The Determination of Orbitsfrom Spacecraft Imaging
A thesis submitted to the University of London
for the degree of Doctor of Philosophy
Michael Wyn Evans
Queen Mary, University of London
November 2000
1
Abstract
Orbits are derived for the saturnian satellites Atlas, Prometheus and Pandora
from images taken by the Voyager 1 and Voyager 2 spacecraft.
The process of geometrical correction of Voyager images, whereby distortions
introduced by the vidicon imaging system are removed, is discussed. The IDL
routines written to perform this process are described.
The mechanics of image navigation, where the precise pointing direction of the
camera at the time a image was taken, are explained. Circular features in the outer
saturnian ring system and Saturn’s limb are used as fiducials for navigating the
Atlas, Prometheus and Pandora images..
Orbits are determined by fitting a uniformly precessing, inclined ellipse to the
observations. A differential correction process is used to minimise the rms residuals
between the calculated and observed locations of the object of interest.
The orbital elements of Atlas, Prometheus and Pandora at the Voyager 1 and
Voyager 2 epochs are derived. Possible explanations for Prometheus’ observed mean
longitude lag in 1995 are discussed. The effect of Pandora’s close proximity to the
3:2 co-rotation eccentricity resonance with Mimas are investigated.
The relationship between the accuracy of an orbit determination and such factors
as number of observations, range of spacecraft at observation time and epoch range
of the observations is investigated. This is then used in the planning of observations
of the small saturnian satellites for the Cassini mission.
2
Acknowledgments
First and foremost my thanks go to Prof. Carl Murray, my supervisor. He saw
something in a candidate with a somewhat checkered academic history. The idea
for the subject of this thesis was all his, although I think his original vision involved
less orbit determination and more on rings. Even when the work was progressing
slowly he remained supportive and provided encouragement. Carl appears to be on
first name terms with everybody working on rings and their associated satellites,
and through him I too have met many of them. There is a saying that “its not what
you know its who you know”, Carl knows both the “what” and the “who” and has
endeavored to pass on both to me. He has attempted to improve my spelling and
grammatical style, reading drafts of this thesis must have been painful for him.
My special thanks to Dr. Terry Arter, my personal postscript consultant. The
better diagrams would not exist in the same form without him acting as a “human
interface”. He also bravely submitted himself to torture by thesis, the current
version makes much more sense due to his puzzlement.
This work would not have been possible without the help of several people.
Dr. Carolyn Porco gave permission for me to use the MINAS software package,
without which the task of geometrical correction and image navigation would have
been much more difficult. Her comments during visits to Queen Mary highlighting
areas where I had previously seen no problems.
Dr. Mark Showalter aided in locating images of Prometheus, providing details
of searches he had performed for that satellite. Mark made available images that
are not currently on the PDS Voyager data CD-ROMs. I spent an enjoyable week
at NASA Ames with Mark as my host, an experience I thank him for.
Though he knows it not, Dr. Mitch Gordon has provided invaluable aid in my
work. Mitch was also Carl Murray’s student and his thesis involved geometrically
correcting and navigating Voyager images from the Saturn encounter in the search
for additional satellites. His thesis showed me the nuts and bolts of performing
many necessary tasks. The C-kernels I used for initial pointing directions for my
images were complied by Mitch from raw data.
My thanks to Prof. Iwan Williams for the opportunity I had to observe the
3
Leonids from La Palma in 1998, tens of fireballs a minute putting your average
meteor shower to shame. I spent a week on top of a volcano in November 1999
observing saturnian satellites with Iwan. It is due to him that I am classed as an
‘experienced’ observer.
Discussions with Dr. Dick French about my work and his on Prometheus and
Pandora have proved useful. Dr. Phil Nicholson has provided data on the orbit
of the F ring and passed on unpublished orbits for Prometheus and Pandora from
Dr. Bob Jacobson. I thank Dr. Don Taylor for his Technical Note which finally
enabled me to fit precessing ellipses to observations.
I was only able to study for a Ph. D. with a studentship from the Particle Physics
and Astronomy Research Council. My thanks to them and a British government
who still provide some funding for astronomy research.
Then there are the people who kept me sane (in no particular order) Kevin,
Tolis, Angela, Helena, Phil, Tom, Agueda, John, Matt, Paul, Chad and Nick. All
either post-docs or postgrads at Queen Mary, if I have forgotten anyone its simply
an oversight.
