .......................................................................................
MONTE CARLO SIMULATIONS OF A MIE SCATTERING RING
5.1
5.2
5.3
5.4
5.5
5.6
5.7
52
52
54
54
56
75
I n tr o d u c tio n .........................................................................................
75
Microwave Scattering by S aturn’s R in g s ..........................................
76
76
5.2.1 In tro d u c tio n .............................................................................
5.2.2 Scattering b a s ic s .......................................................................
77
5.2.3 Optical depth and the absorption coefficient.......................
79
5.2.4 Mie s c a tte r in g ..........................................................................
83
85
Therm al Emission from S atu rn ’s Rings .........................................
5.3.1 In tro d u c tio n .............................................................................
85
5.3.2 Thermal P r o p e r tie s ................................................................
86
5.3.3 Geometry effe cts.......................................................................
87
Monte Carlo S im u la tio n s ....................................................................
88
5.4.1 Underlying a ssu m p tio n s.........................................................
88
5.4.2 Basic equations .......................................................................
88
5.4.3 Scattered R a d ia tio n ................................................................
91
5.4.4 Thermal p hotons......................................................................
94
5.4.5 Cosmic B a c k g ro u n d ................................................................
94
5.4.6 Full m o d e l ................................................................................
95
Comparisons to previous results ......................................................
96
Checking the phase function ............................................................ 100
Looking at the dependencies of Mie scattering on optical depth . 101
VI D ISC U SSIO N ................................................................................................
6.1
6.2
52
109
I n tr o d u c tio n .........................................................................................
The ring physical parameters as seen in the l i t e r a t u r e ...............
6.2.1 Physical dimensions of S a t u r n ............................................
6.2.2 The physical tem perature of the r i n g s ................................
6.2.3 The optical depth and particle size d istrib u tio n ................
6.2.4 Dirt contamination fra c tio n ...................................................
6.2.5 Variables we may c h a n g e ......................................................
R e s u l t s ..................................................................................................
6.3.1 P relim inaries.............................................................................
6.3.2 Case one: isotropic sc a tte rin g ...............................................
6.3.3 Case two: Mie s c a t t e r i n g ......................................................
6.3.4 Case three: Hybrid isotropic/Mie sc a tte rin g ......................
D is c u s s io n ............................................................................................
109
109
109
110
110
Ill
Ill
115
115
121
126
132
137
VII C O N C L U S IO N .............................................................................................
139
6.3
6.4
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7.1
7.2
7.3
7.4
Overview...................................................................................................
The atm osphere......................................................................................
The rings ...............................................................................................
Further w o r k ...........................................................................................
139
139
140
141
R E F E R E N C E S ..............................................................................................................
143
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LIST OF TA BLES
Table
1-1
2-1
3-1
3-2
4-1
4-2
6-1
6-2
6-3
6-4
6-5
Page
The geometry of the main rings....................................................................
VLA primary flux calibrators........................................................................
Components in the Saturn Model f0.7 and 2.0 cm ).................................
Components in the Saturn Model (3.6 and 6.1 cm ).................................
Circumstances of second epoch VLA observations...................................
Components in the Saturn Model for the second epoch)........................
Observed ring te m p e r a tu r e s .......................................................................
Some ring particle param eters.......................................................................
Radii of ring annuli and norm al optical depths used in the model cal­
culations
Summary of the effectiveness of the two m odels.......................................
The fraction of isotropy, Fiso, used for each ring/wavelengths in our
hybrid model.....................................................................................................
viii
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4
14
31
32
53
57
109
Ill
112
130
136
LIST OF F IG U R E S
Figure
1-1
2-1
2-2
3-1
3-2
3-3
3-4
3-5
3-6
3-7
3-8
3-9
3-10
Page
Spacecraft images of Saturn, (top) Voyager 2 image of Saturn, (bottom)
Hubble IR image of Saturn (Karkoschka and NASA 1 9 9 8 .) ................
Visibility plot for the 7 February 1997 at 3.6 cm and 6.1 cm. The axes
are scaled to the wavelength..........................................................................
A schematic of the VLA layout. Each arm is 120° from the other and
the North Arm is displaced 5° west of no rth .............................................
View from E arth of the ring model components in the west ansa. Solid
lines indicate separations between the m ain rings (labeled A, B, and C).
The gap between the A and B rings is the Cassini Division. Dashed
lines indicate separations between azim uthal segments of the model
components.......................................................................................................
Polar view of the ring model components. Solid lines indicate separa­
tions between the main rings (labeled A, B, and C). The gap between
the A and B rings is the Cassini Division. Dashed lines indicate sepa­
rations between the azimuthal segments of the model components. . .
Disk-averaged spectrum of Saturn. Filled circles represent the data
from this work. Open circles refer to d a ta from de Pater and Dickel
(1991) and open triangles refer to data from Grossman (1990).............
Plot of the limb darkening fraction, a 2 /a i , versus wavelength...............
Map of Saturn at 0.7 cm wavelength on 20 November 1995. The restor­
ing beam is 0.57" x 0.46" FWHM, indicated by an ellipse in the upper
right. The wedge across the top gives the transfer function of the
residual in Kelvin............................................................................................
Map of Saturn at 2.0 cm wavelength on 20 November 1995. The restor­
ing beam is 0.85" x 0.4" FWHM, indicated by an ellipse in the upper
right. The wedge across the top gives the transfer function of the
residual in Kelvin............................................................................................
Map of Saturn at 3.6 cm wavelength on 20 November 1995. The restor­
ing beam is 0.8" x 0.64" FWHM, indicated by an ellipse in the upper
right. The wedge across the top gives the transfer function of the
residual in Kelvin. The transfer function has been set to show both
the planet and the rings.................................................................................
Map of Saturn at 6.1 cm wavelength on 20 November 1995. The restor­
ing beam is 1.38" x 1.13" FWHM, indicated by an ellipse in the upper
right. The wedge across the top gives the transfer function of the
residual in Kelvin............................................................................................
Longitudinally averaged latitude structure of the Saturnian atmosphere
at 0.7 (top) and 2.0 (bottom) cm wavelength on 20 November 1995. .
Longitudinally averaged latitude structure of the Saturnian atmosphere
at 3.6 (top) and 6.1 (bottom) cm wavelength on 20 November 1995. .
ix
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3
11
12
28
29
33
33
34
35
36
37
39
40
3-11 Maps of the 0.7 cm wavelength ring structure on the west (top) and
bottom (east) side. Contours indicate temperatures of 5, 7, 9, 12, &
15 Kelvin. The restoring beam is 0.57" x 0.46" FWHM, indicated by
the ellipse in the upper right corner. The thermal noise is 1.6 K. . . . 41
3-12 Maps of the 2.0 cm wavelength ring structure on the west (top) and
bottom (east) sides. Contours indicate temperatures of 5, 7, 9, 12, &
15 Kelvin. The restoring beam is 0.64" x 0.4" FWHM, indicated by
the ellipse in the upper right comer. The thermal noise is 4.1 K. . . . 42
3-13 Maps of the 3.6 cm wavelength ring structure on the west (top) and
bottom (east) sides. Contours indicate temperatures of 5, 7, 9, 12, &
15 Kelvin. The restoring beam is 0.8" x 0.64" FWHM, indicated by
the ellipse in the upper right corner. The thermal noise is 1.1 K. . . . 43
3-14 Maps of the 6.1 cm wavelength ring structure on the west (top) and
bottom (east) side. Contours indicate temperatures of 3.5, 7, 9, & 12
Kelvin. The restoring beam is 1.38" x 1.13" FWHM, indicated by the
44
ellipse in the upper right corner. The thermal noise is 1.2 K.................
3-15 A polar view of Saturn and a ring. Sample ray paths for Saturnian
radio emission scattered towards Earth by ring particles at six different
azimuths are labeled A-F. Particle wakes inclined by ~25° are indicated
50
schematically with bold line segments........................................................
4-1 View from Earth of the ring model components in the west ansa. Solid
lines indicate separations between the main rings (labeled A, B, and C).
The gap between the A and B rings is the Cassini Division. Dashed
lines indicate separations between azimuthal segments of the model
components......................................................................................................
56
4-2 Disk-averaged spectrum of Saturn. Filled circles represent the data
from the first epoch, the filled squares the second epoch. Open circles
refer to data from de Pater and Dickel (1991) and open triangles refer
to data from Grossman (1990).....................................................................
58
4-3 Plot of the limb darkening fraction, a2/a i, versus wavelength. The
solid line indicates data from the second epoch, the dshed line from
the first................................................................................
59
4-4 Map of Saturn at 1.3 cm wavelength on 14 February 1997. The restor­
ing beam is 0.60" x 0.32" FWHM, indicated by an ellipse in the upper
right. The wedge across the top gives the transfer function of the
residual in Kelvin............................................................................................
60
4-5 Map of Saturn at 2.0 cm wavelength on 14 February 1997. The restor­
ing beam is 0.90" x 0.46" FWHM, indicated by an ellipse in the upper
right. The wedge across the top gives the transfer function of the
residual in Kelvin............................................................................................
61
4-6 Map of Saturn at 3.6 cm wavelength on 7 February 1997. The restoring
beam is 0.64" x 0.32" FWHM, indicated by an ellipse in the upper right.
The wedge across the top gives the transfer function of the residual in
Kelvin...............................................................................................................
62
4-7 Map of Saturn at 6.1 cm wavelength on 7 February 1997. The restoring
beam is 1.07" x 0.52" FWHM, indicated by an ellipse in the upper right.
The wedge across the top gives the transfer function of the residual in
Kelvin...............................................................................................................
63
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4-8
4-9
4-10
4-11
4-12
4-13
4-14
4-15
4-16
5-1
5-2
5-3
5-4
Map of Saturn at 18.0 cm wavelength on 7 February 1997. The restor­
ing beam is 3.41" x 1.74" FWHM, indicated by an ellipse in the upper
right. The wedge across the top gives the transfer function of the
residual in Kelvin.............................................................................................
Map of Saturn at 21.3 cm wavelength on 7 February 1997. The restor­
ing beam is 3.41" x 1.80" FWHM, indicated by an ellipse in the upper
right. The wedge across the top gives the transfer function of the
residual in Kelvin.............................................................................................
Residual maps of the 1997 February epoch. The images are of the data
with the appropriate a,\ + a.2 p. removed from the disk. The images are
labeled as follows: top left 1.3 cm, top right 2.0 cm, bottom left, 3.6 cm
bottom right, 6.1 cm........................................................................................
Longitudinally averaged latitude structure of the Saturnian atmosphere
at 1.3 cm (top) and 2.0 cm (bottom) wavelength on 14 February 1997.
Longitudinally averaged latitude structure of the Saturnian atmosphere
at 3.6 cm (top) and 6.1 cm (bottom) wavelength on 7 February 1997.
Maps of the 1.3 cm wavelength ring structure on the west (top) and
east (bottom) side. Contours indicate tem peratures of 5, 7, 9, 12, and
15 Kelvin. The restoring beam is 0.60" x 0.32" FWHM, indicated by
the ellipse in the upper right corner. The therm al noise is 2.0 K. . . .
Maps of the 2.0 cm wavelength ring structure on the west (top) and
east (bottom) side. Contours indicate tem peratures of 5, 7, 9, 12, and
15 Kelvin. The restoring beam is 0.90" x 0.46" FWHM, indicated by
the ellipse in the upper right corner. The therm al noise is 1.0 K. . . .
Maps of the 3.6 cm wavelength ring structure on the west (top) and
east (bottom) side. Contours indicate tem peratures of 5, 7, 9, 12, and
15 Kelvin. The restoring beam is 0.64" x 0.32" FWHM, indicated by
the ellipse in the upper right corner. The therm al noise is 1.6 K. . . .
Maps of the 6.1 cm wavelength ring structure on the west (top) and
east (bottom) side. Contours indicate tem peratures of 5, 7, 9, 12, and
15 Kelvin. The restoring beam is 1.07" x 0.52" FWHM, indicated by
the ellipse in the upper right corner. The therm al noise is 1.9 K. . . .
The extinction efficiency as a function of the size parameter, x =
27ra/A, for an ice sphere with m = 1.78 —i 0.00001..................................
The geometry for the Monte Carlo simulations. The incoming angles
for a photon are designated by 6i and 0i. The outgoing is designated
by the inclination i ..........................................................................................
Simulation results for an isotropic ring. Two million photons were
simulated to impinge the ring plane with an incident angle of 5.5 °from
the ring plane. Results are collected in l°by l°bins and plotted here
between 0 (white) and 40 (black).................................................................
Comparison plot of the thermal dependence on wavelength. We use a
particle size distribution power law number of q=3 for amin = 1-0 cm.
Plot (A) has amax = 1000 cm and plot (B) has amax = 100 cm. Lines
without symbols are from Cuzzi, Pollack, and Summers (1980), while
the line with symbols are Mie calculations (From Fig. 5 of Cuzzi,
Pollack, and Summers (1980)........................................................................
xi
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64
65
67
68
69
71
72
73
74
85
92
96
97
5-5
5-6
5-7
5-8
Ol C
M CM
5-9
-10
-11
-12
5-13
6-1
6-2
6-3
6-4
6-5
6-6
6-7
6-8
A second comparison plot of the therm al dependence on wavelength.
We consider four combinations of q, amin, amax, and F stated in the
figure. Lines without symbols are from Grossman (1990, Fig. 5.6),
while the line with symbols are this work’s Mie calculations..................
98
Comparison plot of the scattering dependence at 3.71 cm. We use
the data from Cuzzi, Pollack, and Summers (1980), Table II, and are
represented by line without symbols, while the fines with symbols are
calculations from this work. Errors from their calculations are on the
order of 10%, while our calculations contain less than 1% uncertainty.
99
Isotropic phase function. Top Basic plot showing phasefunction vs. 9.
Bottom Cumulative sum of p u (cos0) d(cos9)........................................... 101
Mie phase function for 0.7 cm. Top Basic plot showing phasefunction
vs. 9. Bottom Cumulative sum of p n (c o s9) d(cos9)................................ 102
Mie phase function for 6.1 cm. Top Basic plot showing phasefunction
vs. 9. Bottom Cumulative sum of p n (cos 9) d(cos9)................................ 103
The scattering brightness tem perature at various Saturn ring radii. . 104
Scattering brightness tem perature in the optical thin and thick cases. 105
Singly and multiply scattered fight compared in the optically thin (top)
and optically thick (bottom) cases................................................................. 106
A comparison between the diffuse transm itted and reflected fight for
the optically thin (top) and optically thick (bottom) cases.......................... 107
The Radio Science System occultation d ata from Voyager 1. The exact
optical depth of the middle and outer B ring is not known since the
d ata saturate at r = 1 .1 ................................................................................ 113
D ata maps of Saturn for November 1995. Images depict the folded
over residual maps at (from left to right and top to bottom ) 0.7, 2.0,
3.6, and 6.1 cm. The dynamic range of the maps has been set to show
the details of the disk...................................................................................... 116
D ata maps of Saturn for February 1997. Images depict the folded over
residual maps at (from left to right and top to bottom) 1.3, 2.0, 3.6,
and 6.1 cm. The dynamic range of the maps has been set to show the
details of the disk............................................................................................. 117
D ata maps of Saturn for November 1995. Images depict the folded
over residual maps at (from left to right and top to bottom ) 0.7, 2.0,
3.6, and 6.1 cm. The dynamic range of the maps has been set to show
the details of the rings.................................................................................... 118
D ata maps of Saturn for February 1997. Images depict the folded over
residual maps at (from left to right and top to bottom) 1.3, 2.0, 3.6,
and 6.1 cm. The dynamic range of the maps has been set to show the
details of the rings........................................................................................... 119
Model maps of isotropic scattering calculations. Images depict the
folded over residual maps at (from left to right and top to bottom ) 0.7,
2.0, 3.6, and 6.1 cm.......................................................................................... 121
Model maps of isotropic scattering calculation. Images depict the
folded over residual maps at (from left to right and top to bottom )
1.3, 2.0, 3.6, and 6.1 cm .................................................................................. 122
Residual maps of isotropic scattering calculations. Images depict the
folded over residual maps at (from left to right and top to bottom ) 0.7,
2.0, 3.6, and 6.1 cm.......................................................................................... 123
xii
R e p r o d u c e d w ith p e r m issio n o f th e co p y rig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
6-9
6-10
6-11
6-12
6-13
6-14
6-15
6-16
6-17
Residual maps of isotropic scattering calculations. Images depict the
folded over residual maps at (from left to right and top to bottom) 1.3,
2.0, 3.6, and 6.1 cm....................................................................................... 124
Model maps of Mie scattering calculations for the November 1995
epoch. Images depict the folded over residual maps at (from left to
right and top to bottom) 0.7, 2.0, 3.6, and 6.1 cm................................... 126
Model maps of Mie scattering calculations for the February 1997 epoch.
Images depict the folded over residual maps at (from left to right and
top to bottom) 1.3, 2.0, 3.6, and 6.1 cm ..................................................... 127
Residual maps of Mie scattering calculation. Images depict the folded
over residual maps at (from left to right and top to bottom) 1.3, 2.0,
3.6, and 6.1 cm................................................................................................ 128
Residual maps of Mie scattering calculations. Images depict the folded
over residual maps at (from left to right and top to bottom) 1.3, 2.0,
3.6, and 6.1 cm................................................................................................ 129
Model maps of hybrid isotropic/Mie scattering calculations for the
November 1995 epoch. Images depict the folded over residual maps
at (from left to right and top to bottom ) 0.7, 2.0, 3.6, and 6.1 cm. . . 132
Model maps of hybrid isotropic/Mie scattering calculations for the
February 1997 epoch. Images depict the folded over residual maps
at (from left to right and top to bottom ) 1.3, 2.0, 3.6, and 6.1 cm. . . 133
Residual maps of hybrid isotropic/Mie scattering calculation. Images
depict the folded over residual maps at (from left to right and top to
bottom ) 0.7, 2.0, 3.6, and 6.1 cm ................................................................. 134
Residual maps of the hybrid isotropic/M ie scattering calculations. Im­
ages depict the folded over residual maps at (from left to right and top
to bottom) 1.3, 2.0, 3.6, and 6.1 cm............................................................ 135
xiii
R e p r o d u c e d w ith p e r m issio n o f th e co p y rig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
1
CHAPTER I
IN T R O D U C T IO N
1.1
H istorical O verview
The study of Saturn has been a progressive understanding of observation and
theory. Galileo first turned his telescope to Saturn in 1610. Because the rings hap­
pened to be nearly edge-on as Galileo saw them, he interpreted them not as rings but
instead as a pair of fixed satellites (Galilei 1613). In Huygens proposed th at these
were not satellites but a flat, solid ring, detached from Saturn and inclined from the
ecliptic. Twenty years later, Cassini noticed a dark band separating outer from inner
ring. This would be later named the Cassini Division.
In the 19th century, the understanding of the rings improved dramatically.
LaPlace (1802) showed th a t any solid ring around Saturn would break apart un­
der centrifugal forces and Maxwell (1859) show m athem atically th a t in order for this
ring system to remain stable, it must be made up of many sm all independent or­
biting particles. This was confirmed with the Doppler shifted spectra taken in 1895
by Keeler and confirmed by Campbell (Alexander 1962) being consistent with the
particle nature of the rings.
The atmosphere was first scrutinized in the late eighteenth, century by Herschel
(Alexander 1962). He deduced th at the zones, belts and spots seen on Saturn were not
surface features, but atmospheric phenomenon. This also led to the first estimation
of the rotation period of Saturn (10.281 hours) close to the m odern value of 10.6562
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
2
hours (Davies et al. 1989). Throughout the nineteenth century, the zone and belts
were found to change and spots were often transient features which lasted a year or
so (Alexander 1962).
In the twentieth century the advent of multiwavelength astronomy and inter­
planetary spacecraft allowed new and more careful study of Saturn. Radio wave­
lengths allowed the atmosphere to be penetrated to deeper levels than the cloudtops.
Centimetric radar studies (Goldstein and Morris 1973) observed the ring to be highly
reflective, implying that ring particles had sizes of order several centimeters. The
flybys of Pioneer 11 (Dyer 1980), Voyager 1 (Stone and Miller 1981), and Voyager
2 (Stone and Miller 1982) observed the rings at unprecedented resolution, revealing
surprising detail, which is still not fully understood after nearly twenty years of anal­
ysis (see Figure 1-1). Later in the 1980’s, Very Large Array (VLA) measurements of
Saturn (dePater and Dickel 1991 and references within) revealed some of the scat­
tering properties of the rings and shown only a modestly active atmosphere. On the
optical side of things, the Hubble space telescope has taken high resolution images of
Saturn which rival that of Voyager (Figure 1-1).
Parameters used in this thesis are given in Table 1-1. Saturn’s rings are usually
classified into seven rings, each given the letter designations A through G, though we
will only be concerned with the main rings (A, B, C) and the Cassini Division (CD).
Note th a t in Table 1-1 the radius of Saturn (R sat = 60,268 km) is given at the one
bar level, appropriate to radio wavelengths.
1.2
M o tiv a tio n
If one were to select which Jovian planets would be best to study a ring system
and an atmosphere in the microwave regime (0.7 to 21 cm wavelengths), then Saturn
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
3
Figure 1-1: Spacecraft images of Saturn, (top) Voyager 2 image of Saturn, (bottom)
Hubble IR image of Saturn (Karkoschka and NASA 1998.)