For all my computing needs there was always Richard Frewin and STARLINK.
There are good systems managers and there are helpful systems managers, Richard
is both.
And finally my mother. It must be worrying when you have a child (in his late
twenties) who still has not decided what he wants to be when he grows up. Not
only did she not criticise me for lack of direction but supported me both morally
and financially when I had no right to expect it.
4
I know that I am mortal by nature, and ephemeral; but when I trace at my
pleasure the windings to and fro of the heavenly bodies I no longer touch earth with
my feet: I stand in the presence of Zeus himself and take my fill of ambrosia, food
of the gods.
Ptolemy, epigram to Syntaxis
circa 145 A.D.
5
For my mother, and her patience with a son
in his thirties who is still at school.
6
Contents
1 Introduction 16
1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
1.2 The initial research goal . . . . . . . . . . . . . . . . . . . . . . . . 18
1.3 Possible origins of Prometheus’ lag . . . . . . . . . . . . . . . . . . 19
1.3.1 A co-orbital companion . . . . . . . . . . . . . . . . . . . . . 19
1.3.2 Periodic encounters with the F ring . . . . . . . . . . . . . . 20
1.3.3 Cometary impact . . . . . . . . . . . . . . . . . . . . . . . . 21
1.3.4 Gravitational interaction with the F ring . . . . . . . . . . . 21
1.3.5 Other mechanisms . . . . . . . . . . . . . . . . . . . . . . . 21
1.4 Pandora develops a lag . . . . . . . . . . . . . . . . . . . . . . . . . 21
1.5 The accuracy of determined orbits . . . . . . . . . . . . . . . . . . . 22
1.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2 Orbits and Orbit Determination 24
2.1 The geometry of elliptical orbits . . . . . . . . . . . . . . . . . . . . 24
2.1.1 The position of an object in an unperturbed Keplerian orbit 29
2.1.2 Orbital and reference frames . . . . . . . . . . . . . . . . . . 31
2.2 Osculating and geometric elements . . . . . . . . . . . . . . . . . . 34
2.3 Orbit determination . . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.4 Preliminary orbit determination . . . . . . . . . . . . . . . . . . . . 36
2.4.1 Laplace’s method . . . . . . . . . . . . . . . . . . . . . . . . 36
2.4.2 Gauss’ method . . . . . . . . . . . . . . . . . . . . . . . . . 38
2.4.3 The f and g functions . . . . . . . . . . . . . . . . . . . . . . 40
7
2.4.4 The suitability of Laplace’s and Gauss’ methods for planetary
satellites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
2.4.5 Determining the orbits of planetary satellites from spacecraft 42
2.5 Improving the orbit . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
2.5.1 Integration of the full equations of motion for an entire system 43
2.5.2 The precessing ellipse model . . . . . . . . . . . . . . . . . . 44
2.5.3 Generation of an ephemeris . . . . . . . . . . . . . . . . . . 47
2.6 The differential correction of elements . . . . . . . . . . . . . . . . . 47
2.7 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
2.7.1 Integration of the full equations of motion . . . . . . . . . . 50
2.7.2 The precessing ellipse model . . . . . . . . . . . . . . . . . . 51
2.8 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
3 Voyager Images 53
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
3.2 Voyager spacecraft and instruments . . . . . . . . . . . . . . . . . . 53
3.3 Voyager images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
3.4 The geometrical correction of Voyager images . . . . . . . . . . . . 56
3.4.1 Overview of the geometrical correction process . . . . . . . . 58
3.4.2 Reseau marks . . . . . . . . . . . . . . . . . . . . . . . . . . 58
3.4.3 Location of reseau marks . . . . . . . . . . . . . . . . . . . . 60
3.4.4 The detection of reseaus in a raw image . . . . . . . . . . . 62
3.4.5 The generation of ‘pseudo-reseau’ marks . . . . . . . . . . . 62
3.4.6 Mapping image-space locations into object-space . . . . . . . 62
3.4.7 The nearest neighbour pixel mapping technique . . . . . . . 68
3.4.8 Pixel interpolation method . . . . . . . . . . . . . . . . . . . 68
3.4.9 Comparison of the two methods . . . . . . . . . . . . . . . . 68