R e p r o d u c e d w ith p e r m issio n o f th e co p y r ig h t o w n e r. F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
4
Basic Ring Geometry
Rsa£(l atm .) = 60268 km
Ring
Radial distance (km)
(R sat)
inner C
74510
1.236
inner B
92000
1.527
inner CD
117580
1.951
inner A
122170
2.027
outer A
136780
2.270
Table 1-1: The geometry of the m ain rings. Radial distance scale is given by Nichol­
son. Cooke, and Pelton (1990).
might be the best and only choice for both. Saturn clearly has the largest and most
sophisticated ring system; in fact it is the only ring system which had been detected
in the microwave. As for the atmosphere, both Uranus and Neptune provide too small
of targets (3.8” and 2.4” at best) to see much detail. While Jupiter has the angular
size, its disk is obscured by synchrotron emission from its strong magnetic field at
21 cm and by outgassing from Io at 6 cm. Shortward of 4 cm the disk of the planet
become clearly and unambiguously resolved (see dePater 1990). However, due to
the simultaneous influence of both temperature and opacity on the radio brightness,
observatios at a wide range of wavelengths is required in order to unambiguously
interpret any radio map. Saturn has a large enough angular size (~ 20°), to show
the atmosphere, but none of the unusual phenomena th a t one sees at Jupiter. The
only thing th at obscures the atmosphere is, of course, the rings. However, only one
hemisphere is blocked at a time, and if one observes at the right time (i.e., when the
inclination of the planet is small) then the obscuration by the rings can be reduced.
R e p r o d u c e d w ith p e r m issio n o f th e co p y rig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
5
High resolution (< 1.5"), high dynamic range, low inclination (|z| > 5.5 ° maps
of Saturn at microwave wavelengths have never been acquired. D ePater and Dickel
(1982) observed with the VLA Saturn at i = —5.4° (1982), but the poor quality of
their maps made it difficult to see the rings. Since Saturn has a 29.5 year orbital
period it was not until 1995 when observations could once again be taken at low
inclinations. In fact 1995-1996 had three ring plane crossings as seen from the Earth
as well the one as see from the Sun (the Saturnian autum nal equinox).
It is this latter phenomenon which intrigued us the most. Even though the
Sun was in the plane of the rings (and hence not shining on them), the E arth lay
some 2.7° above the ring plane, nearly the maximum angle possible. This gave us
the opportunity to observe the rings while the Sun’s influence was at a minimum.
While it’s true the Sun microwave emission is too weak to see in reflected sunlight,
the Sun’s flux does provide a fair fraction of the physical heating of the rings which in
turn dictates the amount of thermal emission a ring particle may have. Furthermore,
this therm al emission is best seen at shorter wavelengths like 0.7 cm.
This motivates us to observe multiwavelength VLA observations Saturn during
the autum nal equinox. Observations at 0.7 cm have never been done for only recently
have thirteen 0.7 cm been installed at the VLA. These obseravtion would have the
added benefit of being the most sensitive to the ring thermal emission. A follow-up
observation was also done at a modestly larger i = —5°, which can be used as a
sensitive test to geometry changes in the rings and check on any secular changes in
the atmosphere. In the 1980’s, dePater and Dickel (1991) did such a VLA monitoring,
covering inclination angles between 5.4 and 26°.
They found the atm osphere to be
generally unchanging throughout the decade.
As for the rings, various models have been used to describe their brightness
R e p r o d u c e d w ith p e r m issio n o f th e co p y r ig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
6
tem perature.
1.3
O utline o f T h esis
In this thesis we shall examine two VLA data sets taken in November 1995 and
February 1997 and introduce a fairly sophisticated model which will provide further
constraints on the ring and its physical and spatial composition. In Chapter 2, we
explain the basics of interferometric radio astronomy and synthesis imaging as applied
to VLA observations of Saturn. In Chapter 3, we look at the first data set compare
it to older data sets and note the several confirmed observations and new discoveries.
In Chapter 4, we examine the second data set and make comparisons of that to our
first data set. Next, in Chapter 5, we discuss the Mie scattering, Monte Carlo ring
models, examining the fundamental equations and basic assumptions. In Chapter
6, we apply this model to the data. We compare and contrast the consistency of
the data with both isotropic and Mie scattering models, and suggest future work for
determining the single particle phase function. Finally, in Chapter 7, we summarize
our conclusions.
R e p r o d u c e d w ith p e r m issio n o f th e co p y r ig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
7
C H A P T E R II
OBSERVATIONS A N D D A T A R E D U C T IO N
2.1
In trod u ction
Even with, the largest single dish antennas in the world, high resolution imaging
of most objects in the microwave regime would be impossible. Interferom etry allows
observations of compact or highly structured sources, by increasing th e effective res­
olution of the instrument by the cooperative use of multiple antennas separated by
distances much larger than the sizes of the antenna. This is ideal for studying the
zonal structure on the surface of Saturn (or any other planet) as well as distinguishing
the radial and azimuthal structure of the rings.
The main crux of this chapter will be to focus on the issues at hand for interferometric observations. Specifically, we will discuss the Very Large Array (VLA)
and the issues of imaging planetary objects. In Section 2.2 we will discuss the basic
concepts of interferometry and aperture synthesis. In Section 2.3 VLA design and
capabilities. In Section 2.4 we deal with the issues of planetary observations at the
VLA. In Section 2.5 we confront the difficulty of making maps (imaging) of the data.
R e p r o d u c e d w ith p e r m issio n o f th e co p y rig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
8
2.2
In te r f e r o m e tr y an d A p e r tu r e sy n th e sis
2.2.1
Preface
For readers unfamiliar with aperture synthesis, we present this section as an
introduction to interferometry and aperture synthesis. Extensive reviews of this sub­
ject may be found in Taylor, Carilli, and Perley (1999, 1994) and Thompson, Moran
and Swenson (1986).
2.2.2
Visibilities
Incoming radiation from a distant radio source may be denoted by a time vari­
able, frequency dependent complex electric field E t/(r, t ), where the direction of propa­
gation is denoted by r. If we only concern ourselves the scalar properties of the electric
field then, E „(r, t) = -E^r). If two elements (antennas) of an array observe these sig­
nals, then both may be multiplied to produce a visibility, V ( r i,r 2) = E ^v-ijE ^vo),
where the asterisk indicates the complex conjugate. The visibility is the basic observ­
able for any interferometric observation.
The visibility has units of flux density, W m-2 Hz-1 . The standard unit of flux
density is a Jansky (Jy), defined as 10-26 W m -2 Hz-1 . The visibility is a function
of the radial vectors along the line of sight from the antennas to the source. It is
convenient to consider the vector difference or spatial coherence function, r 2 — r 1;
with coordinates u ,v ,w , so th at V (r1:r 2) —>
■V (r2 —iq) = V (u ,v .w ).
An assumption can be made to simplify the visibility function. If we confine our
observations to the sky plane, then the u, v plane lies in the sky plane and the spatial
coherence function may be defined in terms of the wavelength, A = c/u, sothatT2—ri
=
X(u,v,0). Therefore, the visibilities depend only on the separations of the antennas.
R e p r o d u c e d w ith p e r m issio n o f th e co p y rig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
9
2.2.3 Intensity derived
The visibilities do not lead directly to
in
from visibilities
a finished image which must be put
terms of the radio brightness or intensity, Inu, of the object as function of sky
coordinates. The units of intensity are flux density per steradian W m -2 Hz-1 sr-1.
If we confine our observation to a small region in the sky then we can relate the
visibility to the intensity in a recognizable form (see Chapter I of Perley, Frederic
and Bridle 1994):
Vu{u,v) = I J I u{ x ,y ) e ^ 2^ ux+vy)) dx dy
(2.1)
This is just a Fourier transform and may be inverted,
/„(x, y) = I J Vu( u , v ) e ^ ux+Vy)) du dv
(2.2)
Our ability to sufficiently sample the u, v plane is crucial in unambiguously determin­
ing
2.2.4
Discrete sampling and u, v coverage
How an antenna pair reads a signal from a source, depends not only on the
qualities of the antennas, but also their spatial separation and orientation. The more
antenna pairs we have available for an observation the more combinations of r 2 —r x
are available and the better sampling or u, v coverage we have. Furthermore, the
rotation of the Earth over the course of an observation, will alter the u, v coverage of
the a single pair of antennas since the antennas will change their spatial orientation
as seen by the source. This, in effect, will increase the coverage in the u , v enabling
us to better calculate I u in Equation 2.2. In Figure 2-1 we plot the visibilities for the
7 February 1997 observation at 3.6 cm and 6.1 cm scaled to the wavelength. We can
parameterize this u ,v coverage by introducing the sampling function S ( u , v ). Then
R e p r o d u c e d w ith p e r m issio n o f th e co p y r ig h t o w n e r. F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
10
Equation 2.2 becomes,
I ? ( x ,y ) =
jj
Vu(u ,v) S{u,v) e2wi^ x+vy) du dv
(2.3)
This is often called the dirty beam (denoted by the superscript D ) , which is simply
the convolution (denoted by the asterisk) of the true intensity and the synthesized
beam, B (see Chapter 1 of Perley, Frederic and Bridle 1994),
B{x,y) =
JJ
S(u, v) e2^ ux+vy) du dv
(2.4)
The quantity B( x, y ) is also known as the point spread function.
2.2.5
The brightness temperature
Throughout this work, we will discuss the temperature of the planet in terms of
the brightness temperature. This is defined as the tem perature of a blackbody which
is emitting the same intensity at some specified frequency. In the radio regime, the
Rayleigh-Jeans law applies and we have for the intensity at frequency, u,
2 iP
=
T
(2-5)
or,
Tb
=
^
2 ul k
( 2 -6 )
where k is the Stefan-Boltzmann constant and c is the speed of light.
2.3
T h e V ery Large Array
The instrument with which we took all of the observations of Saturn was the
Very Large
Array (VLA), located in Plains of San Augustin in NewMexico and
run by the National Radio Astronomy Observatory (NRAO). Information about
R e p r o d u c e d w ith p e r m issio n o f th e co p y rig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
11
300
200
—
100
—
CT)
>
CO
5
_c
o
if
5
-100
-200
—
—
••» •- *•': \4\. *r
.•• • t, , *t "• /.*.*
s•
•
' * *•
^
I . •T ' t r T y C / . ••
1 VufVlh &'
••
-300 —
Kilo W avlngth
Figure 2-1: Visibility plot for the 7 February 1997 at 3.6 cm and 6.1 cm. The axes
are scaled to the wavelength.
R e p r o d u c e d with p e r m issio n o f th e co p y r ig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
12
N
VLA Schematic
/N
1
o
/
.o'
1
ex,.
/
0
Figure 2-2: A schematic of the VLA layout. Each arm is 120° from the other and
the North Arm is displaced 5° west of north.
the history and detailed functionality can be obtained at VLA web site at URL
h ttp ://w w w .n ra o .e d u /v la /html/VLAhome. shtm l. Here I will outline some of the
principal features of the instrument.
The VLA consists of twenty-seven 25 meter parabolic antennas arranged in a
wye, such th a t nine antennas lie on each arm. The array is slightly rotated (5° towards
west) from pointing to true geographic north (see Figure 2-2). The instrument may
be configured in 4 primary arrays of varying lengths of about 36 km, 11 km, 3 km, and
1
km. These are known simply as the A, B, C, and D arrays. The array configuration
is changed 4 months, moving to the next smallest array. There are also 3 transitional
arrays (A /B , B /C and C /D ), each lasting about a month, which retain a longer North
arm. These configurations are often used to observe more southemly objects since
R e p r o d u c e d w ith p e r m issio n o f th e co p y r ig h t o w n e r. F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
13
those see a forshortened North arm.
Attached to each antenna is a secondary* feed which houses the receiver. All the
VLA receivers are cooled and are sensitive to a number of bands with approximate
wavelengths 1.3. 2.0, 3.6, 6.1, 20, 90 cm. Recently (1994), T hirteen of the antennas
were are also equipped with 0.7 cm receivers. The receivers can tune to two frequen­
cies channels per wavelength. Each channel is sensitive to right and left circularly
polarized light and have maximum bandwidth of 50 MHz. The two frequency chan­
nels may be separated by up to 450 MHz. Each of these d ata streams is referred to
as an I F .
2.3.1
Calibration
In all VLA observations one collects an ensemble of observed visibilities, Vy,
where i j denote one antenna pair. W hat we desire is the true visibility, V^-. Since
the configuration and engineering is well known and well done at the VLA, we can
relate the two quantities by a relatively simple relationship (see Chapter 5 of Perley,
Frederic and Bridle 1994),
Vij = &ij(t)Vij + £ij(t) +
(2-7)
where t is the time of the observation , Qij{t) is a baseline-based complex gain, £ij{t)
is a baseline-based complex offset, T)ij(t) is a stochastic complex noise. In general it
is adequate to neglect the Cij{t) term as small and to represent Gij(t) as the product
of complex antenna-based gains (Gij = GiGj).
The task is then reduced to determining the amplitude and the phase of the
£>(u). A secondary calibrator is used to construct the gain function (Gij(t)) which is
used (via Equation 2.7) to obtain the real visibilities. Equation 2.7 is solvable because
of the number of antennas pair available, N ( N — l)/2 . If all (iV = 27) antennas are
R e p r o d u c e d w ith p e r m issio n o f th e co p y r ig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
14
Prim ary Flux Calibrators
Flux Calibrator
Position (a, 5)J2ooo
F1u x3.6
3C286
13* 31m 08.3s, 30° 45' 58.6"
5.20
3C48
01* 37™ 41.3s, 33° 09' 35.1"
3.15
3C147
OS' 1 42m 36.1s, 49° 51' 07.2"
4.7
(Jy)
Table 2-1: VLA primary flux calibrators. Fluxes of each are given at 3.6 cm.
used, this amounts to 351 baselines per observation of the secondary calibrator. The
prim ary flux calibrator is used in turn to obtain the current flux of the secondary
calibrator.
Therefore, one must choose calibrators in the sky which are suitable for am­
plitude and phase calibration. The best absolute flux calibrators are ones which are
unresolved and have well known flux and do not exhibit secular changes. Three such
calibrators which are used are given in Table 2-1. The flux densities at all VLA
frequencies are given by Baars et al. 1977, and can be found at the VLA web site.
These objects need be observed only once of twice per observing run, preferably when
the elevation of calibrator is equal to th at of the source. Some of these calibrators
are partially resolved, especially at higher frequencies and in the extended arrays and
require adjusting the flux downward to properly apply it to the complex gain.
Also essential for calibration is the phase correction. Signals coming from a
source are essentially in phase unless they impinge on a turbulent medium (such as
the E arth’s atmosphere).
The secondary calibrator should be near the program source. Like absolute
R e p r o d u c e d w ith p e r m issio n o f th e co p y r ig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
15
(primary) flux calibrator, the most suitable secondary calibrators should be unre­
solved. However, they may be slowly variable, provided th a t their period of variation
is much longer than the observation run. Though a strong (>
1
Jy) source is desirable,
sources weaker than this can be adequately used to calibrate the phases. Because our
atmospheric conditions are rapidly changing, we must observe the secondary calibra­
tor often enough to interpolate between gain corrections. The typical time between
secondary calibrator observations is wavelength dependent and ranges from as long
as 30-45 minutes for 21 cm to 5-10 minutes at 0.7 cm.
2.4
M a k in g o b se rv atio n s o f S a tu r n a t t h e V L A
2.4.1
Issues of planetary imaging
Now we turn our attention to the issues and concerns of observing Saturn with
the VLA. Much discussion of this on this topic is given by B utler and Bastian (1999),
with some of these issues discussed in Bastian (1994). Generally, planets are good
emitters of microwave radiation. Most of this is a ttrib u ted to therm al radiation,
especially those planets with little or no atmosphere. In fact planets such as Mars
and Uranus have often been used as calibrators, because of their strong signal and
unresolved surface or atmospheric features. Planets are also extended sources, and
some do show resolvable structure from not only the m ain disk but from other sources
around the disk such as synchrotron radiation (as from Jupiter) and rings (as seen in
Saturn) which are interesting phenomena but complicate the analysis. Furthermore,
planetary satellites may provide another complication in the observation of the planet
if their contribution is not considered.
Concerning the planet of this thesis, the rings of Saturn provide most of the
challenges in the observations. If one is interested in only the planet, the rings are a
R e p r o d u c e d with p e r m issio n o f th e co p y r ig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
16
nuisance to be dealt with. Furthermore, they obscure the disk of the planet allowing
you to observe only one hemisphere fully at moderate to high ring inclinations. One
can avoid dealing with the rings by observing them edge-on, but these are rare (every
13-15 years) occurrences, so more often than not one must consider the contribution
from the rings.
Observing the rings provided their own challenges. The flux em itted from the
ansae is a combination of scattered Saturnian therm al emission and therm al emission
coming directly from the rings. The Rayleigh-Jeans law (Equation 2.5) shows that
therm al emission becomes stronger at smaller wavelengths. Furthermore, the rings
across the planet will also show attenuated direct therm al emission from Saturn. In
general the brightness tem perature of the rings is given by,
T r = Tsc + Tth + TSat e~T'sin^
(2 .8 )
where sc and th refer to the scattered and thermal components of the brightness
tem perature and Sat refers to the thermal emission of Saturn. Also, r is the normal
optical depth and i is the inclination of Saturn’s rings as seen from Earth. By exam­
ining the properties of rings though careful inspection of our data, it may be possible
to distinguish each term in Equation 2.8 from each other.
Putting these issues aside there are other concerns which are unique to the VLA,
Saturn, and time of our observations. Choosing the proper array size and observing
strategy we will deal with presently. In Section 2.5 we will deal with the issues of
synthesis imaging a planet.
2.4.2
Choosing the correct arrays
The array configuration is crucial since this and frequency determine the angular
resolution and u, v coverage of the array. Angular resolution is proportional to the
R e p r o d u c e d with p e r m issio n o f th e co p y r ig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
17
ratio of the wavelength over the antenna separation distance.
Hence the highest
resolution occurs at 0.7 cm using the A array. W ith this configuration, however, a
problem of sufficient u, v coverage becomes im portant. Since the VLA essentially
"sees’" a Fourier transform of the real object in the sky, oblate spheroids like Saturn
appear like a standard 2-D Bessel function in the u , v plane. The vast bulk of the
signal is contained in the central peak. If the u, v coverage is sufficient, then proper
flux is calculated from Equation 2.7. If the u, v coverage is rather poor in this region
then reconstructing the planet may be more difficult.
To map any planet one must deal with the relatively large angular size of the
disk as well as the sharpness of the atmosphere’s edge and in the case of Saturn, the
rings. The first issue is a m atter of observing the total flux of the disk and requires
antennas with short spacings. Observing the sharpness of Saturn’s atmosphere and
rings requires large spacings to give us the necessary resolution. The problem then
lies in the finite configurations of the VLA: there is no single configuration with
very short and very long antenna spacings. One could observe the planet at two
configurations (one extended, one compact) and combine the results. However, since
planets are dynamical (both intrinsically and geometrically) combining results from
two significantly different time is not feasible.
This was of major concern to us in November 1995 since the thirteen 0.7 cm
receivers were arranged as 5 on the North arm and 4 each on the other two arms
(a configuration no longer used). This means that observations done with the outer
14 antennas (2.0, 3.6 and 6.1 cm in the case of the November 1995 epoch) have a
paucity of short spacings and the visibilities may be insufficient to reconstruct the
planet properly.
R e p r o d u c e d w ith p e r m issio n o f th e co p y r ig h t o w n e r. F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
18
2.5
S y n th esis Im a g in g a n d M o d e lin g
2.5.1
Creating the image
Once the d ata have been taken they can be processed with a number of imaging
processing software packages. The one we used was the Astronomical Image Pro­
cessing System (AIPS) which was developed by the NRAO (1996). The ALPS tasks
allow to run a relatively few tasks (subroutines) to calibrate the data. After the data
are calibrated, they can be imaged by taking the Fourier transform as dictated by
Equation 2.3. Since we have a finite number of visibilities, in practice Equation 2.3 is
a discrete sum. Furthermore, each point should be weighted (typically by the inverse
of the variance, see Chapter
6
of Perley, Schwab, and Bridle 1994). Hence, we can
now write for Equation 2.3,
I ? ( x , y) = E M u k ,
k=1
Vk )
W ( u k, vk) S ( u k, vk) e^
«**+««»
(2.9)
where M is the number of visibilities and W{u, v) is the weighting function.
As
noted before, the dirty beam is the convolution (denoted by the asterisk) of the true
intensity and the synthesized beam:
I ? ( x , V) = I* * B (x , y)
(2 -1 0 )
So our dirty image would be convolved with B ( x . y ) . Since B ( x , y ) has significant
sidelobes the resulting map will have unphysical artifacts which must be removed.
To remove these artifacts we use the standard AIPS CLEAN algorithm devel­
oped by Hogbom (1974), modified by Clark (1980) and described in Perley, Schwab,
and Bridle (1994, Chapter
8
). It essentially involves removing the sidelobes by ( 1 )
finding the peak flux in the dirty image; (2 ) subtracting a specified fractional portion
of th a t flux from the map; (3) and either return to step (1) or if the map has been
R e p r o d u c e d with p e r m issio n o f th e co p y r ig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
19
sufficiently cleaned; (4) convolve the removed part with a “clean” beam (usually an
elliptical Gaussian fitted to the central lobe of the dirty beam); (5) and add the resid­
uals to the dirty image to the “CLEAN” image. This in effect removes the sidelobe
artifacts without removing any true sky flux.