3.5 VICAR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
3.6 MINAS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
3.7 Routines for reseau location and geometrical correction . . . . . . . 71
3.7.1 resloc.pro: locating reseau marks . . . . . . . . . . . . . . 72
3.7.2 geoma.pro: Performing the Geometrical Correction . . . . 76
8
3.8 Comparison of VICAR and MINAS results . . . . . . . . . . . . . . 78
4 Image Navigation 80
4.1 Introduction to image navigation . . . . . . . . . . . . . . . . . . . 80
4.2 Methods of image navigation . . . . . . . . . . . . . . . . . . . . . . 81
4.3 Software for image navigation, MINAS . . . . . . . . . . . . . . . . 82
4.4 Navigating an image . . . . . . . . . . . . . . . . . . . . . . . . . . 83
4.5 Calculating the pointing vector to an object in an image . . . . . . 86
5 Atlas 88
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
5.2 Search methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
5.3 Geometrical correction . . . . . . . . . . . . . . . . . . . . . . . . . 91
5.4 Image navigation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
5.5 Generating a set of observations . . . . . . . . . . . . . . . . . . . . 93
5.6 Orbit determination . . . . . . . . . . . . . . . . . . . . . . . . . . 94
5.7 Identified images . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
5.8 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
5.9 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
5.10 The distribution of longitudes at observation mid-times . . . . . . . 100
5.11 The orbit of Atlas in JPL Ephemerides . . . . . . . . . . . . . . . . 103
5.12 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
6 Prometheus 108
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
6.2 Search methodology and orbit determination . . . . . . . . . . . . . 111
6.3 Identified images . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
6.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
6.5 The distribution of longitudes at observation mid-times . . . . . . . 124
6.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
6.7 The orbit of Prometheus in JPL Ephemerides . . . . . . . . . . . . 129
6.8 Comparison of the derived orbits with explanations for the origins of
Prometheus’ lag . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
9
6.8.1 A co-orbital companion . . . . . . . . . . . . . . . . . . . . . 131
6.8.2 Periodic encounters with the F ring . . . . . . . . . . . . . . 132
6.8.3 Cometary impact . . . . . . . . . . . . . . . . . . . . . . . . 132
6.8.4 Gravitational interaction with the F ring . . . . . . . . . . . 133
6.8.5 Other mechanisms . . . . . . . . . . . . . . . . . . . . . . . 133
6.8.6 Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
6.9 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
7 Pandora 135
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
7.2 Search methodology and orbit determination . . . . . . . . . . . . . 135
7.3 Identified images . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
7.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
7.5 The distribution of longitudes at observation mid-times . . . . . . . 146
7.6 Discussion of the orbit fits . . . . . . . . . . . . . . . . . . . . . . . 148
7.7 The orbit of Pandora in JPL Ephemerides . . . . . . . . . . . . . . 149
7.8 The 3:2 near-resonance with Mimas . . . . . . . . . . . . . . . . . . 151
7.8.1 The planetary disturbing function . . . . . . . . . . . . . . . 152
7.8.2 Lagrange’s planetary equations . . . . . . . . . . . . . . . . 154
7.8.3 The effects of the Pandora-Mimas 3:2 CER from theory . . . 155
7.8.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158
8 Planning Future Observations 162
8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
8.2 The error in e from geometry . . . . . . . . . . . . . . . . . . . . . 164
8.2.1 Determining F(SSL) . . . . . . . . . . . . . . . . . . . . . . 167
8.2.2 Other orbital parameters . . . . . . . . . . . . . . . . . . . . 167
8.2.3 Limitations . . . . . . . . . . . . . . . . . . . . . . . . . . . 168
8.3 Expected accuracy of realistic observations . . . . . . . . . . . . . . 168
8.4 Planning Cassini observations of the small saturnian satellites. . . . 169
8.4.1 Introducing synthetic ‘observational errors’ . . . . . . . . . . 170
8.4.2 Fitting an orbit . . . . . . . . . . . . . . . . . . . . . . . . . 170
10
8.4.3 Examples of calculated orbit determination accuracy for Cassini171
8.4.4 The effects of varying the number of observations and as-
sumed observational error . . . . . . . . . . . . . . . . . . . 174
9 Summary and Discussion 177
9.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177
9.2 Accuracy of determinations . . . . . . . . . . . . . . . . . . . . . . 178
9.3 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178
11
List of Figures
2.1 The geometry of an ellipse . . . . . . . . . . . . . . . . . . . . . . . 25
2.2 Definition of the eccentric anomaly E. . . . . . . . . . . . . . . . . 26
2.3 The geometry of an orbit in 3-dimensions. . . . . . . . . . . . . . . 28
2.4 Orbit and reference frame coordinates . . . . . . . . . . . . . . . . . 31
2.5 The orbital frame for a body. . . . . . . . . . . . . . . . . . . . . . 32
2.6 The J2000 reference frame. . . . . . . . . . . . . . . . . . . . . . . . 33
2.7 Laplace’s method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
2.8 Reference frame for precessing elliptical orbit. . . . . . . . . . . . . 46
2.9 Reference frame for orbital elements of a satellite. . . . . . . . . . . 50
3.1 Raw Voyager image orientation. . . . . . . . . . . . . . . . . . . . . 56
3.2 Raw space mapped onto image space . . . . . . . . . . . . . . . . . 57
3.3 A raw Voyager image showing reseau marks . . . . . . . . . . . . . 59
3.4 Object-space reseau mark locations . . . . . . . . . . . . . . . . . . 61
3.5 Object-space pseudo-reseau mark locations . . . . . . . . . . . . . . 63
3.6 Object-space triangular grid . . . . . . . . . . . . . . . . . . . . . . 64
3.7 Image-space triangular grid . . . . . . . . . . . . . . . . . . . . . . 65
3.8 Pixel interpolation 1 . . . . . . . . . . . . . . . . . . . . . . . . . . 67
3.9 Pixel interpolation 2 . . . . . . . . . . . . . . . . . . . . . . . . . . 69
3.10 9 × 9 pixel region around the location of a reseau mark. . . . . . . . 73
3.11 Index numbers for a 4 × 4 pixel image. . . . . . . . . . . . . . . . . 77
4.1 Image displayed by Minas, pre-navigation . . . . . . . . . . . . . . . 84
4.2 Image displayed by Minas, post-navigation . . . . . . . . . . . . . . 85
4.3 Instrument frame for Voyager cameras . . . . . . . . . . . . . . . . 86
12
5.1 Atlas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
5.2 Voyager 1 positions of Atlas . . . . . . . . . . . . . . . . . . . . . . 101
5.3 Voyager 2 positions of Atlas . . . . . . . . . . . . . . . . . . . . . . 102
6.1 Prometheus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
6.2 Wide angle Prometheus . . . . . . . . . . . . . . . . . . . . . . . . 110
6.3 Voyager 1 positions of Prometheus . . . . . . . . . . . . . . . . . . 125
6.4 Voyager 2 positions of Prometheus . . . . . . . . . . . . . . . . . . 126
7.1 Pandora . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
7.2 Voyager 1 positions of Pandora . . . . . . . . . . . . . . . . . . . . 146
7.3 Voyager 2 positions of Pandora . . . . . . . . . . . . . . . . . . . . 147
7.4 The position vectors r and r′ of m and m′ . . . . . . . . . . . . . . 153
7.5 Time variation of the resonant argument of Pandora . . . . . . . . . 157
7.6 Time variation of the mean motion of Pandora . . . . . . . . . . . . 157
7.7 Time variation of ∆Λ for Pandora. Epoch: 2444839.6682 JED . . . 158
8.1 Geometry of observations 1 . . . . . . . . . . . . . . . . . . . . . . 164
8.2 Geometry of observations 2 . . . . . . . . . . . . . . . . . . . . . . 165
8.3 Relationship between F(SSL) and SSL . . . . . . . . . . . . . . . . 168
8.4√N against ∆ values from Table 8.5 . . . . . . . . . . . . . . . . . 176
13
List of Tables
1.1 1995 ring plane crossing mean longitudes . . . . . . . . . . . . . . . 17
1.2 Expected errors in mean longitudes in 1995 . . . . . . . . . . . . . . 18
5.1 Constants used in the image navigation and orbit determination pro-