2.5.2
Hybrid Mapping/Modeling Procedure
In choosing an analysis procedure for our data, we sought a m ethod th at will
deal adequately with the technical issues presented by interferometry of planets as well
as the scientific issues of quantifying both ring and planet brightness tem peratures in
ways conducive to comparison with physical models and with other data sets. In this
subsection, we briefly describe each of these issues and then explain how the hybrid
modeling/mapping procedure we used addresses them.
Making maps from our interferometric data present three interrelated technical
issues. First, as is generally true of interferometric data with very high SNR (as our
shorter spacings have), the accuracy of antenna phases determined from a secondary
calibrator is insufficient. The general solution is to self-calibrate these phases by
allowing them to vary in the mapping process.
Second, standard mapping programs are oriented towards objects th a t can be
represented as a set of point sources, and generally fail to converge for a disk-shaped
object like a planet. The general solution is to subtract an- oblate disk with the
approximate dimensions and brightness tem perature of the planet from the u, v data
first. Then the residuals are mapped, allowing for the possibility of both positive and
negative brightness features. However, once the initial disk is subtracted, the phase
self-calibration can no longer be done as it will not have been applied to the initial
disk.
R e p r o d u c e d w ith p e r m issio n o f th e co p y rig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
20
Finally, estimation of the average brightness tem perature is generally based on
measurements at very short interferometer spacings, ones th a t reflect the total flux
density of the planet. As mentioned before, the subarrays we used in the November
1995 epoch have few measurements at spacings short enough to be on the central
maximum in the Bessel function which is the Fourier transform of the planet disk.
None are short enough to directly measure the to tal flux density.
We use a multistep, hybrid m apping/m odeling procedure to address the issues
raised above. The central obstacle to solving the technical issues in the way previous
authors have is the lack of short spacings in the subarrays we have used. The key to
solving this problem is to realize that longer spacings can be used equally well because
we know the precise shape of the planet and rings in advance. In particular, 98% of
the total flux density in our final maps can be accounted for with a three parameter
model. We represent the brightness tem perature of the unocculted portion of the
planet disk as ai -f- a-iii, where
and a? are model param eters and fi is the cosine of
the angular distance of the Earth from local norm al on Saturn. The third parameter,
&i, represents the brightness of the stripe where the rings are projected against the
planet. First we solve for the model param eters with a linear, least squares fit in the
u, v plane. Then we use this model as a reference for self-calibration of the antenna
phases. We then iterate a few times until the solutions converge. The specificity of
the component shapes allow the longer spacings to be used unambiguously to fix the
total flux density. Also, as this model accounts for the bulk of the total flux, the
phase calibration is already very close to optim al.
After subtraction of the final model, one could now map and clean the small
deviations from the model (i.e., the latitudinal structure and the ring ansae). A
linear least squares fit in the map plane could then be performed to determine the
R e p r o d u c e d w ith p e r m issio n o f th e co p y rig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
21
brightness of each ring versus azimuth. However, we choose here to do the same fit
in the u, v plane instead. The advantage of working directly with the observables is
th at one obtains meaningful correlation coefficients the model param eters. We use
these to help decide how finely to segment the rings, balancing additional azimuthal
information against unacceptable correlations. The disadvantage to fitting in the
u, v plane is the greater computational requirements (since u, v visibilities outnumber
nonzero map pixels), but this was not prohibitive in our case. Our approach is a hybrid
m apping/m odeling procedure in th at we iterate between modeling ring components
in the u , v plane and cleaning the disk of the planet with the standard AIPS task
(IMAGR). The iteration is analogous to the m ajor clean cycles th a t already take place
within IMAGR. The final map would be the same in either approach. We also allow
additional self-calibration of phases within these final iterations, but little change in
the calibration takes place.
This approach was utilized fully for the February 1995 epoch. For the November
1997 epoch, the more open geometry of the rings and higher angular resolution de­
manded finer modeling of the rings. This would require a computationally prohibitive
amount of visibilities to fit. Thus, this portion of the mapping scheme was not done
in favor of the Monte Carlo simulation (cf. Chapter 5).
R e p r o d u c e d w ith p e r m issio n o f th e co p y rig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
22
C H A P T E R III
F IR S T E P O C H
3.1
In trod u ction
Very Large Array (VLA) observations of Saturn have been published by two
groups. Grossman, Muhleman, and Berge (1989) [hereafter GMB] reported on 2 and
6
cm wavelength images made with a ring inclination of 25°. Further details of these
observations are in Grossman (1990). De Pater and Dickel (1991) [hereafter dPD]
reported on a series of observations made from 1981 to 1987 with ring angles between
5 and 26°. Both groups discuss the implications of their images for understanding
the chemistry and dynamics of the 0.8-10 bar layer of Saturn’s atmosphere and the
scattering characteristics of the ring particles.
The dynamics of the zonal flow in Saturn’s atmosphere is not yet well understood
(see review by Ingersoll et al. 1984). Unlike the case of Jupiter, Voyager observations
did not show a tight correlation between the locations of visible belts and zones and
zonal velocity structure. Kerola, Larson and Tomasko (1997) show that the aerosols
th at affect optical and infrared light are probably produced photochemically in the
upper atmosphere, and are not directly connected to the convection cycle below.
GMB and dPD found a long-lived zone of enhanced radio emission in the zone 20-50°
north at wavelengths greater than 2 cm. The enhancement must indicate either an
increase in tem perature or a decrease in opacity in th at zone. They attributed the
enhancement to decreased ammonia opacity caused by large scale downwelling. This
R e p r o d u c e d w ith p e r m issio n o f th e co p y rig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
23
zone correlated with a warm zone seen by the Voyager IRIS experiment, which they
interpreted the same way.
The composition and size distribution of particles in Saturn's main rings have'
been established through the combination of a variety of observations, including es­
pecially the Voyager occultation and imaging experiments (see reviews by Cuzzi et
al. 1984, Nicholson and Dones 1991, and Esposito 1993). On the other hand, there is
little consensus on the details of the spatial distribution of the particles, both verti­
cally and in the plane. Numerous dynamical simulations in recent years have shown
the possibilities of monolayer vertical distribution and of the clustering of particles
in wakes in the plane (Salo 1991, 1992, 1995, Mosqueira 1995, Sterzik et al. 1995,
Schmit and Tscharnuter 1995, Mosqueira, Estrada, and Brookshaw 1996, and Spahn,
Thiessenhusen, and Hertzsch 1996). However, there have been no quantitative obser­
vational tests of these computations as yet.
GMB and dPD show convincingly that the centimetric emission observed from
the rings is therm al emission from Saturn that has scattered one or more times in the
rings. (Thermal emission from the rings themselves begins to contribute at shorter
wavelengths.) This geometry implies that in a single image we see emission from
a wide range of scattering angles.
T hey both find a preference for forward over
backscattering at wavelengths greater than 2 cm. Grossman (1990) confines this
result to just the C ring. Finally, dPD note a slight preference for higher brightness
temperatures in the west ansa than the east. Cuzzi, Pollack, and Summers (1980)
performed the first detailed scattering computations for S atu rn ’s rings, and found
that the ring brightness as a function of inclination was sensitive to the vertical
distribution of the particles. Grossman (1990) computed the single scattering of
randomly distributed particles, and found he could reproduce some of the details of
R e p r o d u c e d w ith p e r m issio n o f th e co p y rig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
24
the azimuthal distribution with an empirically determined (non-Mie) scattering phase
function- However, he was unable to reproduce the C ring forward scattering result.
We note further that the sym m etry of a random spatial distribution precludes the
possibility of reproducing the east-west asymmetry.
By imaging Saturn with high resolution at four wavelengths a t Saturnian equinox,
we extend the observational d a ta base in several ways. For the study of the planet,
increased resolution is im portant to unambiguously identify radio features with op­
tical ones. The equinox also provides the opportunity to simultaneously view both
northern and southern hemispheres (where previous observations observed only the
north). For the study of the rings, the equinox provides a much lower inclination than
previously observed. Finally, our observations include the first high quality maps at
0.7 cm. Hence we can study the thermal properties of the rings at this rare period
when no sunlight reaches them directly.
In Section II, we describe the details of how the observations were made. We
begin Section III with a discussion of the hybrid m apping/m odeling procedure we
use to analyse the data. We then describe the application of this procedure to the
equinox data, and present the results of the analysis. In Section IV we discuss the
implications of our results for both the planet and the rings, as well as suggestions for
further work, both observational and theoretical, to resolve the outstanding questions.
3.2
O bservations
We observed Saturn for one full track with the VLA in the B configuration
(Thompson et al. 1980). The observation took place on 20/21 November 1995 at
21:50-7:40 UT. It was scheduled to coincide with the passage of the sun through the
ring plane (autumnal equinox). During the observation T itan was near quadrature
R e p r o d u c e d w ith p e r m issio n o f th e co p y rig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
25
(170" from Saturn), so the effect of its emission on our Saturn maps is negligible. The
pointing was set to track the motion of Saturn, which was at (a,
£ )j2 o o o = (2 3 A1 9 m ,
—6 °47'). Saturn’s equatorial angular diameter was 18.03"; its rotation axis was in­
clined 4-2.68° to the plane of the sky (showing the north side of the rings) and had a
projected position angle 5.02° east of north.
We divided the array into two subarrays. The first subarray consisted of the
13 antennas with 0.7 cm wavelength receivers (the inner 5 on the north arm and the
inner four on the east and west arms), and was used at 0.7 cm (43.3149 GHz) for the
whole run. The second subarray consisted of the remaining 14 antennas, and was set
at 2.0, 3.6, and 6.1 cm wavelengths (14.9649, 8.4149, 4.8851 GHz) in alternation. All
observations were made in dual circular polarization in each of two 50 MHz channels.
The synthesized beams in these arrays were 0.5", 0.4", 0.7", and 1.2" at 0.7, 2.0, 3.6
and
6 .1
cm wavelengths, respectively.
The primary calibrator, 3C286, was observed when it was at an elevation similar
to th at of Saturn. The absolute flux density scale a t 2.0, 3.6, and
6 .1
cm wavelengths
was set by assuming the flux of 3C286 is 3.45, 5.20, 7.41 Jy (Baars et al. 1977),
respectively, and is good to 3%. The absolute flux density scale at 0.7 cm assumed
1.47 Jy for 3C286, which was based on a comparison with Mars and is probably good
to 10% (B. Butler, private communication). For a secondary calibrator we observed
J2323—032, which was 3.7° from Saturn. These observations determine preliminary
phases and final amplitudes of the complex antenna gains.
The basic observing schedule at 0.7 cm wavelength consisted of 10 m inute time
blocks split between a 7.5 minute scan of Saturn and a 2.5 minutes scan of the
secondary calibrator. In addition we made reference pointing scans of the secondary
calibrator a t 3.6 cm wavelength once each hour. This procedure improves the antenna
R e p r o d u c e d w ith p e r m issio n o f th e co p y rig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
26
pointing from the nominal 10" accuracy to 3". No jumps could be seen in the antenna
gains following these scans, indicating that the incremental corrections were small.
The basic observing schedule in the second subarray consisted of 20 minute time
blocks with 14 minutes on Saturn
6 .1
cm) and
6
(6
minutes at 2.0 cm and 4 minutes each at 3.6 and
minutes on the secondary calibrator
(2
minutes at each wavelength).
Both subarrays used a 10 second integration time.
3.3
3.3.1
A n a ly sis
Hybrid Mapping/Modeling Procedure
In Section 2.5.2 we described the procedure used to map the d ata by consid­
ering the technical issues of the November 1995 data. In addition to solving these
technical issues, we also seek a procedure th a t will quantify brightness temperatures
in ways conducive to comparison with physical models and to other d ata sets. While
the earlier analyses of dPD and GMB adequately treat the technical issues presented
by their data sets, they have a variety of lim itations in their quantitative presen­
tations. Brightness temperature variations across the disk of the planet arise from
limb darkening and latitudinal variations. (Any real longitudinal variations will be
smeared out by planet rotation during the observation.) To show these, GMB and
dPD present east-west and meridional scans. This presentation does not fully sepa­
rate the two phenomena, and it only uses a portion of the data Since splitting our
observations among four wavelengths reduced our sensitivity, make optimal use of
the d ata we have will be important. Also, since the limb darkening is convolved with
the restoring beam, which varies from observation to observation, intercomparison
between data sets is not possible. Grossman (1990) gives a more general treatm ent,
simultaneously fitting in the map plane for limb darkening function and latitudinal
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
27
dependences, but the limb darkening param eters found are not reported.
The brightness tem perature of the rings varies with Saturnicentric radius and
azimuth. dPD present only scans through the rings. These indicate qualitative be­
havior, but fail to use the whole d ata set. Also, no attem pt is m ade to deconvolve
the separate rings. GMB present average values for the main rings based on linear
least squares fits of the maps. However, the azimuthal variation is not quantified.
Grossman (1990) plots ring brightness as a function of azimuth (both ansae averaged
together). However, neither the effect of mixing between adjacent rings and nor the
correlation between adjacent azimuths is addressed. Comparison of the uncertainties
with the variation in the data suggests few of the points are statistically independent.
Information about the average planet disk (average brightness tem perature and
degree of limb darkening) is contained in the param eters ai and a 2. Brightness de­
viations versus latitude may be determined by averaging over longitude in the clean
map with the model components removed. Since this residual map has nearly a zero
mean, rolloff effects near the planet limb and the ring stripe are minimal.
3.3.2
Application to Equinox D ata
We fix the precise shape of planet and the value of /z as a function of sky
coordinate by computing an equipotential surface with an equatorial radius equal to
the
1
bar radius, 60286 km, and a rotation period equal to the magnetic field period,
38362.4 s (Davies et al. 1995). We also use the gravitational mass and moments of
Campbell and Anderson (1990).
We modeled the rings with 10 components in each ansa. Figure 3-1 shows the
geometry of the west ansa as viewed during the observation. Azim uthal segment
boundaries are marked with dashed lines. The segments of the ring directly adjoining
R e p r o d u c e d w ith p e r m issio n o f th e co p y r ig h t o w n e r. F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
28
20 N o v em b er 1995
CV2
O
<D
VI
O
O
cd
CV2
10
12
14
18
16
20
a rc sec
Figure 3-1: View from Earth, of the ring model components in the west ansa. Solid
lines indicate separations between the main rings (labeled A, B, and C). The gap
between the A and B rings is the Cassini Division. Dashed lines indicate separations
between azimuthal segments of the model components.
the planet could not be resolved into the separate main rings. The A and B rings
are broken into three segments: near, quadrature, and far. The Cassini Division
and the C ring are represented by single components. At
6
cm wavelength, where
the resolution was poorest, the Cassini division components were merged into the
neighboring A ring components. Figure 3-2 shows the component geometry as it
would be seen from above the ring plane.
Before mapping we edit d ata taken near anomalous calibrator measurements,
less than 1% of the total. After a few modeling iterations we also clip d ata with
residuals many times the random noise, again a small percentage of the total.
We use modified uniform weighting (with robustness param eter set to zero) in
the mapping. We also reduce the weight of baselines in the central Bessel spot so that
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
29
— --- '--------- 1--------- '------
r
O
C \2
O
CD
U\
a
cO
r ,
c
C \2
I
_!______ i ■ ■
20
I■
0
_
1
-2 0
arc sec
Figure 3-2: Polar view of the ring model components. Solid lines indicate separations
between the main rings (labeled A, B, and C). The gap between the A and B rings
is the Cassini Division. Dashed lines indicate separations between the azimuthal
segments of the model components.
they have identical SNRs, equal to the greatest SNR elsewhere. This is appropriate as
these data are limited by the uncertainty in the phase calibration, while the remaining
d ata are limited by therm al noise. We weight the 0.7 cm data further, based on the
average rms of the calibrator data with time. This is im portant in weighting down
the first and final hours of data, which were taken at high airmass.
Finally we apply appropriate corrections to both the models and the maps for
the primary beam of the VLA antennas. At 2 cm wavelength, this does not affect the
planet, but is a 2% correction at the furthest point of the A ring. At 0.7 cm, this is
a 4% correction at the limb of the planet and a 25% correction at the furthest point
of the A ring.
All maps are 1024 by 1024 grids with 0.05" pixels, small enough to avoid roundoff
R e p r o d u c e d w ith p e r m issio n o f th e co p y r ig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
30
at all wavelengths. The maps are oriented with Saturn’s north pole directed up.
3.3.3
Results
The final model values along with their uncertainties are presented in Tables 3-1
and 3-2. For brevity we do not reproduce the tables of correlation coefficients. How­
ever, the values indicate reasonable statistical independence. The greatest values
among ring components is only 0.56, and these are rare. The m ajority are less than
0 . 02 .
The correlation between the two planet parameters, ai and a2, is large (—0.85).
An orthogonal pair of parameters could be defined by a linear combination: a i + ^ a o ,
and a2, where (/i) is the spatial average of n and has a value of 0.652 a t our epoch.
However, both parameters are well determined and the definition is adequate for our
purposes. In Figure 3-3 we plot as a function of wavelength the disk-averaged bright­
ness tem perature ((Tp) = ai + (fj.)a2+2.7+(TR), where (T r ) is the mean tem perature
of the residual map, and the last term corrects for the microwave background). For
.comparison we also plot the same quantity from GMB, Grossman (1990), and dPD.
The close agreement at 2.0, 3.6, and
6 .1
cm wavelengths is a verification th a t our
procedure overcame the lack of short spacings. (The GMB value at 6.1 cm lies di­
rectly under our value.) The disagreement of 0.7 cm is probably within the limits of
the absolute calibration uncertainty. In Figure 3-4 we plot the ratio a 2 /aj., the frac­
tional importance of limb darkening, versus wavelength. It shows a strong, positive
correlation as expected qualitatively from earlier observations.
Figures 3-5 to 3-8 are the final maps at each of the four wavelengths. In order
not to lose the subtle ring and latitudinal structure in the strong contrast with the
bright planet, we have subtracted a3 + a2ju from the planet, where a3 is a significant
R e p r o d u c e d w ith p e r m issio n o f th e co p y r ig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
31
Band
0.7 cm
Component
T b (K)
ai
115.7(0.2)
129.4(0.3)
0-1
0 1 0 6
. ( . )
11.2(0.5)
bi
24.2(0.5)
41.3(0.7)
2 .0
cm
East
West
East
West
Al
4.0(2.7)
9.1(2.7)
10.0(3.5)
8 .6
A2
8
.3(1.3)
3.3(1.3)
3.9(1.6 )
1.9(1.6 )
16.5(2.7)
7.9(3.5)
3.6(3.5)
-1.6(4.6 )
5.7(4.6)
.9(2.6 )
14.8(2.6)
6 8 1
. ( .7)
3.6(1.6 )
Ring:
(3.4)
A3
5.8(2.7)
CD
-3.8(3.9)
3.2(3.9)
Bl
14.2(1.9)
13.2(1.8)
B2
9.3(1.1)
7.3(1.1)
B3
10.3(1.8)
11.5(1.8)
9.4(2.6 )
5.3(2.6 )
C
14.1(1.2)
15.0(1.2)
15.4(1.7)
19.0(1.7)
North
15.0(1.3)
10.9(1.3)
South
14.5(1.3)
19.1(1.3)
6
.9(2.0)
12 6 2 0
15.7(2.0)
12 2 2 0
6
. ( . )
. ( . )
Table 3-1: Components in the Saturn Model for 0.7 and 2.0 cm. One sigma errors
are given in parenthesis next to the ring segment temperature. The abbreviations
used in the Segment are described in Figure 3-1. The letters A, B, and C refer to
the corresponding ring CD refers to the Cassini Division. The numbers attached to
A and B refer to eh subsection of th a t ring: (1) far. (2) middle, and (3) near, the
terms North and South refer to the segments closest to the planet in Figure 3-1.
R e p r o d u c e d w ith p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
32
3.6 cm
6 .1
cm
Component
T b ( K)
at
149.3(0.1)
153.9(0.3)
do
19.9(0.2)
28.8(0.4)
bi
56.7(0.4)
37.6(1.2)
Ring:
East
Al
West
East
7.8(8.8)
0.3(8.8 )
-14.5(8.3)
-9.1(8.2)
A2
1.7(8.8 )
7.2(8.8)
3.9(2.8 )
5.9(2.8 )
A3
-1.3(8.8 )
3.3(8.8 )
17.6(8.3)
31.0(8.4)
CD
6.3(8.8 )
12.5(8.8)
Bl
1.3(8.8 )
13.9(8.8)
B2
4.0(8.8)
5.0(8.8 )
B3
9.5(8.8 )
12 8 8 8
C
14.1(8.8)
11 6 8 8
8
North
7.0(8.8 )
South
16.8(8.8)
N /A
West
N /A
12.0(7.3)
21.9(7.2)
.7(4.2)
9.9(4.2)
-2.1(7.2)
-5.8(7.3)
.9(4.8 )
10.6(4.8)
10.5(8.8)
2.8(3.4)
5.3(3.5)
22.9(8.8)
15.1(3.5)
19.2(3.5)
. ( . )
. ( . )
8
Table 3-2: Components in the Saturn Model for 3.6 and 6 . 1 cm. One sigma errors
are given in parenthesis next to the ring segment tem perature. The abbreviations
used in the Segment are described in Figure 3-1. The letters A, B, and C refer to
the corresponding ring CD refers to the Cassini Division. The numbers attached to
A and B refer to eh subsection of th at ring: (1 ) far, (2) middle, and (3) near, the
terms North and South refer to the segments closest to the planet in Figure 3-1.