cess. Where f and σ are the focal length and scale factor of the
camera as used in Eqns. 4.3-4.5 . . . . . . . . . . . . . . . . . . . . 92
5.2 Voyager 1 Images of Atlas . . . . . . . . . . . . . . . . . . . . . . . 96
5.3 Voyager 2 Images of Atlas. . . . . . . . . . . . . . . . . . . . . . . . 97
5.4 The orbital elements of Atlas . . . . . . . . . . . . . . . . . . . . . 98
5.5 Voyager 1 orbital elements of Atlas . . . . . . . . . . . . . . . . . . 100
5.6 Orbital elements of Atlas from vg2 sat.bsp . . . . . . . . . . . . . . 105
5.7 Orbital elements of Atlas from sat081.4.bsp . . . . . . . . . . . . . . 106
5.8 The adopted orbital elements of Atlas: epoch 2444839.6682 . . . . . 107
6.1 Voyager 1 Images. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
6.2 Voyager 2 Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
6.3 Voyager 1 Results: epoch 2444839.6682 JED. . . . . . . . . . . . . . 120
6.4 Voyager 2 Results: epoch 2444839.6682 JED. . . . . . . . . . . . . . 120
6.5 Voyager 1 Results using Synnott et al. ’s 18 images . . . . . . . . . 121
6.6 Voyager 2 Results using Synnott et al.’s 27 images . . . . . . . . . . 121
6.7 Voyager 1 Results: epoch 2444513.5 JED . . . . . . . . . . . . . . . 122
6.8 Voyager 1 Results using Synnott et al.’s images at Voyager 1 epoch 122
6.9 Combined Voyager 1 & 2 Results: epoch 2444839.6682 JED . . . . 123
6.10 Combined Synnott et al.’s (1983) Voyager 1 & 2 Results . . . . . . 123
6.11 Mean longitudes for Prometheus from various authors . . . . . . . . 128
14
15
6.12 Orbital elements for Prometheus from vg2 sat.bsp . . . . . . . . . . 130
6.13 Orbital elements for Prometheus from sat081.4.bsp . . . . . . . . . 131
7.1 Voyager 1 Images of Pandora . . . . . . . . . . . . . . . . . . . . . 138
7.2 Voyager 2 Images of Pandora . . . . . . . . . . . . . . . . . . . . . 140
7.3 Voyager 1 Results: epoch 2444839.6682 JED. . . . . . . . . . . . . . 143
7.4 Voyager 2 Results: epoch 2444839.6682 JED. . . . . . . . . . . . . . 143
7.5 Voyager 1 Results using 28 of Synnott et al. ’s 32 images . . . . . . 144
7.6 Voyager 2 Results using 37 of Synnott et al. ’s 39 images . . . . . . 144
7.7 Combined Voyager 1 & 2 Results: epoch 2444839.6682 JED. . . . . 145
7.8 Combined 65 of Synnott et al. ’s(1983) Voyager 1 & 2 Results . . . 145
7.9 Orbital elements for Pandora from vg2 sat.bsp . . . . . . . . . . . . 150
7.10 Orbital elements for Pandora from sat081.4.bsp . . . . . . . . . . . 151
7.11 The orbital elements of Pandora including the 3:2 resonance with
Mimas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
7.12 The orbital elements of Pandora including the 3:2 resonance with
Mimas, best fit phase . . . . . . . . . . . . . . . . . . . . . . . . . . 160
7.13 The orbital elements of Pandora of French et al. (2000) . . . . . . . 161
8.1 90 test observations of Prometheus . . . . . . . . . . . . . . . . . . 171
8.2 90 test observations of Pandora . . . . . . . . . . . . . . . . . . . . 172
8.3 90 test observations of Epimetheus . . . . . . . . . . . . . . . . . . 173
8.4 Observations of Prometheus, constant number of images . . . . . . 174
8.5 Observations of Prometheus, constant σ . . . . . . . . . . . . . . . 175
Chapter 1
Introduction
1.1 Background
The smaller satellites in the saturnian system are generally not detectable using
ground-based observations. They were discovered using images from the Voyager
1 spacecraft. They should however be visible during ring plane crossings. Close to
the time when the Earth crosses the saturnian ring plane the brightness of the rings
drops and satellites that are normally too faint to be observed can be detected.
The only Sun crossing since the Voyager encounters occurred on November 19
1995, with Earth crossings on May 22 1995, August 10 1995 and February 11 1996.
During the May event all observed satellites were at their expected positions except
for Prometheus (e.g. Nicholson et al. 1996). An object subsequently identified as
Prometheus was detected although it was 19.7 from the position expected (Bosh
and Rivkin 1995, Bosh and Rivkin 1996). Prometheus orbits between the narrow
F ring and the outer edge of the A ring.