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
33
A
v
W a v e le n g th (c m )
Figure 3-3: Disk-averaged spectrum of Saturn. Filled circles represent the data from
this work. Open circles refer to data from de P ater and Dickel (1991) and open
triangles refer to data from Grossman (1990).
w
o ..
U3 '
ca
CM
ca
d■
o
d
10
10 °
l
W a v e le n g th (c m )
Figure 3-4: Plot of the limb darkening fraction, a 2 / a 1, versus wavelength.
R e p r o d u c e d w ith p e r m issio n o f th e co p y r ig h t o w n e r. F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
34
o
r*
m
o
ini
o
t—
OHS OUV
Figure 3-5: Map of Saturn at 0.7 cm wavelength on 20 November 1995. The restoring
beam is 0.57" x 0.46" FWHM, indicated by an ellipse in the upper right. The wedge
across the top gives the transfer function of the residual in Kelvin.
R e p r o d u c e d with p e r m issio n o f th e co p y r ig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
35
O
t*
LO
O
LO
i
O
■
oas oav
Figure 3-6: Map of Saturn at 2.0 cm wavelength on 20 November 1995. The restoring
beam is 0.85" x 0.4" FWHM, indicated by an ellipse in the upper right. The wedge
across the top gives the transfer function of the residual in Kelvin.
R e p r o d u c e d w ith p e r m issio n o f th e co p y rig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
36
Figure 3-7: Map of Saturn at 3.6 cm wavelength on 20 November 1995. The restoring
beam is 0.8" x 0.64" FWHM, indicated by an ellipse in the upper right. The wedge
across the top gives the transfer function of the residual in Kelvin. The transfer
function has been set to show both the planet and the rings.
R e p r o d u c e d w ith p e r m issio n o f th e co p y rig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
37
Figure 3-8: Map of Saturn at 6.1 cm wavelength on 20 November 1995. The restoring
beam is 1.38" x 1.13" FWHM, indicated by an ellipse in the upper right. The wedge
across the top gives the transfer function of the residual in Kelvin.
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
38
fraction of ei].. The restoring beams are indicated by ellipses in the comer of each
figure, and the values of a3 are given in the figure captions. The noise seen away from
the planet is near the expected thermal value at all wavelengths; one-sigma values
for a single beam are indicated in the figure captions. In Figures 3-9 and 3-10 we
plot longitudinally-averaged residual brightness tem perature (with respect to the two
component planet model) as a function of planetographic latitude. The data points
in the 0.7 and 2.0 cm plots are separated by one beam and should be completely
independent. The points in the 3.6 and 6.1 cm plots are separated by one half a
beam, so that immediately neighboring points are not independent.
Figures 3-5-3-10 show a range of latitudinal structure on the planet. The most
significant structure at 0.7 cm is a 3 K equatorial enhancement in the range sine lati­
tude —0.2 to +0.2. This band is broken by a dip at +0.05 th a t is well separated from,
and therefore not caused by, the ring stripe. There are also smaller enhancements
th at peak at sine latitude —0.85, +0.48 and +0.66. And finally there is a gradual drift
towards lower brightness in the northern hemisphere. The 2.0 cm wavelength emis­
sion show no statistically compelling structure. However, as the sensitivity is less here
than at 0.7 cm, we cannot rule out the presence of similar structure at this wavelength
with these data. The 3.6 cm emission shows an even more pronounced equatorial en­
hancement, now well separated into two peaks. Other distinct features are peaks at
sine latitude —0.7, +0.68, and the North Polar Cap. W ithin the lower resolution of
the 3.6 cm data, the peaks at —0.7 and +0.68 seem to correlate with peaks at 0.7
cm. Finally, the 6.1 cm figures are once more dominated by the two equatorial peaks.
Mild increases in the north and south are consistent with the 3.6 cm structure, but
the still lower resolution at 6.1 cm washes out fine structure, and precludes definitive
identifications.
R e p r o d u c e d w ith p e r m issio n o f th e co p y r ig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
39
o -
-
- 0 .5
1.0
0.0
0 .5
1.0
sin (latitud e)
M -ki
■.'jM
-i-1r ••
iI i '1
!fJ-- t
ca
E-
<1
-L
I
_i
-
1.0
.
L
- 0 .5
0 .0
0 .5
1.0
sin(latitude)
Figure 3-9: Longitudinally averaged latitude structure of the Saturnian atmosphere
at 0.7 (top) and 2.0 (bottom) cm wavelength on 20 November 1995.
R e p r o d u c e d w ith p e r m issio n o f th e co p y rig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
sin (latitu d e)
£3
o
LO
-
1.0
- 0 .5
0.0
0 .5
1.0
sin (latitud e)
Figure 3-10: Longitudinally averaged latitude structure of the Saturnian atmosphere
at 3.6 (top) and 6.1 (bottom) cm wavelength on 20 November 1995.
R e p r o d u c e d w ith p e r m issio n o f th e co p y r ig h t o w n e r. F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
41
0
10
10
12
14
0
10
16
ARC SEC
20
18
10
12
14
16
ARC SEC
20
20
18
20
Figure 3-11: Maps of the 0.7 cm wavelength ring structure on the west (top) and
bottom (east) side. Contours indicate temperatures of 5, 7, 9, 1 2 , & 15 Kelvin. The
restoring beam is 0.57" x 0.46" FWHM, indicated by the ellipse in the upper right
corner. The therm al noise is 1 . 6 K.
R e p r o d u c e d w ith p e r m issio n o f th e co p y r ig h t o w n e r. F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
42
0
10
10
12
14
0
10
16
ARC SEC
20
18
10
12
14
16
ARC SEC
20
20
18
20
Figure 3-12: Maps of the 2.0 cm wavelength ring structure on the west (top) and
bottom (east) sides. Contours indicate tem peratures of 5, 7, 9, 12, & 15 Kelvin. The
restoring beam is 0.64" x 0.4" FWHM, indicated by the ellipse in the upper right
corner. The therm al noise is 4.1 K.
R e p r o d u c e d with p e r m issio n o f th e co p y r ig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
43
0
I_____________
10
5
15
;_I____________________ I_____________________ !_________________________
12
0
10
10
14
5
12
14
16
ARC SEC
16
ARC SEC
- ____________I—
18
10
20
20
15
18
20
20
Figure 3-13: Maps of the 3.6 cm wavelength ring structure on the west (top) and
bottom (east) sides. Contours indicate tem peratures of 5, 7, 9, 12, Sz 15 Kelvin. The
restoring beam is 0.8" x 0.64" FWHM, indicated by the ellipse in the upper right
corner. The therm al noise is 1.1 K.
R e p r o d u c e d w ith p e r m issio n o f th e co p y rig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
44
0
10
5
12
0
10
10
14
5
12
14
16
ARC SEC
15
18
10
16
ARC SEC
20
20
15
18
20
20
Figure 3-14: Maps of the 6.1 cm wavelength ring structure on the west (top) and
bottom (east) side. Contours indicate temperatures of 3.5, 7, 9, & 12 Kelvin. The
restoring beam is 1.38" x 1.13" FWHM, indicated by the ellipse in the upper right
corner. The thermal noise is 1.2 K.
R e p r o d u c e d w ith p e r m issio n o f th e co p y rig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
45
Figures 3-11 to 3-14 show just the portions of the maps with the ring ansae. The
east ansae have been flipped for direct comparison with their western counterparts.
The scale is the same as in Figure 3-1, which shows the ring geometry. The planet,
which is shown at its full intensity and is saturated here, extends beyond its geometric
bounds by a full beam. The emission at 0.7 cm wavelength traces well the ring
geometry seen in 3-1, including the gap inside the C ring. The ring brightness steadily
declines with increasing radius, a feature repeated at all wavelengths. The east and
west ansae look similar except for the A ring/Cassini Division. However, given the
low brightness at this radius it is not clear how significant this difference is. At 2.0 cm
the gap inside the C ring is seen again. But here the emission in the west ansa is
stronger than the east from the middle of the B ring inward. At 3.6 cm the emission
near the planet clearly favors the southern (near) side over the northern (far) side,
so the gap is no longer well defined. The north-south resolution is insufficient to
determine which ring (or rings) contributes most to the southern enhancement. The
emission in the west ansa now appears stronger than in the east over the whole ring
system. At 6.1 cm the increasing beam size causes the planet to fill in more of the
gap. Nonetheless, the ridge of emission adjoining the planet still favors the southern
side as at 3.6 cm. And once more the emission in the west ansa is stronger than in
the east, at least for the C and B rings. The A ring is near the noise level in both
ansae.
Comparison of the maps to the model components in Table I indicates a good
correlation, as expected, and confirms quantitatively the trend for the western ansa
to dominate at wavelengths greater than 0.7 cm. Two minor improvements in the
selection of component boundaries are possible, however. The back and front portions
of the A and B rings are not sufficiently resolved at
6
cm wavelength, where the
R e p r o d u c e d w ith p e r m issio n o f th e co p y r ig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
46
resolution is poorest, and should probably be merged into a single A /B back and
another A/B front component. This is seen at its worst in the back of the west
ansa, which gives a (meaningless) negative value for the A ring and an implausibly
high value for the B ring. The sum of the two, however, would be a good estimate
of the average of both rings. On the other hand, the components at quadrature at
the shorter wavelengths have enough resolution to be split into two radially. This
would quantify the effect visible in the maps that the brightness tem perature tends
to decrease with radius within individual rings.
3 .4
Discussion.
3.4.1
Atmospheric Dynamics
The latitudinal structure described above (two bright bands straddling the equa­
tor with a scattering of fainter, narrow bands elsewhere) is directly opposite the results
consistently found in the 1980s (a single broad band in the northern midlatitudes).
The appearance of structure at 0.7 cm is likewise counter to previous experience, as
no structure had been seen at 2 cm. This is the first indication of a m ajor change in
the global circulation pattern.
Comparison with Hubble Space Telescope images obtained three days before our
observations (from the HST web site) show the Equatorial Zone boundaries to be sine
latitude —0.21 and +0.29. This coincides with the latitude range th at includes the
two equatorial bands observed in the radio. There is no sign of an optical counterpart
to the radio dim zone sitting directly on the equator.
The common interpretation of the Equatorial Zone is th a t it is a region of
upwelling material. It has been associated with a number of storm systems in recent
years (e.g., Barnet et al. 1992), which is consistent with that interpretation. However,
R e p r o d u c e d w ith p e r m issio n o f th e co p y r ig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
47
following the interpretation that radio structure is due to opacity variations (dPD
and GMB), one expects an upwelling region to have high opacity and therefore low
observed brightness. We consider therefore the alternative hypothesis: radio structure
is due to tem perature variations. Such gradients may be stably sustained in the
presence of vertical velocity gradients (e.g., Ingersoll et al. 1984).
We can now suggest a self-consistent scenario for the time variable structure of
the Equatorial Zone by pursuing an analogy of the zone of cool w ater th a t sits directly
on the E arth ’s equator in the Pacific Ocean during the La Nina portion of the ENSO
cycle. This strip arises when steady easterly winds drive a westward motion of surface
water throughout the tropics. The Coriolis effect adds a force directed towards the
poles. The divergence at the equator draws cool, deep ocean w ater up to maintain
dynamic equilibrium.
S aturn’s Equatorial Zone has a wide belt of wind with large velocities to the
east. Presumably at its base, however, the zonal velocities go to zero. Therefore there
must be a significant range of layers w ith wind velocity increasing with elevation.
In these layers the Coriolis effect adds a force directed towards the equator. The
convergence at the equator should force gas downwards, where it will be cooler than
its surroundings. This is observed as the radio dim zone.
If at another time the velocity gradient lessens, the dim zone on the equator
will vanish, analogous to the El Nino portion of the ENSO cycle on E arth. On Earth
the oscillation between these two phases is thought to have an indirect, but signifi­
cant impact on the atmospheric dynamics at midlatitudes as well (e.g., Guetter and
Georgakakos 1996). How this might work on Saturn is a completely open question.
But it suggests that it is plausible th a t the disappearance of the m idlatitude radio
bright zone and the simultaneous appearance of the Equatorial Zone structure was
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
48
not coincidental.
A test of this scenario would be to look for direct evidence of the change in
vertical wind shear with time. Barnet et al. (1992) used HST data from 1990 to
determine a vertical wind shear and found velocity decreasing with elevation. This
should correspond with the El Nino phase. Van der Tak et al. (1997) reported that the
m idlatitude radio band was still strong at that epoch, as required by the hypothesis.
New optical observations to look for a change in the wind shear to go along with the
radio state now observed would be a strong further test.
3.4.2
Rings
Both GMB and dPD found the B ring to be the brightest ring at all wavelengths,
followed in order by the A ring, the C ring, and lastly the Cassini Division, which was
not definitively detected. The variations were attributed to varying distance from
Saturn (the source of the scattered light) and varying optical depths. The projected
optical depth of the A ring is always greater than unity, but it is further from Saturn
than the B ring. The C ring is closer to Saturn than the B ring, but has a projected
optical depth less than unity. The Cassini Division is both further and thinner than
the B ring.
In the equinox observations, the rings are foreshortened by a factor of
21
. The
Voyager RSS d a ta (which was obtained at a ring inclination of 5.9°) showed a 3.6 cm
wavelength optical depth of ~ 0.10 for the C ring near the B /C boundary (Tyler et
al. 1983). This implies a contrast of 12% at that boundary, which would be too small
for us to see. East-west traces along our 0.7, 2.0, and 3.6 cm d a ta in both ansae show
a brightness peaked in the inner C ring that gradually diminishes with distance from
Saturn, in good agreement with the prediction.
R e p r o d u c e d with p e r m issio n o f th e co p y r ig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
49
At longer wavelengths the ring opacity drops as the smallest particles become
transparent. The contrast predicted by the Voyager at 13 cm is 23%. B oth ansae
of the
6
cm image (Figures 19 and 20) show declining brightness with radius within
the B and C rings, but a 50% contrast at the B/C border. This is greater than the
predicted value. We suggest, however, that the contrast is an opacity effect. The
quantitative discrepancy may lie in a nonrandom vertical distribution of particles
which would alter the simple application of the foreshortening factor.
While the resolution is adequate to separate the Cassini Division from its sur­
roundings, our sensitivity to such a small region is inadequate to distinguish a bright­
ness equal to the A ring from a nondetection.
We found above the ring brightness is enhanced close to the planet on the near
side at 3.6 and
6
cm wavelengths. This is consistent with the forward scattering
reported by GBM, Grossman (1990), and dPD. Comparisons of the front and back
components of the A and B rings give conflicting results, so it is indeterm inate whether
the effect continues to greater angular distances from the planet. Grossman (1990)
noted this effect is predominantly in the C ring. We note th at as the A and B rings
ansae display a wider range of scattering angle than the C ring, this observation
implies a fundamental difference between the scattering properties of the C ring on
the one side and the A and B rings on the other.
DPD cited a slight east-west asymmetry favoring the west ansa in their data.
We find a clear asymmetry with the same sign at 2.0, 3.6, and
6 .1
cm. This not
only confirms the earlier result, but also suggest the effect increases with decreasing
inclination. As described in the introduction, this asymmetry cannot be accounted
for by single (or multiple) scattering off randomly distributed particles. We propose
th at multiple scattering in the anisotropic particle distribution of wakes predicted by
R e p r o d u c e d w ith p e r m issio n o f th e co p y rig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
50
East
West
E
c
D
▼
Figure 3-15: A polar view of Saturn and a ring. Sample ray paths for Satum ian
radio emission scattered towards E arth by ring particles at six different azimuths are
labeled A-F. Particle wakes inclined by ~25° are indicated schematically with bold
line segments.
theory may account for the asymmetry. We illustrate how these wakes may induce
asymmetries heuristically by marking them in Figure 3-15, which illustrates (from a
viewpoint high above the orbital plane of the rings) the ray paths of Saturn radiation
scattering off a ring towards the Earth.
Radiation from Saturn can be scattered directly toward the Earth by the par­
ticles in a wake on the western side of the rings (azimuth E). On the eastern side
of the rings (azimuth B), however, incident radiation must be scattered through the
wake before it can reach the E arth. These effects should operate at all wavelengths,
but may diminish with increased inclination. If this hypothesis is born out by a real
radiative transfer calculation, then maps of ring brightness versus ring azim uth and
inclination contain direct information about anisotropies in the spatial distribution
of the particles on size scales ju st larger than the particles themselves, and may be
R e p r o d u c e d w ith p e r m issio n o f th e co p y r ig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
51
used to constrain the particle dynamics.
The 0.7 cm maps (Figure 3-11) show higher average brightnesses in the A and
B rings. This was predicted by dPD, who argue for a significant therm al contribution
at this wavelength.
The lack of observed east-west asym m etry here may simply
be a dilution effect of the additional emission. For therm al emission, one might
expect a diurnal effect, with the ring generally warmer on the side of Saturn that
receives reflected sunlight. The emission peak might also be shifted by a phase lag
due to thermal inertia. The east-west symmetry, particularly at the points nearest
the planet, sets an upper limit on any diurnal effect.
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
52
C H A P T E R IV
SE C O N D E P O C H
4.1
In trod uction
We took a second set of data, fifteen months after our first data set (described
in C hapter III). This was long enough to notice any mid-term temporal variations
on the disk or rings, but not so far removed in time, as to make these observations
completely uncomplimentary. Other secular, high resolution observations of Saturn
at the Very Large Array (VLA) have been done by both de Pater and Dickel (1991)
[hereafter dPD] and Grossman, Mulhleman, and Berge (1989) [hereafter GMBj. It
should be noted that except for their 1982 observing run, dPD observed at only
one or two different wavelengths. Additionally, dPD had no observations with ring
inclination below 5.8° and no secular observations have been taken in the 1990’s.
In Section 4.2 we describe the observation and their similarities and differences
from the first epoch. In Section 4.3 we present our results and make comparisons to
the previous data including the first epoch. We defer further discussion of the rings
until Chapter VI after we have described our ring modeling technique.
4.2
O bservations
In our second epoch, we observed Saturn for two full tracks with the VLA.
We observed Saturn on two nonconsecutive days, 7/8 February (17:00-4:00 UT) and
14/15 February 1997 (16:30-2:30 UT). For brevity, we will refer to the 7/8 February
R e p r o d u c e d w ith p e r m issio n o f th e co p y rig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
53
Date
Array
Position
Equatorial
Inclin.
Angle
Angle
(1997)
(arcsecs)
16.41
-4.91
3.78
16.28
-5.20
3.73
B-C
00A 19m, -0 0 ° 20'
14/15 February
CnB
00
22
m, -
00
°
02
'
(degrees)
(degrees)
(a;, £) .72000
7/8 February
*
Position
Table 4-1: Circumstances of second epoch VLA observations. The array B-C indicates
the array was set in a transistion configuration between the standard B and BnC
arrays. The negative inclinations indicate the south pole is tipped toward Earth.
The position angles are measured east of north.
observation as the first track and the 14/15 February observation and the second
track. The details of the observation are given in Table 4-1. Notice th a t the first
track was in a different configuration from the second track (B-C vs. BnC), and
that the first track was the middle of a transition between the B array and BnC.
We attem pted to observe Saturn as close to the same absolute inclination as the
first epoch (+2.68°). However, we were allocated tim e when Saturn’s inclination was
somewhat larger than in November 1995. Titan this tim e was closer to Saturn (~ 60"
on 7/8 February and ~ 22" on 14/15 February), but posed no problem in our analysis.
We used all 27 antennas for each observation. T he wavelengths for the first
track were set of 3.6, 6.1, ~20 cm (8.48601, 4.8601, ~ 1.6 GHz). The second track’s
wavelengths were set at 1.3 and 2.0 cm (22.4601 and 14.1940 GHz). All observa­
tion were in dual circular polarization in each of two 50 MHz channels except the
one at 20 cm. For this wavelength the channels were split into one 25 MHz chan­
nel centered at 18 cm and one 50 MHz channel centered a t 21.3 cm (1.6650 and
1.3851 GHz, respectively). This was done to avoid radio frequency interference (RFI)
R e p r o d u c e d w ith p e r m issio n o f th e co p y rig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
54
which plagues this regime and nearly prevent any observation with the two 50 MHz
channels. This has the consequence of increasing signal to noise, but does allow us to
observe at two distinct wavelengths. The synthesized beams for the first track were
0.6" x 0.3", 1.1" x 0.5", 3.8" x 1.7", and 3.4" x 1.8" at 3.6, 6.1, 18, and 21.3 cm, respec­
tively. For the second track the synthesized beams were 0.6" x 0.3" and 0.90" x 0.46",
for 1.3 and 2.0 cm, respectively. The m ajor axis of the beam was roughly along the
east-west direction in the Saturnian reference frame.
For both tracks, the primary calibrator, 3C286, was observed when it was at
an elevation similar to th at of Saturn. The absolute flux density scale a t 1.3, 2.0 3.6,
6.1, 18, and 21.3 cm is 2.53, 3.45, 5.20, 7.41, 13.6, and 14.8 Jy (Baars et al.
1997),
respectively. For a secondary calibrator we observed J0016-032, which was 0.7° from
Saturn in the first epoch and 2.0° for the second epoch.
The basic observing schedule for the first track consisted of 21 m inute time
blocks divided between 3.6, 6.1, and 18/21 cm. For each wavelength, five minutes
were spent on Saturn and two were spent on the secondary calibrator. For the second
track, the basic observing schedule also consisted of 2 1 minute time blocks. The scans
consisted of 16 minute scans of Saturn at 1.3 cm and 2.0 cm (two 4 m inute blocks
for each wavelength) and 5 minutes on the secondary calibrator (three m inute scans
at 1.3 cm and two minute scans at 2.0 cm). As in the first epoch, all scans used 10
second integration time.
4.3
4 .4
A nalysis
Issues w ith our second ep och d ata
The technical issues which need to be addressed have been outlined in Sections
2.5.2 and 3.3. W ith all 27 antennas available for our observations, the lack of short
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
55
spacings was not an issue as it was with our first epoch. Some of the antenna spacings
may be sufficiently short enough to measure directly the total flux density of the
planet’s disk. Nevertheless, due to our success in mapping the first epoch data, we
follow the same general hybrid m apping/modeling procedure for the planet as outlined
in Section 2.5.2.
As in our first epoch, we represent the brightness tem perature of the unocculted
portion of the disk as ai + a 2 ju, where ai and a2 are model parameters and n is
the cosine of the angular distance of the E arth local normal on Saturn. However,
the model for the rings across the disk must be reconsidered. The planet’s larger
inclination (5.0° as compared to 2.7°) coupled with generally better resolution in
the vertical direction, demands th at each individual ring (A, B, C) be modeled. The
Cassini division was merged into the neighboring A ring. These parameters are labeled
bi, b2 and b3. For the poorest resolution data of 18 and 21 cm, a single component was
used to represent the entire brightness tem perature of the rings across the planet.
Unlike the first epoch, the ring ansae are not modeled in this way. The geometry
of the rings is shown in Figure 4-1. Because of the increased projected area of the
rings, the number of independent model components would be greater than the first
epoch and be computationally prohibitive. In this case the rings were cleaned with
the standard AIPS task (IMAGR) along with any residual planetary flux after the
planetary components are removed. We defer the discussion of the modeling of the
ansae until we outline our Monte Carlo simulations in C hapter V.
The data were edited to take anomalous calibrator measurements amounting to
less than 1% of the total data, except for 18/21 cm. In these cases some 5% of the
data needed to be edited due to strong R FI’s. Even after this editing, the 18/21 data
still suffered from anomalous d ata probably due to unexpected R FI’s (cf below).
R e p r o d u c e d w ith p e r m issio n o f th e co p y rig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
56
February 1997
CM
O
<D
W
O
t-i
O
8
10
14
12
16
18
20
arc sec
Figure 4-1: View from E arth of the ring model components in the west ansa. Solid
lines indicate separations between the main rings (labeled A, B, and C). The gap
between the A and B rings is the Cassini Division. Dashed lines indicate separations
between azimuthal segments of the model components.
Though we used modified uniform weighting (robustness param eter set to zero)
in our maps, no additional weighting of was necessary due to the higher quality of
d ata over the first epoch.
We also applied the appropriate primary beam corrections, which even at 1.3 cm
is only 0.3% on the planet and 5% on the outer A ring. T he maps are 1024 by 1024
grids with 0.05” pixels. Because of the lower resolution a t 18/21 cm, these maps are
256 by 256 grids of 0.20” pixels.
4.4.1
Results
The final model values along with their uncertainties are presented in Table 4-2.
For brevity we do not reproduce the tables of correlation coefficients. Few of these
were above 0.50. However, the correlation between the two planet parameters, a x and
R e p r o d u c e d w ith p e r m issio n o f th e co p y rig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
57
Band
1.3 cm
Cmpnt.
T b (K)
ar
126.8(0.1)
CLo
cm
21
160.2(0.1)
179.1(0.3)
177.1(0.6)
16.8(0.1)
23.6(0.2)
67.4(2.5)
72.5(3.2)
31.5(0.7)
50.6(0.4)
53.1(1.6)
124.8(4.3)
110.0(5.3)
28.7(0.6)
28.6(0.7)
17.3(0.4)
23.0(1.7)
-53.6(18.3)
33.6(21.4)
97.7(0.6)
100.9(0.7)
94.8(0.4)
102 1 1 6
6 .1
142.3(0.1)
151.8(0.1)
19.4(0.2)
13.9(0.2)
bi
32.6(0.6)
bo
63
cm
cm
18 cm
3.6 cm
2 .0
. ( . )
Table 4-2: Components in the Saturn Model for the second epoch. One sigma errors
are given in parenthesis next to the component brightness tem perature. Component
ai and Go are the uniform and limb darkened disk components, repectively. Except
for 18/21 cm, the ring components, bi, b2: and 6 3 are the A, B, and C rings segments
across the planet, respectively. For 18/21 cm, bi and b2 are the entire ring across the
planet and a limb darkened ring across the planet, repsectively. The ansae were not
fit by the u, u data.
a2 were quite large (> —0.90). If we choose an orthogonal pair, ai + (fi) a2, a 2 then
the correlations are minimal, but the brightness tem peratures derived are essentially
the same. So for our purposes, the definitions of ai and a 2 are adequate.
In Figure 4-2 we plot the disk-averaged brightness tem perature as function of
wavelength. Though similar to Figure 3-3, it plots (To) = ai + (n) a2 + 2.7 4- (T r )
where
( T r)
for both epochs and the works of GMB, Grossman (1990) and dPD . At
3.6, 6.1, 18, and 21.3 cm there is close agreement amongst observations. At 1.3 and
2.0 cm, however, there more than a 3-sigma disagreement with both the our first
epoch and previously published data. This may be indicative of some of the ongoing
change in the atmosphere since the early 1990’s. In Figure 4-3, we examine the ratio
of the limb-darkened component to th a t of the constant disk, a2/a \ and compare to
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
58
I
J
» •'' o
,- v f
u' y
•' y
_t____________________________
10 °
i
i
101
W a v e le n g th (c m )
Figure 4-2: Disk-averaged spectrum of Saturn. Filled circles represent the data from
the first epoch, the filled squares the second epoch. Open circles refer to data from
de Pater and Dickel (1991) and open triangles refer to data from Grossman (1990).
the first epoch. In general the agreement is good, and we continue to get a positive
correlation longward of 6 cm. The largest discrepancy comes at 1.3 cm, where the
ratio is much higher than we might expect. The smaller offsets at 3.6 and 6.1 cm may
simply be due to the better quality of data we are presented in second epoch.
Figures 4-4 to 4-9 are the final maps at each of the six wavelengths. Since it is
difficult to show details of the ring and the planet simultaneously we plot two other
sets of map which show the subtleties in both the planet and the rings. Figure 4-10
show the disk of the planet with ax + a2^ subtracted from the disk. The thermal
noise seem away from the planet is near the expected value except in the case of 18
and 21 cm, where the noise is considerably worse. One-sigma values are indicated at
the figure captions. Figures 4-11 to 4-12 show the longitudinally-averaged residual
brightness tem peratures as a function of planetographic latitude. In each figure the
R e p r o d u c e d w ith p e r m issio n o f th e co p y r ig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
59
O
m
o'
O
_
<C
CO
©
w
B
o
CV2
o
o
o
o
o
o
10°
io ‘
W avelength, (c m )
Figure 4-3: Plot of the limb darkening fraction, a2/a i, versus wavelength. The solid
line indicates data from the second epoch, the dshed line from the first.
points are separated by one half a beam, so that immediately neighboring points are
not independent. A gap is present in each figure where the rings occult the planet.
Because of the high noise, we choose not to show latitude maps at 18.0 and 21.3 cm.
Figures 4-4 to 4-12 show a range of latitudinal structure on the planet. These
generally confirm the results of Section 3.3.3 for the atmosphere, but the finer reso­
lution allows us make other observations. At 1.3 cm there is an enhancement at sine
latitude near —0.1 and possibly at —0.5 and +0.65, with a deficit near the equator.
These four features get stronger at 2.0 cm which likely confirm their validity. This
structure does not compare well with the first epoch 0.7 cm data (Figure 3-9). W ith
the poor single-to-noise with the first epoch 2.0 cm data, direct comparisons with the
latitude maps are quite difficult. Meanwhile, the 3.6 and 6.1 cm d a ta set compare
quite well with their first epoch counterparts. Like the first epoch, the enhancements
around the equatorial region are clearly present a t both wavelengths. In the 3.6 cm
R e p r o d u c e d w ith p e r m issio n o f th e co p y r ig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
60
Figure 4-4: Map of Saturn at 1.3 cm wavelength on 14 February 1997. The restoring
beam is 0.60" x 0.32" FWHM, indicated by an ellipse in the upper right. The wedge
across the top gives the transfer function of the residual in Kelvin.
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
Figure 4-5: Map of Saturn at 2.0 cm wavelength, on 14 February 1997. The restoring
beam is 0.90" x 0.46" FWHM, indicated by an ellipse in the upper right. The wedge
across the top gives the transfer function of the residual in Kelvin.
R e p r o d u c e d w ith p e r m issio n o f th e co p y rig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
62
O
t—
1A
O
03 s
tfl
i
O
t—
i
oav
Figure 4-6: Map of Saturn at 3.6 cm wavelength on 7 February 1997. The restoring
beam is 0.64" x 0.32" FWHM, indicated by an ellipse in the upper right. The wedge
across the top gives the transfer function of the residual in Kelvin.
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
63
Figure 4-7: Map of Saturn at 6.1 cm wavelength, on 7 February 1997. The restoring
beam is 1.07" x 0.52" FWHM, indicated by an ellipse in the upper right. The wedge
across the top gives the transfer function of the residual in Kelvin.
R e p r o d u c e d w ith p e r m issio n o f th e co p y r ig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
64
Figure 4-8: Map of Saturn at 18.0 cm wavelength on 7 February 1997. The restoring
beam is 3.41" x 1.74" FWHM, indicated by an ellipse in the upper right. The wedge
across the top gives the transfer function of the residual in Kelvin.
R e p r o d u c e d w ith p e r m issio n o f th e co p y rig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
65
Figure 4-9: Map of Saturn at 21.3 cm wavelength on 7 February 1997. The restoring
beam is 3.41" x 1.80" FWHM, indicated by an ellipse in the upper right. The wedge
across the top gives the transfer function of the residual in Kelvin.
R e p r o d u c e d w ith p e r m issio n o f th e co p y r ig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
66
map, there are minor enhancements a t —0.7 and +0.62, and at 6.1 cm there is a
minor enhancement which lies at sine latitude —0.6. B oth also very clearly show a
deficit at the equator; the rings no longer obscure the issue here. Other deficits are
see at —0.4 and 0.18 at both 3.6 and 6.1 cm. All of these features can be seen at both
epochs (though are sharper at the second epoch owing to the higher resolution).
Figures 4-13 to 4-16 show just the portions of the m aps with the ring ansae. As
with those ansae found in 3.3.3, the east ansae have been flipped for direct comparison
with their western counterparts. The scale is the same as in Figure 4-1, which shows
the ring geometry. The planet, which is shown at its full intensity and is saturated
here, extends beyond its geometric bounds by a full beam . The maps at 18.0 and
21.3 cm are not depicted since they no significant signal (above the noise) on the
ansae.
As with the first epoch 0.7 cm figure, the emission at 1.3 cm wavelength traces
well the ring geometry seen in Section 4-1, including the gap inside the C ring, which
clearly seen at all wavelengths at this epoch. Again, we see the ring brightness steadily
decline w ith increasing radius, a characteristic repeated a t all wavelengths. W ith the
higher resolution and more open geometry, we are now able to see the specific location,
namely the C ring, of the emission enhancement on the near side (which is now on
the top part of the rings). This is somewhat stronger on the west ansae than the
east, perhaps suggesting an asymmetry. Furthermore, we can see an enhancement at
the inner B ring, something noted with first epoch data,though its location is better
determined here. At 2.0 cm we see only a hint of an asym m etry in the C ring and
the inner B ring enhancement has diminished possibly due to decreased resolution.
At 3.6 cm the inner B ring enhancement is as strong as it was at 1.3 cm. Also, while
no asym m etry exists, the near (top) p art of the C ring is stronger near the edge of
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
67
Figure 4-10: Residual maps of the 1997 February epoch. The images are of the data
with the appropriate ai + a2/x removed from the disk. The images are labeled as
follows: top left 1.3 cm, top right 2.0 cm, bottom left, 3.6 cm bottom right, 6.1 cm.
R e p r o d u c e d w ith p e r m issio n o f th e co p y rig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
68
1.3 c m
LO
o
-
1.0
0.0
- 0 .5
1.0
0 .5
sin (latitud e)
“i—1—i—■
—i—1—s—1—|—1—i—'—i—1—i—1—r~
2 .0 cm
I
'
I
'
i
'
[
I
i
in
A
m
I
- 1 .0
- 0 .5
0 .0
0 .5
1.0
sin (latitud e)
Figure 4-11: Longitudinally averaged latitude structure of the Saturnian atmosphere
at 1.3 cm (top) and 2.0 cm (bottom) wavelength on 14 February 1997.
R e p r o d u c e d w ith p e r m issio n o f th e co p y rig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
69
3 .6 c m
o
m
-
- 0 .5
1.0
0.0
0 .5
1.0
sin(latitude)
6.1 c m
23
e—
1
o
<3
-
1.0
- 0 .5
0.0
0 .5
1.0
sin(latitude)
Figure 4-12: Longitudinally averaged latitude structure of the Saturnian atmosphere
at 3.6 cm (top) and 6.1 cm (bottom) wavelength on 7 February 1997.
R e p r o d u c e d w ith p e r m issio n o f th e co p y rig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
70
planet than at either 1.3 or 2.0 cm. At 6.1 cm the inner B ring seems diminished
perhaps due to poorer resolution, but near C ring is even more enhanced then 3.6 cm
We shall summerize the observations here. First, we generally do not see any
asymmetry at the second epoch. This is presumably due to the higher inclination of
this epoch. Second, the enhancement on the near part of the rings is present again
and can be associated with the C ring. Third, we again see the enhancement in the
B ring. In the second epoch, it is present at all wavelengths and is determined to be
in the inner B ring.
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
8
10
12
14
16
18
20
ARC SEC
Figure 4-13: Maps of the 1.3 cm wavelength ring structure on the west (top) and east
(bottom) side. Contours indicate temperatures of 5, 7, 9, 12, and 15 Kelvin. The
restoring beam is 0.60" x 0.32" FWHM, indicated by the ellipse in the upper right
corner. The thermal noise is 2.0 K.
R e p r o d u c e d w ith p e r m issio n o f th e co p y r ig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
8
10
12
14
16
18
20
ARC SEC
Figure 4-14: Maps of the 2.0 cm wavelength ring structure on the west (top) and east
(bottom) side. Contours indicate tem peratures of 5, 7, 9, 12, and 15 Kelvin. The
restoring beam is 0.90" x 0.46" FWHM, indicated by the ellipse in the upper right
corner. The thermal noise is 1.0 K.
R e p r o d u c e d w ith p e r m issio n o f th e co p y rig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
73
0
5
10
15
20
15
20
ARC SEC
0
5
10
ARC SEC
Figure 4-15: Maps of the 3.6 cm wavelength ring structure on the west (top) and east
(bottom) side. Contours indicate temperatures of 5, 7, 9, 12, and 15 Kelvin. The
restoring beam is 0.64" x 0.32" FWHM, indicated by the ellipse in the upper right
corner. The thermal noise is 1.6 K.
R e p r o d u c e d w ith p e r m issio n o f th e co p y r ig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
74
0
5
10
15
20
ARC SEC
Figure 4-16: Maps of the 6.1 cm wavelength ring structure on the west (top) and east
(bottom ) side. Contours indicate tem peratures of 5, 7. 9, 12, and 15 Kelvin. The
restoring beam is 1.07" x 0.52" FWHM, indicated by the ellipse in the upper right
corner. The therm al noise is 1.9 K.
R e p r o d u c e d w ith p e r m issio n o f th e co p y rig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
75
CHAPTER V
M O N TE CARLO SIM U LA TIO N S OF A MEE S C A T T E R IN G R IN G
5.1
Introduction
Due to limited resolution and sensitivity, early studies of Saturn at radio wave­
lengths drew few conclusions about the nature of the rings {e.g., Berge and Read 1968,
Berge and Muhleman 1973, Briggs 1974). The first m easurement which allowed sig­
nificant modeling (Cuzzi and Van Blerkom 1974, Cuzzi and Dent 1975) of the rings
came with the first high resolution radar observations (Goldstein and Morris 1973).
Cuzzi and Van Blerkom (1974) use a ring model which consists of a plane-parallel
layer of constant optical thickness typical of the B ring. They furthermore used a
Monte-Carlo approach and collected the results in very low (45°) resolution bins. All
the ring particles had the same phase function (i.e., identical particles). Cuzzi and
Dent (1975) modeled the A and B ring separately and fit that to the visibilities of
the radar data.
Multi-wavelength radar observations (Goldstein et al. 1977), led to new mod­
els. Cuzzi and Pollack (1978) used the doubling method for radiative transfer, but
included a particle size distribution and a consideration of non-sphericity in the parti­
cles. Later, Cuzzi, Pollack, and Summers (1980) reverted back to Monte Carlo simu­
lations, using a model similar to Cuzzi and van Blerkom (1974), but which included a
particle size distribution, and radial structure in the rings. Cuzzi, Pollack and Sum­
mers (1980) constructed higher resolution models (20° in azimuth) and considered
R e p r o d u c e d with p e r m issio n o f th e co p y r ig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
76
several wavelengths to compare with the data. However, the data were insufficient to
resolve the azimuthal brightness variations.
W ith the advent of high resolution VLA d a ta (de Pater and Dickel 1982, 1983; de
Pater 1985; Grossman, Muhleman and Berge 1989) more sophisticated models could
be made to compare to the higher quality data. Grossman (1990) presented the Mie
theory equations appropriate to a distribution of spherical particles consisting of dirt
and ice, although he neglected to include the consequences of multiple scattering. He
stopped short of making synthetic images of the rings based on these models once he
realized th a t Mie theory cannot possibly account for the observed orientation of the
polarized emission. He suggested a semi-empirical alternative to Mie scattering, but
made no quantitative comparisons.
In Section 5.2, we introduce the basics of radiative transfer and scattering.
This leads to a discussion of Mie theory. Next, in Section 5.3 we discuss the thermal
emission of the rings. In Section 5.4, we outline the procedure for our Monte Carlo
simulations.
In Section
5.5, we compare some basic results to those previously
published.
5.2
M icrow ave S c a tte rin g b y S a tu r n ’s R in g s
5.2.1
Introduction
Scattering of microwave radiation off particles of size similar to the incoming
radiation precludes the use of either Rayleigh scattering (small particle sizes) or ge­
ometric optics (large particle sizes). Mie theory treats correctly the intermediate
regime for the case of spherical particles. For a particle of radius, a, scattering light
at wavelength, A, the size param eter, x = 27ra/A, determines which of the three
regimes is relevant. Mie theory is most useful when x is approximately between 0.01
R e p r o d u c e d w ith p e r m issio n o f th e co p y rig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
77
and 100.
5.2.2
Scattering basics
Before pursuing a brief description of Mie theory, we will discuss some of the
basic equations of radiative transfer and scattering. Much of the theory can be found
in Chandrasekhar (I960), van de Hulst (1981), and Bohren and Huffman (1983).
Suppose we consider a parallel beam of radiation of net flux irF impinging upon a
plane-parallel layer of scatterers of optical depth r L (cf. below) with some specified
direction yu0, 9?o, (/i = cos d). We define fi to be the cosine of the angle relative
to outward normal and
is measured from a suitable defined x-axis (see Figure).
Then we can define the scattering and transm itting functions to be, respectively
(Chandrasekhar 1960),
(o . l)
T(n;n,(p’,iio,<Po),
The amount of energy, dE in a beam of frequency interval u.u -f- dv and confined to
a solid angle dQ, transported across a surface area element dui, at an angle d, over a
time interval dt can be expressed
dEu = I u cos(d) dA du dQ. dt,
where I u is defined as the intensity, and dA is the surface area element.
(5.2)
In the
plane-parallel case, we may express this
Iu =
(5.3)
so th at the reflected and transm itted intensities are given by (Chandrasekhar 1960)
I{0,+fJL,<p) =
I {ru —fj,, ip) =
fj., tp\ fj,o, <p0).
R e p r o d u c e d w ith p e r m issio n o f th e co p y rig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
78
Photons which impinge upon the layer will do one of two things when interacting
with m atter: (1) scatter in some direction (p., tp), or (2) get absorbed by the material.
We can quantify this by considering the cross-sectional areas of scattering (crsca) and
absorption (cra6s) of the material. Their sum is the total cross-section of extinction,
O'ex t — O sca -{- O a bs .
(O .O )
This is just a statement of the conservation of energy. It is convenient to introduce the
extinction, scattering and absorption efficiency factors, the ratio of a to the geometric
cross section (G)
Qext — Oext/Gi
(o.6a)
osca/ G ,
(5.6b)
—Oabs/G.