Further observations made during the August and November events again sho-
wed that Prometheus was lagging behind its predicted location, by 18.74 and
18.82 respectively (Nicholson et al. 1996). An error in the Voyager ephemeris was
deemed unlikely. Nicholson et al. (1996) showed that for the August and November
events the fitted orbits for Mimas, Janus, Epimetheus and Pandora all agreed with
their previously determined elements. Furthermore their mean longitudes were all
within 1 of their expected values, consistent with the propagation of small errors
16
CHAPTER 1. INTRODUCTION 17
in the mean motions in the 15 years since the Voyager encounters. Table 1.1 shows
our reconstruction of the expected positions of Mimas, Prometheus and Pandora
as compared to their observed locations during the Earth crossing of the saturnian
Table 1.1: Comparison of predicted and observed mean longitudes at epoch, λ0,for satellites during the saturnian ring plane crossing of August 1995. The epochis 2449940.0 JED. The observed mean longitudes, λ0,obs are from Nicholson et al.
(1996), the calculated mean longitudes, λ0,calc, for Prometheus and Pandora arefrom Jacobson (private communication) while that for Mimas is from Harper andTaylor (1993).
Satellite Predicted Observed ∆λ
λ0,calc λ0,obs (λ0,calc − λ0,obs)
Mimas 176.07 177.11 ± 0.05 −1.04
Prometheus 358.32 339.23 ± 0.10 −19.09
Pandora 95.59 95.68 ± 0.18 −0.09
ring plane on August 10th 1995. Janus and Epimetheus have not been included in
Table 1.1 due to the difficulty in calculating their expected longitudes accurately
in 1995 because of their co-orbital configuration. Nicholson et al. (1996) state that
∆λJanus = −0.33 and ∆λEpimetheus = 0.76.
The error associated with the calculated mean longitude of a satellite at time t
can be estimated from Eqn. 2.12 (Section 2.1),
λ = n (t− t0) + λ0
where t0 is the epoch, λ0 the mean longitude at epoch and n the mean motion
determined at epoch. The relationship between the standard error in λ and the
standard errors in n and λ0 is given by
σλ =√
(σnt)2 + (σλ0
)2 (1.1)
Eqn. 1.1 is based on the calculation of errors from Squires (1976). Table 1.2 is
constructed using the mean motions and mean longitudes at the Voyager 2 epoch
of Prometheus and Pandora. Eqns. 2.12 and 1.1 are used to calculate the expected
mean longitudes during the August 10th 1995 ring plane crossing to illustrate the
maximum expected size of the deviations between the two epochs.
CHAPTER 1. INTRODUCTION 18
Table 1.2: Illustration of expected errors in mean longitudes at 2449940.0 JEDbased on Voyager values of mean motion and mean longitude at epoch. The valuesfor λ0 and n are from Jacobson (private communication) the error in n is fromSynnott et al. (1983) and the error in λ0 is an estimate based on Smith et al. (1981)and Smith et al. (1982)
Satellite λ0 n λ
Prometheus 188.54 ± 0.20 (587.2892 ± 0.0005)/day 358.32 ± 2.56
Pandora 82.15 ± 0.20 (572.7889 ± 0.0005)/day 95.59 ± 2.56
Table 1.2 clearly shows that the errors in the mean longitudes for Prometheus
and Pandora in 1995, based on Voyager epoch elements, could be as high as ∼2.6. The maximum expected errors in the mean longitudes of the other satellites
should be of approximately the same size. Table 1.1 along with ∆λJanus = −0.33
and ∆λEpimetheus = 0.76 (Nicholson et al. 1996) shows that the observed satellites
were all within the error bars of their expected mean longitudes during the ring
plane crossing event in August 1995, except for Prometheus. The lag between
Prometheus’ predicted and observed position was ∼ 19, a difference of ∼ 8σ.
Analysis of Hubble Space Telescope (HST) observations during the oppositions
of 1996, 1997 and 1998 showed that this lag was increasing by approximately
0.6 deg /year (French et al. 1998). Additional observations obtained in 1994, 1999
and 2000 when combined with the ring plane crossing data from 1995 and the 1996,
1997 and 1998 HST observations indicate that Prometheus’ lag is increasing by
0.57year−1 (French et al. 2000).
With the observed positions of Pandora, Janus, Epimetheus and Mimas all
agreeing with their previously determined elements (being well within the expected
errors (Table 1.2)) the ∼ 19 lag in Prometheus’ mean longitude from the expected
value based on the Voyager orbit was real and required explanation.
1.2 The initial research goal
There are two possible explanations for the observed change in the orbit of Prome-
theus:
CHAPTER 1. INTRODUCTION 19
1. The initial orbit determinations of Synnott et al. (1983) and Jacobson (private
communication) for the Voyager epoch are incorrect.
2. The Voyager epoch elements are correct and Prometheus’ orbit had changed
between Voyager and the ring plane crossing in 1995.
The initial goal of this research is to address the problem presented by the anomalous
motion of Prometheus. The task of determining which of the two explanations
are correct is approached by re-examining the Voyager data set and obtaining
the elements of Prometheus’ orbit at the Voyager 1 and Voyager 2 epochs both
separately and for a combined Voyager result. It is hoped that the inclusion of
additional images, not used by Synnott et al. (1983), will lead to a more accurate
determination of Prometheus’ orbital elements. If the derived orbits are comparable
with those of Synnott et al. (1983) and Jacobson (private communication) then the
simplest explanation would be that Prometheus’ orbit had indeed changed. If the
Voyager epoch orbit is indeed accurate then the mechanism responsible for changing
the orbit of Prometheus could have operated in the interval between the Voyager
1 and Voyager 2 encounters. Such a change in orbital elements would make the
use of combined data ill-advised. In this case any significant difference between
the elements at the two epochs would place important constraints on any model
explaining the origin of the lag.
1.3 Possible origins of Prometheus’ lag
What mechanisms could account for the observed difference in the orbit of Prome-
theus between the Voyager epoch and HST observations from 1994 to 2000?
Any model for explaining the origin of Prometheus’ lag must reproduce both
the observed mean longitudes and mean motions at all epochs simultaneously. Just
fitting the mean longitudes or the mean motions is insufficient.
1.3.1 A co-orbital companion
Prometheus could be maintaining a small co-orbital companion satellite in a horse-
shoe orbit (Murray and Giuliatti-Winter 1996, Nicholson et al. 1996). In such a
CHAPTER 1. INTRODUCTION 20
configuration each satellite has two distinct values for its mean motion, n. For half
the time the companion satellite orbits exterior to Prometheus’ orbit with
ncompanion = n0 − ∆ncompanion (1.2)
and the other half of the time interior to the orbit of Prometheus with
ncompanion = n0 + ∆ncompanion (1.3)
The periodic changes in ncompanion would occur near the times of closest approach
between Prometheus and the companion. Conservation of orbital angular momen-
tum leads to Prometheus experiencing similar changes in mean motion. When the
companion is exterior to Prometheus’ orbit
nPrometheus = n0 + ∆nPrometheus (1.4)
and when it is interior
nPrometheus = n0 − ∆nPrometheus (1.5)
The values of ∆nPrometheus, ∆ncompanion and n0 are constants. The value of nPrometheus
could remain constant for years before changing to a different value, remaining
constant at this new value for years before switching back once again to the original
value. With the whole cycle repeating on a regular basis. If Prometheus’ mean
motion switched to a lower value after Voyager and back again just before 1995 the
∼ 19 lag could be accounted for. For a full discussion of co-orbital satellites see
Dermott and Murray (1981a, 1981b).
1.3.2 Periodic encounters with the F ring
Murray and Giuliatti-Winter (1996) pointed out that Prometheus should experience
encounters with the F ring every 19 years. The apocentre of Prometheus’ orbit being
exterior to the pericentre of the F ring’s orbit with differential precession bringing
the two into alignment every ∼ 19 years.
A collision with an object, or objects, in the F ring between 1981 and 1995 could
quite easily lead to a reduction in Prometheus’ mean motion thus accounting for
the observed lag.
CHAPTER 1. INTRODUCTION 21
1.3.3 Cometary impact
The impact of a comet, or asteroid, between 1981 and 1995 could have resulted in a
reduction in Prometheus’ mean motion, accounting for the observed lag (Nicholson
et al. 1996). The end result is identical to that from the F ring encounter scenario
(subsection 1.3.2).
1.3.4 Gravitational interaction with the F ring
Prometheus’ could be undergoing gravitational interactions with F ring material,
either clumps or embedded moonlets, producing a ‘pseudo-random’ walk in its orbit
(Showalter 1999).
1.3.5 Other mechanisms
Gross errors in the Voyager ephemeris, back reaction from density wave torques
and secular effects are all considered by Nicholson et al. (1996).