(o.6c)
Q sca =
Q abs
These are related in the same way,
Q e x t — Q sca
Q abs •
(^ * 0
If multiple scattering is involved, it is useful to define the ratio between scattering
and extinction efficiencies known as the single scattering albedo,
roo = ^
Wext
(5.8)
Another im portant param eter in the scattering process is the phase function ,
p(fj.) wherep = cos d. This simply shows the dependence
of the polar direction of
the outgoing radiation to that of the incoming radiation.Forexample, the
simplest
case, when p{p) is constant, indicates isotropic scattering.
We can parameterize the shape of the phase function with the asymmetry pa­
rameter (Bohren and Huffman 1983),
R e p r o d u c e d w ith p e r m issio n o f th e co p y rig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
79
i
9 = (p ) = ~ r
47r Jo
rl
/ 9 p(a*. <p) dfj. dip.
j —i
(5.9)
A perfect forward scatterer (i9 = 180°) has g — 1, a perfect backscatterer (i9 = 0°),
g = —1, an isotropic scatterer, g = 0. If the scattering is azimuthally symmetric then
1 r1
g = - J p(/i) IMdp.
5.2.3
(5.10)
Optical depth and the absorption coefficient
5.2.3.1
Definitions
In this section, we consider the form of the optical depth, r . The mass absorp­
tion coefficient, a„, representing the loss of intensity in a beam as it travels a distance
ds may be used to define the optical depth, rabS
t(v,s)
=
f a u(s' 1i/) ds'
(5-11)
where s0 is an arbitrary starting point. We can relate a u to the absorption cross
section of aabs (a) by
roc
a(a, y) = / aabs(a) n(a’) da .
Jo
(5.12)
The size distribution, n(a), is defined as n(a ) = dN(a)/da, where iV(a) is the cumu­
lative distribution of all particles of radius less than of equal to a existing within a
column of unit cross-sectional area perpendicular to the ring plane. The equation for
r(i/, s) (5.11) becomes,
ro o
J
rs
da1 / crabs(a') n(a') ds'.
0
Jso
(5.13)
If we express Eq. 5.13 in terms of Q and include analogous equations for the extinction
and scattering counterparts, we have
R e p r o d u c e d with p e r m issio n o f th e co p y r ig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
80
rOQ
rext(a ,v,s) =
J
rS
da' /
/
0
G QCxt{o!.u') n{a') ds'.
(5.14a)
G Qabsia',^') n(a') ds'.
(5.14b)
da'
G QACa(a', r/) n(a') ds'.
J so
(5.14c)
J sq
rs
ro o
Tobs{a,u,s) =
da'
J0
J
sq
roo
TSCa{a,v,s) =
5.2.3.2
J0
rs
The form of the optical depth
For a slab of spherical particles whose sizes range from amin to amax at wave­
length A of vertical thickness H
amax, the extinction optical depth becomes:
r tl
r& m ax
rext(a, A) = / ds'
Jo
J
where
Q e x t ( a ,
A) =
Q
e x t{ a
ira Qext{a', A') n(a)da
(5.15)
O -m in
, u) from above. We assume the particles are distributed by
a power law model of the form
n(a) = n(aQ)[aQ/a]q: if amin < a < amax,
(5.16)
where q0 is a reference radius. For convenience we define
Mo = n (a 0)[a0]9.
(5-17)
By integrating over the vertical thickness of the rings we have
fO -m a x
rext(a: A) =
ttM q
/
n
a 9 Qeit(a,A).
(5.18)
J Q -m in
Thus, for the absorption optical depth we can write,
fQ -m a x
Tabs ( a ,
A) =
7tA/o
_
/
a
**^mtn
~
q
Q a b s ( a ,
A).
R e p r o d u c e d with p e r m issio n o f th e co p y r ig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
(5.19)
81
If we integrate over s, we can also solve for Mo in term s of Qe(a, A),r0, and the layer
thickness, H,
H
M
°
=
*■
c r
5.2.3.3
“
2 -
x
”
)
' '
( 5
' 2 0 )
The form of the absorption coefficient
The value of Qext(a, A) depends on the complex index of refraction:
m = mr —irrii
(5-21)
For ice, we take the real part of the index of refraction, m r , to be 1.78 (see Grossman
[1990] and references within). The imaginary part is given by,
f-(o.2 2)
4?rmi
= —j
—,
where a \ is the absorption coefficient. Cuzzi et al. (1980) use a simple form for
imaginary index of refraction, m* = 7.5 x 10_5/A, which is based on theoretical curve
for ice at T = 100 K (W hattley and Labbe 1969). However, in Grossman’s thesis he
used the laboratory results of Mishima et al. (1983):
An
Q f f 30 ice =
where u iswaveumber
Y
etlcy°/kT
Y
C^ ' k T -
I)1
Vo +
’
(
}
in cm-1. The best fit th a t Mishima etal.(1983)obtained for
the coefficients were uQ = 233cm-1, A 0 = 1.188 x 105 cm-1 K, and
B = 1.11 x 10-6
cm.Pl (Note typo in Grossman [1990] and Mishima [1983 et a/.]).
One can relate the complex index of refraction with the complex dielectric con­
stant:
m 2 = e' — ie",
R e p r o d u c e d w ith p e r m issio n o f th e co p y rig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
(5-24)
82
where e' = m T2 —m 2 and e" = 2m Trrii . We can then parameterize the d irt fraction,
F , in ring particles (as in Grossman [1990]):
- rrii2] (1 - F ) + 3.0F
=
[m r2
=
[2mrmi] (1 - F) + 0.03F.
4 ./d ir t
(5.25)
(5.26)
From the definitions of e' and e" below Equation 5.24 we have
m i = J ~ (Ve'2 + e"2 — e ').
(5.27a)
(Ve'2 4- e"2 4- e'J.
(5.27b)
mr =
5.2.3.4
Determining the path length of the photon
Now we turn our attention to determining the path length, s, of the photon
before it encounters another ring particle. Looking back at Equation 5.15 we now
integrate the distance from 0 to s,
rs
r&max
_
re(a .X )= / ds'
7Ta Qefj(a, A) n(a)da.
J0
JQ.min
(5.28)
We note that the photon path length and the size of the particle encountered are
coupled. Therefore, we must consider three regimes:
0
[0, d-rnin 4“ &old\
4“ & old i ^ m a x 4- aold]
dm ax
(5-29)
\ d m a x + d,0ld , s ] j
where d0id is radius of the previous particle. Let the second integrand of Equation
5.28 be defined as I. Then Eq. 5.28 becomes,
rs'
/ ■ U m a i t s 'l + a o w
re(a, A) = 0 + /
"^rO’old.
ds'
^
rs
I dd +
r a max
ds
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
I dd.
(5.30)
83
Notice, the first term is zero since the photon m ust travel at least amin + a0id. For
the third term the integration over a is simply between amax to amiTl because amax is
no longer a function of s'. Recalling Eq. 5.17 and calculating the first integral in the
second term,
T ’-(CL A.1
'j— =
TryVO
f&max ( s ') - r a 0j<i
^&min
ds
rs‘
rs
I da -+- s — {am -+- amax) /
^&min.
5.2.4
I da .
(5.31)
Qmax~b~Q‘old
Mie scattering
We turn our attention to the form of Qext. W ith simple reasoning one might
conclude th at a typical particle may absorb and scatter only the amount of light
from a beam equal to its cross-sectional area (at least for large x ). However, there is
an equal amount of light which is diffracted (if even a tiny amount) in an identical
way th at a hole of area G would. In this case the amount of light removed would
correspond to a cross section twice the geometric cross section.
rrex£ = 2G,
(5.32)
so th at Qext = 2. This is known as the extinction ■paradox and is further discussed in
van de Hulst (1981, Chapter 8) and Bohren and Huffman (1983, Chapter 4).
For particles much smaller than the wavelength (x
1,Rayleigh scattering) we
typically have Qexl <C 1 (van de Hulst 1981, C hpater 6).
Mie scattering covers the midgroud between these two extremes. Mie theory is
discussed in several books (e.g. van de Hulst 1981, Borhen and Huffman 1983), so
we will cover only the main points of the subject. Solving the Maxwell equations for
the scattering of plane waves on a homogeneous sphere, Mie (1908) calculated infinite
series sums for Q
abs, Q s c a
and g and the complex scattering am plitude functions S\{[x)
R e p r o d u c e d w ith p e r m issio n o f th e co p y r ig h t o w n e r. F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
84
and
(van de Hulst 1981) ,
2 A
Q abs =
(5.33a)
■*■) R e ( ° n + b n ) .
~ ~ 2
n=l
Q sca — ~ j E ( | a n |2 +
X
N
9 = x 2Q
sea
E
n=i
n{n + 2)
„
n + 1 Re(anan+1 +
S ifa) =
S o (a O =
E
71= 1 n \n
E
(5.33b)
| bn |2 ) ,
71=1
T^
„ 2^ T
t^ i n (n + 1)
(2n + l)
bn bn + l ) ^
+ ^
Re(an6n)
(5.33c)
+ bn r n (fi)]
(5.33d)
[a n T „ ( ^ ) + b n TTn (fJ,)}
(5.33e)
ia n^n ( m )
where an and bn are the Mie coefficients which depend on x and the complex index
of refraction, m = m r —irrti. The angular eigenfunctions of / i are denoted by 7vn and
rn. To solve these equations we use the algorithms developed by Wiscombe (1980,
1997), which are currently the most computationally efficient available.
Figure 5-1 shows the extinction of a single ice sphere. We see th at Qext grossly
varies with a on semi-regular period with finer structure superposed. We also note
th at Qext asymptotically approaches 2 for large x, and reduces to very small values
of x , consistent with those extremes.
An often used approximation to the single scattering phase function is the
Henyey-Greenstein (HG) function :
pM
1
1- g 2
= 47T (1 + g2 —2g /x)3/2
(5.34)
The g in this function is the asymmetry parameter described above. However, the
HG function gives zero probability of pure backscattering, while Mie theory predicts
an enhancement near 180° . Furthermore, Mie theory predicts a narrower and conse­
quently stronger forward part of the phase function than the HG function.
R e p r o d u c e d w ith p e r m issio n o f th e co p y rig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
85
X
93
Or
10 =
10 °
Figure 5-1: The extinction efficiency as a function of the size parameter, x = 'lira/A,
for an ice sphere with m = 1.78 —i 0.00001.
Since we wish to explore the applicability of Mie scattering to our scenario, we
shall refrain from using this approximation.
5.3
T h e rm a l E m issio n fro m S a t u r n ’s R ings
5.3.1
Introduction
The ring particles also emit radiation due to their intrinsic (physical) temper­
ature, composition, and size. This is parameterized in the absorption coefficient. In
a layer of ring particles, the effective optical depth, ring thickness, and particle size
distribution are crucial in determining what therm al emission is observed. Thus, we
expect therm al emission to also be a function of viewing geometry and wavelength.
R e p r o d u c e d w ith p e r m issio n o f th e co p y r ig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
86
5.3.2
Thermal Properties
For an optically thick blackbody, the intensity of therm al emission is given by
the Planck function
B " (T) = exPS
r ) - r
( 5
-3 5 )
For the radio regime hi/ <C k T , and Equation 5.35 becomes the Rayleigh-Jeans law:
91/2
B ™ {T) = —
~ kBT,
&
(5.36)
where k B is Boltzm ann’s constant. Thus we expect to observe more therm al emission
at higher frequencies.
In our scenario we do not have blackbody radiation. Instead, we consider par­
ticles with absorption coefficient a v. The optical depth of some scattering layer of
height H is given by Equation 5.11:
Tabs{^: s) = [ ctl / ds' = aH .
JSQ
(5.37)
Kirchhoff’s law relates the emission coefficient, j , to the Planck function and the
absorption coefficient,
j = aB„(T).
(5.38)
Combining Equations 5.37 and 5.38 we have for emission from a column of ring
material
H j = rabsB{T).
(5.39)
Suppose P photons are em itted isotropically over 4 7 t ( ^ ) 2 square degrees. Then the
brightness tem perature per photon is given by
R e p r o d u c e d w ith p e r m issio n o f th e co p y rig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
87
Tb = J f * 1* B
ttP/129600
where T r
(5.40)
v
'
isthe physical tem perature of the rings. Fortherm al
emissionwe also
consider the effects of geometry.
5.3.3
Geometry effects
To show the effects of geometry, one can consider a layer of isotropic emitters
of normal optical depth, r 0. Let the layer lie in the x-y plane and let the layer range
from z equal zero to unity. Define 6th to be the angle between the 4-2-axis and the
em itting thermal radiation. Including the foreshortening of the rings at seen from an
observer viewing at angle #, the intensity, 1(d), is proportional to
1(6) oc t q sin(#)e-Te//,
where r e/ /
(5-41)
isthe effective optical depth of the outgoing photon.This is given by,
T'/r = t0C +£ -/ 2 ); 0 < 90”.
(5.42)
If we then let z' equal (z 4- H / 2 ) / H and integrate Equation 5.41 over the height of
the layer, we get
rz' = 1
1(6) oc rQ [
sin(#) e c°*"a>2 d z'.
Jz'=o
(5.43)
This leads to,
1(6) oc cos# sin# e
.
(5.44)
Hence, we see that a factor of sin # cos # must be considered to obtain the brightness
tem perture from the flux.
R e p r o d u c e d w ith p e r m issio n o f th e co p y r ig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
88
5.4
M onte Carlo Sim ulations
5.4.1
Underlying assumptions
In this model we will assume the rings to be a uniform, plane-parallel layer of
vertical thickness H of ring particles with a particle size distribution described by a
power law with both a minimum and maximum size cutoff. Furtherm ore, we assume
these particles are spheres composed of pure ice or an ice-silicate m ixture and perform
radiative transfer according to Mie theory.
The source of all microwave photons impinging upon the rings is thermal ra­
diation coming solely from Saturn and the cosmic background. T he rings are also
radiating thermally as described in Section 5.3 characterized by th eir physical tem­
perature and radius.
5.4.2
5.4.2.1
Basic equations
Implementation of equations
We now have four equations (5.18, 5.19, 5.20, & 5.31). To code them into
FORTRAN, we must change integrals into sums. The integral,
(5.45)
■mtn
is seen in all four equations. As a single integral (as in Eq. 5.18, 5.19, & 5.20) this is
simply solved:
sum
J
Q(a, A) cq q Sa{
(5.46)
Given the particle size distribution, we use logarithmically spaced da-i to better sample
the behavior of the ensemble
5a = a 5(log a)
R e p r o d u c e d w ith p e r m issio n o f th e co p y rig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
(5.47)
89
Equation 5.31 is somewhat more complicated, but can also be written in terms
of sums:
,
j= a m a x (s ')-r a ald
,,
=
.
i= s '
E,
„
Z
3 — Q’m i ‘n.~T~Go[d
Qext{of,X) a!?~q Sa'i
i— ^ m ;n
(5.48)
{fl'old ~b ^ m o i ) ]
“F
Q e x t{ P - > A )
Z
6a^.
*"= a m i n .
Now we are ready to consider the random processes in the scattering of photons
in our slab of material. There are five events which require a “dice roll.” First, as
mentioned above, we need to determine how far the photon travels before striking
another photon (or leaves the ring altogether). Second, we need to determine the size
of the particle. Third, does the photon get absorbed? If it does not, we then must
consider what direction the photon goes next and determine the polar and azimuthal
angles.
Using the uniform random deviate subroutine of Press et al. (1990), denoted by
R[symboi], we have for rext
Text
= —ln(l -
(5.49)
H r ) .
Incorporating this into Eq. 5.48 and solving for s , we obtain
.
/,
~ ' nf c
o
\
s'
E
a m o x ( s , )+ O o /< i
E
-
M
Q,Tnin~^~Q’ot<i
„
QM ( a ! , \ ) a ' 2- ’ 6a'
&min
C lm ax
&m a x )
s =
« __
Qext(a',A) a’ q 6a .
(5.50)
*m ax
£
Z
Q e x t(a
, A) a2~i 6a'
Note we have dropped the subscripts on a and s.
The size of the particle can be determined by considering not only the particle
size distribution, but also the extinction cross-sectional area, aext(a). Therefore, the
R e p r o d u c e d w ith p e r m issio n o f th e co p y rig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
90
number of particles that exist of which the photon can scatter is
f
o-ext( a ) n { a ! ) d a '
JAmin
If we normalize this function over all particle sizes, then we can write for the random
number R a,
(5.51)
a ex t ( a ' ) n ( a ' ) d a '
S a Z an
We note now th a t the extinction cross section is sim ply the geometric cross section
times the extinction efficiency,
t ( ^ ) — G ( n ) Q e i t ( c i, A )
( 5 .5 2 )
Putting this w ith Eq. 5.51 we obtain,
Ra =
fa
Q e x t i a ' . A ) a ' 2 -'? d a '
m,7 '
c r (’
K
A)
;----------
(5.53
W ritten as sums and recalling (5.46),
R * h sv m
X)
=
Q e x t ( a ' , X ) a n ~ q5 a '
(5.54)
Since I qsutti is a constant for a given particle size distribution, we need to only deter­
mine what particle size a satisfies Eq. 5.54.
The condition of photon absorption can be simply expressed:
R m < (1 —tz70).
(5.55)
If instead the photon is not absorbed, we then determ ine what direction it will
scatter. The azimuthally symmetric phase function p(/x) intergrated over all [i is
normalized:
1 ri
2
j
P ( v ) dV = 1>
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
(5 -5 6 )
91
Therefore the R$ can be expressed in an analogous fashion to R a (Eq. 5.53)
Rfl ~ I l
j
I
-
i
dfJ'''
^5'57^
In terms of sums this is
4
/ M
% = £ pW ) W / £ p(A )
/
1=1
where
= —1 and
(5-58)
1=1
= 1. To getbetter coverage of P. each term in our sums
are
spaced equally in i? and not fj. so that
Slfj.'i) = —sin(/d)6,d.
(5.59)
Wiscombe’s Mie scattering algorithm (Section 5.2.4) outputs p(i9) for each a, (for a
given \ , m r .rrii). A table of p(d) must be generated for each a considered. We can
then find the cos d which satisfies Eq. 5.58.
At last the azimuthal direction is determined randomly.
ip = 27rRv .
5.4.3
(5.60)
Scattered Radiation
We must now employ the Monte Carlo statistics to the above equations. We
first need to collect statistics about the scattering distribution upon the ring. To
accomplish this we need to generate N photons from many incoming angles relative
to the ring. However, we can save much time by considering only the photons which
actually reach the Earth.
Essentially we reverse the arrow of time
and consider
photons approaching the rings at an angle, i, the inclination of ring plane to the
E arth ’s line-of-sight.
R e p r o d u c e d w ith p e r m issio n o f th e co p y rig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
92
To Earth
-* x
Figure 5-2: The geometry for the Monte Carlo simulations. The incoming angles for
a photon are designated by 9X and <px. The outgoing direction is designated by the
inclination i.
We gather the necessary statistics by generating N photons with an incoming
angle of i. The geometry of this is shown in Figure 5-2. The photons scatter until
they are either absorbed or leave the ring. In the latter case we determine their
outgoing polar and azimuthal directions. The outgoing polar angle, 0 X, is defined
from the outward normal above the ring plane. The azim uthal angle, <px, is defined
from the axis in the ring plane which points back to the E arth. The 0 Xand ipx of each
surviving photon are collected in bins of some specified angular size. Furthermore,
photons scattered by differing degrees (zero, single, multiple) are stored in separate
tables. These tables are numerical representations of the scattering and transmission
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
93
functions seen in Equations 5.1 and 5.4.
After gathering these statistics, we apply them by considering the reverse pro­
cess. We can think of the all the incoming angles as output angle and vice versa
(see Chandrasekar Chap IV, Sec. 31 and Chap VII, Sec. 52). We therefore have
generated a table which tells the probability of scattering an incoming photon from
direction 0 : and ipi into the direction specified by i and towards the E arth. To get
the relevant input angles, consider the angular size of Saturn as seen from the rings.
Each location on Saturn can be thought of as a source w ith an incoming angle @i
and (fli. The 4tt steradians around the rings is broken up into an n by n grid. Each
grid cell’s center is either off the planet and contributes nothing or is on the planet
and we must consider its contribution to the scattering. Each incoming photon from
the planet makes an incoming polar angle relative to the ring plane and an azimuthal
angle relative to the line between the E arth and sub-E arth point on Saturn. These
angles are precisely the angles defined as ©i and ipi and so it becomes ju st a m atter
of looking up the appropiate ( 0 i, <pi) in the table. This contribution is weighted
by two factors: (1) the number of the N photons which were used to generate the
table and (2) the temperature of the planet. For the planet tem perature we use the
observed limb darkening laws described in the fits in Chapters 3 and 4.
For these calculations we assume a featureless planet, so that we can assume an
East-West as well as North-South symmetry. We then sum all the statistics and make
a second table from these sum m ations which consists of photon number frequency vs.
ring radius and ring azimuth.
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
94
5.4.4
Thermal photons
The thermal photon statistics are generated in a slightly different way. Since
thermal photons come from inside the ring particle, there is no external source as in
the scattered photon case. We represent the therm al emission by representing the
slab of scatterers by 1 /N sub-layers (we could have also chosen N random vertical
heights within the rings). We launch a photon with random fj, and tp directions. As
before the photon scatters until it either absorbed or leaves the ring and its outgoing
direction recorded. This is used to generate another table of photon number frequency
vs. polar and azimuthal angles.
We now construct a second table by considering the photons we actually ob­
served: those with outgoing angles relative to the plane, i. We then have another
table with photon number frequency vs. ring radius and ring azimuth.
5.4.5
Cosmic Background
We have one final source of photons, the 2.7 K of the cosmic background radation
(CBR). Since you expect these source photons to enter the rings isotropically, one
might expect that these exit isotropically. This is indeed the case. There is one
caveat, the disk of Saturn blocks a fraction of the CBR. However, if the disk is
treated as a combination of 2.7 K uniform disk and the ‘normal’ disk with a 2.7 K
uniform disk subtracted out, then we see that isotropic scatter has no net effect on
the brightness temperatures of the rings. As a consequence, we must subtract 2.7 K
from the physical brightness tem perature of the disk in order to obtain the true flux
coming from the planet. (Note, however, th a t since an interferometer is only sensitive
to differences in brightness, this subtraction is already done in the maps we made.
This 2.7 K must be sutracted from the ai component in Tables 3-1, 3-2 and 4-2
R e p r o d u c e d w ith p e r m issio n o f th e co p y rig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
95
5.4.6
Full model
The final model is the sum of the scattered component (described above) and
the therm al one. To obtain the tem perature from therm al photons consider th at
isotropic scattering then on average we will have for each output bin is N/Qcmtbin
photons. This should correspond to a therm al brightness tem perature (Tt/l) of raTR,
where TR is the physical tem perature of the rings. For the therm al photons, we must
also consider the foreshortening of the ring as seen by th e observer as well of the
latitudinal dependence of the output bins. If we consider nom inal output bins of M x
by My square degree, then the number of bins in 4tt steradians is given by 4^ r80/^' -We have for the brightness tem perature
T
t T
th
“
rMx
47^v '(180/7r)2
My N sin 9 cos 9 '
^
^
Now the scattered and thermal photons are added together, and we can con­
struct a table of tem peratures as a function of the ring radial distance r R and azimuth
(pR. However, to produce a model to compare to the data, we need the change our cur­
rent coordinates into ones relative to Saturnian North and West. All that is required
here is the ring inclination:
r R = y /x * + (y„/sin i)2
(5.62a)
ipR = arctan[(yn/sin i) /x w],
(5.62b)
where x w, yn is the distance west and north, respectively, of Saturnian center. Now
this d ata may be imported into AIPS and compared to the d a ta directly.
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
96
5.5
C om parisons to previous results
Before we apply this scheme and compare it to the data, we test the general
behavior over a range of conditions with either well-known results or results which
may be compared to previous works.
The simplest case to compare to is the isotropic case. As previously mentioned
in Section 5.2.2, the phase function here is a constant. We can consider the equations
in Section 5.4.2, we simulate the distribution of scattered photons. Figure 5-3 shows
the number distribution of m ultiply scattered photons as a function of ©out,
incident on the rings at 5.5°. As expected, the distribution is azimuthally symmetric.
The variations in Qaut are a consequence of smaller solid angle (at 0° and 180°)
and a low probability of emerging at small angles relative to the ring plane (0 =
90°). We also made quantitative comparisons with isotropic computations reported
in Chandrasekhar (1960) which showed good agreement.
Next, we look at the therm al component. The thermal emission from a layer of
particles should also be isotropic, even if the ring particles have non-constant phase
functions. Furthermore, from the Rayleigh-Jeans law (Eq.
5.36 the therm al flux
from the rings should increase when observed at shorter wavelengths. Cuzzi, Pollack,
and Summers (1980) do a Mie scattering calculation of thermal radiation and plot
brightness tem perature (Tg) vs. wavelength for the B ring considering q= 3 and amax
= 100 cm and 1000 cm. In Figure 5-4, we compare his results to Mie scattering
calculations under the same conditions. For case B the agreement is good. For case
A, which has a greater amaxj there is a mild discrepancy. This likely arises from the
approximation used by Cuzzi et al. for the forward lobe in the case of large values of
x.
We can also compare to Grossman (1990) who used Mie scattering to calculate
R e p r o d u c e d w ith p e r m issio n o f th e co p y rig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
97
0
Iso. P h o t o n s
( n = 2 - r ,
50
150
to o
1.3
c m . A, (3 = 5
200
250
.2 )
30 0
350
A z im u th a l A n g le
Figure 5-3: Simulation results for an isotropic ring. Two million photons were sim­
ulated to impinge the ring plane with an incident angle of 5.5 “from the ring plane.
Results are collected in l°by l°bins and plotted here between 0 (white) and 40 (black).
R e p r o d u c e d w ith p e r m issio n o f th e co p y rig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
98
C uzzi e t al. (1 9 8 0 ): Fig. 5
Cuzzi A
C uzzi B
o
in
•
Mie A
• - —Mie B
o
00
r0a3
o
CO
o
CU
w
w
w
ww
\\
10°
io :
Wavelength (cm)
Figure 5-4: Comparison plot of the therm al dependence on wavelength. We use a
particle size distribution power law number of q=3 for amin — 1.0 cm. Plot (A) has
a-max = 1000 cm and plot (B) has amax = 100 cm. Lines w ithout symbols are from
Cuzzi, Pollack, and Summers (1980), while the line with symbols are Mie calculations
(From Fig. 5 of Cuzzi, Pollack, and Summers (1980).
R e p r o d u c e d w ith p e r m issio n o f th e co p y r ig h t o w n e r. F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
99
o
Grossman (1990): Fig. 5.6
A
1-100 cm. q
B: 10-100 cm, q
C:
1-100 cm, q
D: 1-100 cm . q
a
o-—
a
Mie
Mie
Mie
Mie
= 2.6,
= 2.6,
=» 2.6,
= 3.0,
F=
F=
F«
F =
0.00
0.00
0.02
0.05
A
B
C
D
o
sfei
o
10"
’
10°
101
Wavelength (cm)
Figure 5-5: A second comparison plot of the therm al dependence on wavelength. We
consider four combinations of q, amtn, amai, and F stated in the figure. Lines without
symbols are from Grossman (1990, Fig. 5.6), while the line with symbols are this
work’s Mie calculations.
T q vs. wavelength for a variety of combinations of q, amjTl, amax, and dust fraction.
F , on combined A and B ring model set with physical tem perarture 68 I< with r =
0.8. These results are compared to our Mie calculations with the same param eters.
The results are depicted in Figure 5-5. In general, our results show good results at
all wavelengths when F = 0.0. At non-zero F , we th a t our values are somewhat too
high, though quite similar in shape to th a t of Grossman (1990). Exact agreement
should be expected here, as the same equations are used, but Grossman leaves some
uncertainty as to the values used for the viewing angle and the physical tem perature
of the rings. This could be indicative of a m isrepresentation of F on his part.
We can also test the rings modeling of Cuzzi, Pollack, and Summers (1980).
R e p r o d u c e d w ith p e r m issio n o f th e co p y rig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
100
n u z z i e t a l. (1 9 8 0 ): T a b le II
Mie Reflected
Mie Transmitted
Mie All
C u z z i R e f le c te d
C uzzi T ra n s m itte d
C u z z i All
Q
E-
50
100
150
A zim uthal A ngle (deg.)
Figure 5-6: Comparison plot of the scattering dependence at 3.71 cm. We use the
data from Cuzzi, Pollack, and Summers (1980), Table II, and are represented by line
without symbols, while the lines with symbols are calculations from this work. Errors
from their calculations are on the order of 10%, while our calculations contain less
than 1% uncertainty.
They use a Mie phase function which is truncated at forward lobe for ease of calcu­
lation. We compare our the diffuse transm itted, reflected and total transmited light
to theirs in 5-6. Notice that our values are generally highter then theirs. This is to
be expected since our Mie phase function is not truncated and therefore should have
more forward scattering. Cuzzi, Pollack, and Summers (1980) cite their error to be
~ 10%, while our calculations are good to < 1%.
5.6
C hecking th e phase function
We can also examine the behaviour of the phase function. In this section we’ll
examine two cases: isotropic scattering and a typical Mie scattering case. It will be
R e p r o d u c e d w ith p e r m issio n o f th e co p y rig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
101
instructive to show the phase function in two different ways. First, by just showing
the phase function dependence on phase angle, we can examine the general shape.
Second, we can examine the cumulative sum o f pu(cos(9)) d cos(0), which tells us
probability of scattering between two angles. Figure 5-7 show both types of plots for
isotropic scattering. Notice th at the phase function behaves as expected, a constant
Pn and constantly rising cumulative probability. If we look instead at a typical Mie
scattering case (q = 3.0,
amin = 1 cm, amax = 500 cm) in Figures 5-8 and 5-9,
we notice that the phase function is highly forward scattered and contains a fairly
significant backscattering. The extreme forward scattering peak, corresponds to the
diffraction lobe. In the case of 0.7 cm, this corresponds to nearly half of all scatters,
as expected from Mie theory at large x = 2iva[X. At 6.1 cm the phase function is
not as forward scattered, as one might expect. Additional computations of this sort
match exactly those of Grossman Figure 5.5 (properly normalized).
5.7
Looking at th e depend en cies o f M ie scatterin g on o p tica l depth
Another test of the code involves looking a t dependences of Mie scattering on
optical depth. Since we expect our brightness tem peratures to be symmetric, we shall
only plot azimuths between 0 and 180° . We use the second epoch Saturn at 3.6 cm as
our source (ai = 151.8 K; a2 = 16.8 K). Figure 5-10 shows the radial distance influence
on the brightness of the rings. As we naturally expect, the brightness temperature
falls off as we recede from the planet.
Figure 5-11 shows a comparison of the scattered light from an optically thin ring
to th at of an optically thick ring placed at the same distance from Saturn. Notice
th at the optically thin ring suddenly increases in 7# as we see the ring in front of the
planet (~ 54° at 1.24 R sat)• This is a consequence of the heavy forward lobe of the
R e p r o d u c e d w ith p e r m issio n o f th e co p y r ig h t o w n e r. F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
102
CO
°<->
2
O \r
100
150
50
Iso tro p ic
S3
o
o
co
All W a v e l e n g t h s
CO
co
£
3
o
c\j
- 0 .5
0.0
0 .5
1.0
cos(0)
Figure 5-7: Isotropic phase function. Top Basic plot showing phasefunction vs. 8.
Bottom Cumulative sum of pn(cos0) d(cos8).
R e p r o d u c e d w ith p e r m issio n o f th e co p y rig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
103
O
\ (c m )
0 .7
a o n [1 .0 , 5 0 0 c m ] ; q = 3.0
Tr = 8 5 K; F = 0 .0 0 2 5
a.
o
o
150
50
100
9 (d e g )
Mie
K = 0 .7 c m
co
3
p
5
o
150
100
50
9
Figure 5-8: Mie phase function for 0.7 cm. Top Basic plot showing phasefunction vs.
9. Bottom Cumulative sum of p u (cos 9) d(cos9).
R e p r o d u c e d w ith p e r m issio n o f th e co p y rig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
104
O
X (cm )
6.1
a o n [1 .0 , 5 0 0 c m ]; q = 3 .0
T b = 8 5 K; F = 0 .0 0 2 5
cC
o
o
150
50
100
B (deg)
Mie
6.1 c m
o
o
C\2
o
o'
150
100
50
0
6
Figure 5-9: Mie phase function for 6.1 cm. Top Basic plot showing phasefunction vs.
6. Bottom Cumulative sum of p u (cos0) d{c.osQ).
R e p r o d u c e d w ith p e r m issio n o f th e co p y rig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
105
CO
S c a tte r e d lig h t
t = 0.13
R = 1.24 to 1 .5 2 5 Rs*i
co
o
C\2
o
50
10 0
150
A z i m u t h a l A n g le
Figure 5-10: The scattering brightness tem perature at various Saturn ring radii.
phase function. The optically thick ring shows a more fluid increase at that point.
Also, notice the small surge in backstatter. This is indicative of the gegenschein.
The importance of multiple scattering is addressed next. Figure 5-12 shows the
degrees of scattering (single, multiple) for the optically thin and thick rings. Notice
how much more important (even dom inant) multiple scattering is in the optically
thick case, which is expected since the m ean free path in a optically thick ring is
much shorter than optically thin ring coupled with the very high albedo of ice.
Finally, we show the contribution of diffuse reflected and transm itted light.
Diffuse reflection is defined to be scattered radiation which enters and exits on the
same side of the ring, while diffuse transm ission is that radiation which emerges on the
other side. In Figure 5-13, we plot the contribution of diffuse reflected and transm itted
light for both the optically thin and thick cases. Again as we might expect, the
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
106
o
r = 0.13
= 1.0
T
S c a tte re d , lig h t
0J
3
i = 4.91 d eg .
r
R = 1.24 Rs*t
\
= 3.6 c m
o
o
50
100
150
A z im u th a l A n g le
Figure 5-11: Scattering brightness tem perature in the optical thin and thick cases.
transm itted light has a greater (and dominant) contribution in the optically thin
case. However, because of the high albedo of the ring particles, transm itted radiation
still has a significant contribution in the optically thick rings.
R e p r o d u c e d w ith p e r m issio n o f th e co p y rig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
107
o
CD
S in g ly S c a tte r e d
M u ltip ly S c a tte r e d
t
= 0 .1 3
i = 4.91 d e g .
R = 1.24 Rs.t
\ = 3 .6 c m
on
o
50
150
100
A z im u th a l A n g le
CO
S in g ly S c a tte r e d
M u ltip ly S c a tte r e d
O
1.0
R = 1.24 Rs.t
K - 3.6 c m
a
C\2
O
o
50
100
150
A z im u th a l A n g le
Figure 5-12: Singly and multiply scattered light compared in the optically thin (top)
and optically thick (bottom) cases.
R e p r o d u c e d w ith p e r m issio n o f th e co p y r ig h t o w n e r. F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
108
i
t
•i •• t ••
------ D iffuse R e f le c te d
------ D iffuse T r a n s m itte d
(K)
------
r = 0 .1 3
Temperature
i = 4.91 d e g .
R = 1.24 Rs.t
-
\ = 3 .6 c m
Brightness
-
0
, . . . . 1.
50
100
150
A z i m u t h a l A n g le
co
Brightness
Temperature
(K)
D iffuse R e f le c te d
D iffuse T r a n s m itte d
T
=
1.0
/ = 4.91 d e g .
o
\
- 3.6 c m
C\2
o
o
50
100
1 50
A z i m u t h a l A n g le
Figure 5-13: A comparison between the diffuse transm itted and reflected light for the
optically thin (top) and optically thick (bottom) cases.
R e p r o d u c e d w ith p e r m issio n o f th e co p y rig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
109
CH A PTER VI
D IS C U S S IO N
6.1
I n tr o d u c tio n
Now that we have established the equations in Chapter 5, we now wish to use
the Monte Carlo m ethod to simulate the ring maps and test the two extreme cases:
isotropic scattering and Mie theory. We then want to comment on the effectiveness
of each and model an alternative case th a t involves a linear combination of both.
The chapter is organized as follows. In Section 6.2, we look at the physical
properties of the rings and Saturn. In Section 6.3 we do three simulations with the
Monte Carlo code. In Section 6.4 we discuss ramifications of the results and direct
the reader to different methods.
6.2
T h e rin g physical p a ra m e te r s as seen in th e lite r a tu r e
6.2.1
Physical dimensions of Saturn
We shall now discuss the many param eters which determine the physical propeties
of both the disk and the rings. We shall then follow this with a discussion of what
values for each param eter is appropriate for our simulations.
The first param eter th at we should established is the shape, dimensions and
boundaries of the planet and the ring. These have been fairly well known for some
time (Alexander 1962), but were greatly refined with the Voyager missions. A table
(Table 6-3) of ring name, and inner and outer radius was given in the Introduction
R e p r o d u c e d w ith p e r m issio n o f th e co p y r ig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
110
Observed Ring Temperatures
Ring
Illuminated Side (K)
Unilluminated Side (K)
C
85 ± 1
85
B
68 ± 1
50
10 to 75
CD
85 ± 2
A
69 ± 1
50 to 60
Table 6-1: Observed ring temperatures. The numbers in italics are derived from
Voyager 1 (Hanel et al. 1981) those in plain text are from Voyager 2 (Hanel et al. 1982).
to this thesis. The planet shape is well known and is determined by the gravitational
harmonic coefficients measured by Voyager (Campbell and Anderson 1990).
6.2.2
The physical tem perature of the rings
The next param eter needs to be more carefully considered. The physical tem­
perature of the rings is im portant not only for therm al processes also but it plays
a role in determining the albedo. Both Voyager 1 and 2 measured the temperature
of the illuminated side of Saturn, while Voyager 1 measured the tem perature of the
unilluminated side using infrared instruments on both spacecraft (Hanel et al. 1981,
1982). The results are compiled in Table 6-1.
6.2.3
The optical depth and particle size distribution
To obtain information about the microwave optical depth, the best data comes
for the radio occultation experiment from Voyager 1 (Marouf, Tyler, and Rosen 1986
and references w ithin). This gave an optical depth profile of the rings as a function of
radius at both 3.6 cm and 12.6 cm. Bistatic scattering by rings at these wavelengths
R e p r o d u c e d w ith p e r m issio n o f th e co p y rig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
I ll
also allow an estimation of the particle size density (Marouf et al. 1983, Zebker, Tyler,
and Marouf 1985). Both and maximum and minimum macroscopic particle size (amin
and amax) can be derived as well as their spectral index, g, assuming a power law in
the particle size distribution.
Recent studies have also shed some light on these parameters. By using the
Earth-based observations of the occultation of the ring by star 28 Sagittarii in con­
junction with the Voyager 2 optical occultation data, Nicholson and French (1998)
have also determined amin, amax> and q. While the g’s derived are similar to th at
of Zebker, Tyler and Marouf (1985), amin and amai are often quite different . These
results are summarized in Table 6-2.
6.2.4
Dirt contamination fraction
Equation 5.25 parameterizes the mass fraction of contamination of silicates with
ice. W hat has been observed to be the fraction of contaminates? By examining the
reflected sunlight off the rings in the Voyager images at a variety of phase angles,
Dones, Cuzzi, and Showalter (1993) show the dust fraction of the rings to be no
greater than 1%. Grossman (1990) concludes th at his models for therm al emission
also restrict the mass fraction of contaminants to 1%, though this may be even too
high, since Grossman’s calculations for F may be faulty.
6.2.5
Variables we may change
We have just described some of the physical parameters of the rings. Some of
these are well known, while others are only estimates. So when running simulation
of the ring, what variables are we allowed to changed and by how much?
The physical dimensions of each ring (and the planet) are well known. In Table
6-3 we characterize the ring parameters in five regions. Each will be assumed to have
R e p r o d u c e d w ith p e r m issio n o f th e co p y rig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
112
Ring Particle Param eters
Ring
T
C
0.05-0.2
B
CD
A
Q-min (dm)
Qmax (CE3.)
3.1
1
100-500
3.1
1
1000-2000
1.0 (inner)
2.7-3.0
1
500
~2.0 (outer)
2.7-3.0
2.75
30
1000-2000
2.8
1
750
0.1
1000-2000
2.7-3.0
1
500
2.75
1
1000-2000
0.0-0.25
0.65-1.0
Q.
Table 6-2: Some ring particle parameters. The optical depth r is dervied by looking
at the Voyager radio occultation data at 3.6 cm. The dust fraction F is determined in
Dones, Cuzzi, and Showalter (1993). Values in italics are from Nicholson and French
(1998).
R e p r o d u c e d w ith p e r m issio n o f th e co p y rig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
113
Ring Model Annulus
Radial Range
C ring
1.24-1.525
0.13
inner C ring
1.24-1.29
0.08
middle C ring
1.29-1.43
0.15
outer C ring
1.43-1.525
0.10
inner B ring
1.525-1.64
1.0
outer B ring
1.64-1.95
2.0
Cassini Division
1.95-2.025
0.10
A ring
2.025-2.27
0.7
(R sat)
TO
Table 6-3: Radii of ring annuli and normal optical depths used in the model calcula­
tions. The finer gradation in the C ring is used in the final model calculations.
a uniform optical depth, particle size distribution and tem perature throughout the
region (these are to be discussed below). For purposes of our calculations, the planet
will be spherical with brightness tem perature described by two components ai and
ao as given in Tables 3-1, 3-2, and 4-2.
The optical depth is fairly well established by the Voyager radio occultation
experiment, but is still open to some interpretation. The Radio Science occulatation
experiment determined the optical depth as function of radial distance ad depicted
in Figure 6-1 Both the C and A rings vary their optical depth as a function of radial
distance and so to model the entire ring annulus one takes the average over the entire
ring annulus. Furthermore, the outer B ring occultation d a ta is saturated and only
a lower limit may be determined by Voyager. By using the optical occultation data
we can estim ate what r might be in this region. The values for r which were used as
R e p r o d u c e d w ith p e r m issio n o f th e co p y r ig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
114
CO
CD
t
8 0 0 0 0
9 0 0 0 0
1 0 0 0 0 0
1 1 0 0 0 0
1 2 0 0 0 0
1 3 0 0 0 0
Radius (km)
Figure 6-1: The Radio Science System occultation data from Voyager 1. The exact
optical depth of the middle and outer B ring is not known since the d a ta saturate at
T = 1.1 .
R e p r o d u c e d w ith p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
115
our initial estimated param eters is also given in Table 6-3.
The physical tem peratures we used are highly epoch dependent. For the first
epoch, since the rings are not receiving direct sunlight, we shall use the unilluminated
tem peratures from Table 6-1. We assume the same tem perature for the Cassini divi­
sion as the C ring, but we do not use the Voyager 1 number, because the C ring and
Cassini division are not receiving direct sunlight and are therefore not transm itting
light. In this case the only sources of radiation impinging on these rings is the thermal
radiation of Saturn and reflected sunlight off Saturn. For the optically thick A and
B rings, the unilluminated side is essentially devoid of sunlight and so we adopt the
Voyager 1 numbers. For the second epoch, we simply use the numbers obtained from
Voyager 2.
The power law indices and particle size cutoffs are even less well established, so
we have the most freedom here. Still, q only ranges from 3.3 to 2.7. For size cutoffs,
consideration of Voyager d a ta suggests the amin = 1 cm and amax the order of a few
meters. The Nicholson and French (1998) analysis of the 28 Sagittarii d ata yield quite
larger particle sizes for amax. Since this is preliminary work, we will restrict ourselves
to the “classical” Voyager numbers.
Finally, the dust fraction F is clearly small and we shall only consider values
ranging from 0% to 1%.
6.3
6.3.1
R e s u lts
Preliminaries
Now we are ready to use our choice of parameters to the Monte Carlo simulation,
construct models of the rings, and compare them to the data. Before we proceed we
will outline the implementation of the code.
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
116
As noted in Chapter 5, we constructed our model in three steps. The first is
to generate the statistics. To beat down the random noise, we run the simulation
with 2 x 106 photons for each epoch/wavelength/ring combination. The output goes
into bins of one square degree. Second, we add the statistics by considering M x by
My planetary grids. Grids should be small enough to avoid roundoff error. We chose
this bin size to m atch the outputs bins - one square degree as seen from the rings.
This gave smooth results and took about eight hours on a PC 300 MHz Pentium
II running LINUX. The output table generated consists of brightness temperature
as a function of ring radius and azimuth. The final step converts the output into
Saturnian coordinates as viewed in the sky.
Since the model is east-west symmetric, we may fold our d ata and model about
the central meridian and compare. We present the data, folded this way, with two
transfer functions. Figures 6-2 and 6-3 highlight the details of the disk and Figures
6-4 and 6-5 show the details of the rings. The synthesized beam for each map is given
in the upper right hand corner. The benefits of folding our maps result in not only
smaller images but also a \/2 improvement on our noise.
The ring-emphasized Figures 6-4 and 6-5 show a high degree of structure in the
ring. Notice that all maps (where the resolution perm its) show a general trend of
highter T b closer to the planet, which gradually diminshes as an increasing function
of radial distance. The forward part of the inner C ring shows an extra enhancement.
This is clearly seen in the second data set and is suggested in the first. Notice also
th a t at 1.3 cm (second epoch) that both the inner C and inner B rings are enhanced.
The latter enhancement is also seen at 2.0 cm and 3.6 cm.
For the cases we examined, the ring annuli and normal optical depths used are
listed in Table 6-3. Furthermore, we used a value for the particle size power law
R e p r o d u c e d with p e r m issio n o f th e co p y r ig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
117
-10
ARC SEC
•10
ARC SE C
Figure 6-2: Data maps of Saturn for November 1995. Images depict the folded over
residual maps at (from left to right and top to bottom) 0.7, 2.0, 3.6, and 6.1 cm. The
dynamic range of the maps has been set to show the details of the disk.
R e p r o d u c e d w ith p e r m issio n o f th e co p y rig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
118
Figure 6-3: D ata maps of Saturn for February 1997. Images depict the folded over
residual maps a t (from left to right and top to bottom ) 1.3, 2.0, 3.6, and 6.1 cm. The
dynamic range of the maps has been set to show the details of the disk.
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
119
0
S
-5
10
>10
ARC SEC
15
*15
20
*20
0
0
S
*S
10
*1 0
A RC SEC
is
-1 5
20
*20
Figure 6-4: Data maps of Saturn for November 1995. Images depict the folded over
residual maps at (from left to right and top to bottom ) 0.7, 2.0, 3.6, and 6.1 cm. The
dynamic range of the m aps has been set to show the details of the rings.
R e p r o d u c e d with p e r m issio n o f th e co p y r ig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
120
15
20
>10
ARC SEC
A RCSEC
Figure 6-5: D ata maps of Saturn for February 1997. Images depict the folded over
residual maps at (from left to right and top to bottom ) 1.3, 2.0, 3.6, and 6.1 cm. The
dynamic range of the maps has been set to show the details of the rings.
R e p r o d u c e d w ith p e r m issio n o f th e co p y rig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
121
index of q =3.0, and the particle size minima and m axim a were 1 cm and 500 cm,
respectively. The dirt fraction was set to a F — 0.0025.
The d ata and model maps in turn may be subtracted from one another to
produce a folded residual maps. These will allow us to examine the success of our
models and enable us to determine which param eters need to be changed.
6.3.2
Case one: isotropic scattering
The first case we modeled was the simple isotropic case. The models are given
in 6-6 and 6-7. The coressponding residuals, the d a ta minus the model, calculated
for the first second epoch are given in Figures 6-8 and 6-9, respectively.
L et’s examine the second epoch first.
The m ost striking residuals come at
1.3 cm. The A and B rings across the planet and out in the ansae are well fit. The
only problems lie in the C ring where the model is too faint at least 15 K across the
planet and slightly (5 K) too bright on the backscattering side of the ansae. As we
examine the other residuals at second epoch for the longer wavelengths, the A and
B ring ansae remain remarkably steady (and a forward/backward problem starts in
the C ring), while the rings across the planet become worse. Specifically, the B ring
becomes progressively more underrepresented at 2.0, 3.6 and 6.1 cm (20 to 30 K too
low), the A ring is marginally well modeled at 2.0 cm, but suffers the same fate as
the B ring at 3.6 and 6.1 cm.
The first epoch show similar trends, though w ith the shallower geometry, the
individual rings are more difficult to discern. Nevertheless, in the ansae we again
see the C ring backscattering model surplus. The residuals at 0.7 cm, like those at
1.3 cm show remarkable agreement, with the only problem being in the C ring across
the planet (10 to 15 K too faint). The longer wavelengths at the first epoch show
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
0
*5
*10
ARC SEC
-1 5
-2 0
0
*5
> 10
ARC SEC
-IS
-2 0
Figure 6-6: Model maps of isotropic scattering calculations. Images depict the folded
over residual maps at (from left to right and top to bottom) 0.7, 2.0, 3.6, and 6.1 cm.
R e p r o d u c e d w ith p e r m issio n o f th e co p y rig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
ARC S E C
123
•10
ARC SEC
•10
ARC SEC
Figure 6-7: Model maps of isotropic scattering calculation. Images depict the folded
over residual maps at (from left to right and top to bottom) 1.3, 2.0, 3.6, and 6.1 cm.
R e p r o d u c e d w ith p e r m issio n o f th e co p y rig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
•10
ARC SEC
•1 0
A R C SEC
Figure 6-8: Residual maps of isotropic scattering calculations. Images depict the
folded over residual maps at (from left to right and top to bottom ) 0.7, 2.0, 3.6, and
6.1 cm.
R e p r o d u c e d w ith p e r m issio n o f th e co p y r ig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
0
-6
>10
ARC SEC
> 16
>20
0
>5
*10
ARC SEC
-1 5
-2 0
Figure 6-9: Residual maps of isotropic scattering calculations. Images depict the
folded over residual maps at (from left to right and top to bottom ) 1.3, 2.0, 3.6, and
6.1 cm.
R e p r o d u c e d w ith p e r m issio n o f th e co p y rig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
126
progressively more model deficit (10 to 20 K) along the rings across the planet, with
the apparent improved agreement at 6.1 cm being due to the smearing by a very large
beam.
6.3.3
Case two: Mie scattering
The second case we ran was the Mie scattering case. The models are given in
6-10 and 6-11. The corresponding residuals, again being the data minus the model,
calculated for the first second epoch are given in Figures 6-12 and 6-13, respectively.
We first turn our attention to the higher resolution second epoch maps. Notice
th at at all wavelengths the A and B rings are too bright by ~ 10 K in the model in
the forward (upper) half of the rings and too faint (by 5 K) in the back (lower) half.
This suggests the phase function is too steep in the forward lobe. The C ring is fit
much better by the model, especially at 3.6 and 6.1 cm, though the back half does
show a distinctive deficit (5 K). Meanwhile, the rings across the planet generally show
good agreement at 6.1 cm (don’t confuse planetary structure with ring structure) and
mostly at 3.6 cm with some overrepresentation of the model in the B ring (5 K). At
2.0 and 1.3 cm the model fares worse (10 to 15 K too high). Both the A and B rings
are overrepresented, and even the C ring is does not show good agreement.
Upon examining the first epoch we reach similar conclusions. The ansae show
the model to be too forward scattering as we saw above. The rings across the planet
are very well modeled at 3.6 and 6.1 cm. At 0.7 cm and somewhat at 2.0 cm the ring
flux is overestimated by ~ 5 K.
The results of the quality of fit are summarized in Table 6-4. In general whatever
fits well with the isotropic model does not fit well in the Mie model and vice versa.
R e p r o d u c e d w ith p e r m issio n o f th e co p y r ig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
127
Figure 6-10: Model maps of Mie scattering calculations for the November 1995 epoch.
Images depict the folded over residual maps at (from left to right and top to bottom)
0.7. 2.0, 3.6, and 6.1 cm.
R e p r o d u c e d w ith p e r m issio n o f th e co p y rig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
ARC S E C
•10
ARC SEC
Figure 6-11: Model maps of Mie scattering calculations for the February 1997 epoch.
Images depict the folded over residual maps at (from left to right and top to bottom)
1.3, 2.0, 3.6, and 6.1 cm.
R e p r o d u c e d w ith p e r m issio n o f th e co p y rig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
•10
ARC S E C
Figure 6-12: Residual maps of Mie scattering calculation. Images depict the folded
over residual maps at (from left to right and top to bottom ) 1.3, 2.0, 3.6, and 6.1 cm.
R e p r o d u c e d w ith p e r m issio n o f th e co p y rig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
•10
ARC S E C
•10
A RCSEC
Figure 6-13: Residual maps of Mie scattering calculations. Images depict the folded
over residual maps at (from left to right and top to bottom) 1.3, 2.0, 3.6, and 6.1 cm.
R e p r o d u c e d w ith p e r m issio n o f th e co p y r ig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
131
Wavelength
Ring
RAP
ABA
FS
1.3/0.7 cm
A
I
I
I
B
I
I
I
C
N
N
N
A
I
I
I
B
M
I
I
C
M
M
N
A
M
I
I
B
[M]
[I]
I
C
M
M
N
A
M
I
I
B
M
I
I
C
M
M
N
2.0 cm
3.6 cm
6.1 cm
Table 6-4: Summary of the effectiveness of the two models. We examine three aspects
of the rings and indicate which (if any) model produced little or no reduals in either
or both epochs. The three catagories we examined were the rings across the planet
(RAP), the average brightnes of the ansae (ABA), and the degree of forward scattering
(FS). An entry of “M” indicates the Mie scatter worked better, while “I” indicates
th at istropic scattering worked better. If neither suitably match the data and “N” is
dentoed. Brackets around such designations, indictate a dubious agreement in one or
both of the epochs.
R e p r o d u c e d w ith p e r m issio n o f th e co p y rig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
132
6.3.4
Case three: Hybrid isotropic/M ie scattering
Neither isotropic nor Mie scattering correctly describe the ring brightness tem­
perature. Nevertheless, they are often complementary to each other, one type scat­
tering fits the d a ta well while the other fits poorly (see Table 6-4).
Directing our attention to this problem, we propose a hybrid isotropic and Mie
scattering phase function which can address some of these issues. Futhermore, we wish
to apply different phase functions to different rings at different wavelengths instead
of trying one set of param eters to all the rings. There also is clear structure in the
C ring, which suggest th at one r may not be enough to describe the ring. Therefore,
we break the C ring into three sections of appropiate r based on the Voyager Radio
Science data (see Table 6-3). This will likely give us a better fit.
We parameterize the isotropic/Mie combination with FiSO, the fractional amount
of isotropic scattering in the phase function. Therefore, we can describe the phase
fucntion as a linear combination of Mie scattering and isotropic scattering:
p'li = (1 - Fiao)pii + Fiso,
(6.1)
where p u is the Mie scattering phase function. Table 6-5 describes the fraction of
isotropy for each ring and wavelength. Figures 6-14 and 6-15 show the models of first
and second epochs, respectively and Figures 6-16 and 6-17 show the corresponding
residuals. Notice the residual map transfer function ranges from —10 to 10 K, instead
of —15 to 15 K to show more subtle detail.
The results are a remarkable fit. In the first epoch, the rings across the planet
are nearly gone for each ring/wavelengths. T he model ansae are too bright by no
more than ~ 5 K. In the second epoch, we see th a t subtle structure is mapped well.
The rings across the planet is quite good. Much of the structure across the planet
R e p r o d u c e d with p e r m issio n o f th e co p y r ig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
133
0
0
S
-5
to
>10
ARC SE C
15
-1 5
20
>20
0
0
5
>5
10
>10
15
>15
20
>20
ARC SEC
Figure 6-14: Model maps of hybrid isotropic/Mie scattering calculations for the
November 1995 epoch. Images depict the folded over residual maps at (from left
to right and top to bottom) 0.7, 2.0, 3.6, and 6.1 cm.
R e p r o d u c e d w ith p e r m issio n o f th e co p y rig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
134
Figure 6-15: Model maps of hybrid isotropic/M ie scattering calculations for the Febru­
ary 1997 epoch. Images depict the folded over residual maps at (from left to right
and top to bottom) 1.3, 2.0, 3.6, and 6.1 cm.
R e p r o d u c e d w ith p e r m issio n o f th e co p y rig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
•6
>10
ARC SEC
>15
*20
a
*6
-10
-15
*20
ARC SEC
Figure 6-16: Residual maps of hybrid isotropic/Mie scattering calculation. Images
depict the folded over residual maps at (from left to right and top to bottom ) 0.7,
2.0, 3.6, and 6.1 cm.
R e p r o d u c e d w ith p e r m issio n o f th e co p y rig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
ARC 8 E C
136
Figure 6-17: Residual maps of the hybrid isotropic/M ie scattering calculations. Im­
ages depict the folded over residual maps a t (from left to right and top to bottom)
1.3, 2.0, 3.6, and 6.1 cm.
R e p r o d u c e d w ith p e r m issio n o f th e co p y rig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
137
Wavelength (cm)
Ring
A
B
C
1.3/0.7
1.0
1.0
0.25
2.0
1.0
0.5
0.0
3.6
0.5
0.5
0.0
6.1
0.5
0.25
0.0
Table 6-5: The fraction of isotropy, FiS0, used for each ring/wavelengths in our hybrid
model.
and 3.6 and 6.1 cm is planet and not rings. The model ansae are again too bright by
no more than ~ 5 K. The errors are mostly concentrated in the forward part of the
ansae.
6 .4
D is c u s s io n
We have seen that neither a pure isotropic nor Mie scattering phase function
will accurately and completely map the rings of Saturn. Instead, we must consider
not only a linear combination of these two parameters, but also we must apply a
different linear combination for each ring and wavelength. The param eters which
qualitatively matched the d ata (Table 6-5) show some revealing patterns. First, we
see a general trend from isotropic to Mie scattering from short to long wavelengths.
Why could the be so? If the particles have irregularities of order a <1 cm, then only
the short wavelengths would be sensitive to these irregularities. This would promote
a more isotropic phase function at short wavelengths. This effect should diminish
with increasing wavelength and the phase function should be more Mie-like. This is
born out in the data.
R e p r o d u c e d w ith p e r m issio n o f th e co p y rig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
The C ring seems to be a wrinkle in this reasoning. The isotropic models do
not fare well at all, while Mie based models do better. It does seem clear th a t inner
C ring seem different than the rest of the C ring, justifying our sub-annuli in this
region. The C ring very well could be different type of ring. Perhaps the particles
are inherently smoother. The lower optical depth ring suggests th at collisions are not
so common so that conglomeration is less common than what you would find in the
A and B rings (Weidenschilling et al. 1984). Likewise, greater the tidal forces here
also suppress the conglomeration of smaller particles. This is seen to certain extent
between the A and B rings where the A ring is slightly more isotropic than the B
ring.
R e p r o d u c e d w ith p e r m issio n o f th e co p y r ig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
139
C H A P T E R V II
C O N C LU SIO N
7.1
O verview
We have successfully imaged Saturn’s disk and rings using radio interferometric
techniques at two different epochs. These were the first high resolution images ever
taken at such small inclinations and first ones at any type taken at 0.7 cm. We
have also presented a model which tests some of the physical parameters of the ring
particles. We outline these results.
7.2
T he atm osphere
We find that the atmospheric latitudinal structure has changed significantly
since the 1980’s. At both epochs we find that there are two bright bands straddling
a dimmer zone centered on the equator, especially at 3.6 and 6.1 cm. There is no
sign of an optical counterpart to the fainter radio zone. In general, the radio picture
seems uncorrelated with the optical one. The appearance of planetary structure in the
radio may be indicative of a sort of El Nino effect at several atmospheres on Saturn
where zone and belt become differentiated, and may be tied in with the changing of
the seasons. Between our two epochs (where we can compare), the atmosphere has
changed very little.
R e p r o d u c e d w ith p e r m issio n o f th e co p y rig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
140
7.3
T h e rings
Through our analysis of the rings we have found some unexpected results. First,
there was no significant detection of the Cassini division in the first data set. We
expect this ring to be optically thick due to foreshortening and hence be as bright
as any ring. The small angular size of the ring may make modeling the Cassini
Division difficult. Second, the rings have shown an east-west asym m etry during our
first (lower inclination) epoch. This has also been seen in data taken a t high (z <20°)
inclinations (de Pater and Dickel 1991), but was not seen at our second epoch
(z =
5°). Since single scattering cannot lead to asymmetries, does this suggest some sort
of nonuniform spatial distribution of the rings particles (such as wakes)?
Our maps of the rings also confirm what was seen by several previous observa­
tions. First, our 1995 November epoch results generally showed the B ring to be as
bright as the C ring. This makes sense considering that foreshortening turns even the
C ring into a optically thick ring. We clearly see the C ring brighter in the forward
direction than the B ring in second epoch with its higher (5°) inclination. Second,
the rings are generally brighter in the forward part of the ring th a n it is on the
back, generally suggesting th at something similar to a Mie scattering phase function
is appropriate (see below).
Finally, we ran Monte Carlo simulations for both isotropic and Mie scattering,
in order to model the rings. Unlike previous works (i.e. Grossman 1990 and Cuzzi
et al. 1980), our model was of high angular resolution and included the effects of
multiple scattering. We found that neither isotropic nor Mie theory applied to the
nominal values completely fit the data. However, certain rings/wavelength combina­
tion modeled very well to isotropic or Mie scattering. The A and B rings generally
showed a tendency to be isotropic, especially at the longer wavelengths. The C ring,
R e p r o d u c e d w ith p e r m issio n o f th e co p y rig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
141
on the other hand, had a tendency to contain Mie scatterers. We also found th at the
C ring had radially finer structure than we thought, which led us subdivide this ring
into three distinct annuli. The difference between the rings may very well be a result
of tidal forces suppressing particle conglomerations in the C ring.
The phase function we obtained have many broad similarities to non-spherical
phase functions. Pollack and Cuzzi (1980) examined the phase function of nonspherical particles. Their best semi-empirical model show's reduced at the back and forward
scattering extremes and enhanced scattering for middle and moderately backscatter
angles as compared with Mie scattering. This is the same general trend we get when
we combine Mie scattering with isotropic scattering.
7.4
Further work
We have obtained another epoch of data in O ctober 1998, when Saturn ring
inclination was i = —15°. At this inclination the rings will have a greater solid angle
than our other epochs. Will the rings look the same as we have seen them in 1997?
Also, will the atmosphere continue to look like it did in 1995 and 1997, or will the
planet return to a featureless appearance as it did in the 1980s?
We also could go back and look historic data, such as the ones cited in de P ater
and Dickel (1991). These can be modeled in same wray as the data sets in this thesis
to look for subtle changes in the rings and atmosphere.
One param eter which was not addressed but could add valuable insight is po­
larization. The rings have a moderate polarization on the ansae as seen by Grossman,
Muhleman, and Berge (1989). The polarization could very well be indicative of wake
structure in rings. We may therefore expect to find a measurable asymmetry in the
first epoch data, if the asymmetry is due to wakes in the rings.
R e p r o d u c e d w ith p e r m issio n o f th e co p y rig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
142
O ur code must be further generalized to address some of these issues. The ring
model can be expanded to include anisotropic particle distributions such as wakes.
This is necessary to test the asym metry in the rings. A general phase function which
includes polarization can be incorporated, giving the user greater freedom to test and
scattering property.
A qualitative analysis of the data would make our conclusions more robust. Our
analysis has been mostly quantitative, based on our visual inspection of the residual
maps and have not included any least squares fitting of the data. W ith our current
setup we should be able to fit for the Fiso parameter for each ring annulus. This
could eventually lead to a simultaneous fitting of radar, Voyager occultation, optical,
as well as microwave data taken by us and others.
R e p r o d u c e d w ith p e r m issio n o f th e co p y rig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
143
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