1.4 Pandora develops a lag
While Pandora occupied its expected mean longitude during the ring plane crossings
of 1995 (Nicholson et al. 1996), data published after the start of this research project
indicated that its motion too was anomalous (French et al. 1998, French al. 2000).
Observations using the Hubble Space Telescope in the period 1994-2000, along with
the 1995 ring plane crossing data, indicated that Pandora was lagging behind its
Voyager predicted position by −1.27year−1 (French et al. 2000). There was also
an additional oscillatory component to the lag with a period of ∼ 600 days and
amplitude ±0.78, consistent with the perturbation due to Pandora’s proximity
to the 3:2 co-rotation eccentricity resonance (CER) with Mimas (see for example
Murray and Dermott 1999).
In light of this new information it was decided to extend the investigation of
the orbit of Prometheus to include Pandora. An orbit changing from its calculated
orbital elements implies the action of some ‘unseen’ mass which hasn’t been allowed
for in the orbit model used. If Prometheus is being perturbed by an ‘invisible’ mass
CHAPTER 1. INTRODUCTION 22
in the F ring region, the mass could have observable effects on Pandora as well.
It seems logical to further extend the goals to include the orbit of the other small
satellite in the F ring region, Atlas. There have been no observations of Atlas since
Voyager 2 in 1981, although Bosh and Rivkin (1996) did identify an object as Atlas
which was unsupported by other authors.
French et al. (2000) have stated that the observed lag in the orbit of Pandora
is entirely consistent with the effects of the 3:2 co-rotation eccentricity resonance
with Mimas. The perturbations due to this resonance are investigated in detail in
Section 7.8.
1.5 The accuracy of determined orbits
When planning a series of observations of satellites it is essential to know how
many images will be needed to achieve a desired result. For the Cassini spacecraft
with many instrument packages and even more investigators competing for limited
resources (time and memory space) observations have to be prioritised.
Is the scientific case for making a particular set of observations strong enough?
Does another instrument team want to make observations at the same time but
pointing in a different direction? Is there enough memory space available on the
spacecraft’s solid state recorders?
The case for a particular set of observations is strengthened if it can be shown
exactly how many individual observations are required to achieve a desired result.
In the case of orbits, this is how many observations are necessary to achieve a
specific accuracy for determined orbital elements.
We investigate the dependency of the accuracy of a determined orbit with a
number of factors. We specifically use proposed Cassini observations of small sat-
urnian satellites to investigate this dependency.
1.6 Summary
We will search the Voyager dataset for images of Atlas, Prometheus and Pandora.
Orbits will be determined for each of these satellites using the Voyager 1 and Voy-
CHAPTER 1. INTRODUCTION 23
ager 2 images both separately and in combination. Any differences in the derived
orbital elements at the two Voyager epochs will be discussed.
In light of the elements derived, possible explanations for the observed lags in
the orbits of Prometheus and Pandora will be evaluated. The determinations of
the orbits of Atlas, Prometheus and Pandora from Voyager images are discussed
separately in Chapters 5, 6 and 7.
The precessing ellipse model of Taylor (1998) will be used to investigate the rela-
tionship between the accuracy of determined orbital elements for saturnian satellites
and various parameters. These parameters include semi-major axis, observational
error, distance from Saturn, number of images and sub-spacecraft latitude. The
theoretical accuracy of determined elements using observation sets proposed for the
Cassini mission will be discussed.
Chapter 2
Orbits and Orbit Determination
2.1 The geometry of elliptical orbits
To a good approximation, all orbiting objects move in elliptical paths around their
centres of motion (Newton 1687). Objects that are not in bound orbits have
parabolic or hyperbolic trajectories e.g. some comets. We are only interested in
bound and thus approximately elliptical orbits. Fig 2.1 illustrates the geometry
of an ellipse in a plane. The term longitude is used in celestial mechanics for an
angle which has been measured with respect to a reference line which is fixed in
inertial space. In Fig 2.1 the angle θ is called the true longitude and is measured
with respect to a fixed reference direction. The polar equation for an ellipse is
r =a (1 − e2)
1 + e cos (θ −)(2.1)
where r is the distance from the ‘filled’ focus to the point on the ellipse with true
longitude θ. The pericentre and apocentre are respectively the points on the ellipse
that are closest to and furthest from the ‘filled’ focus. The angle f , called the true
anomaly, is simply θ −, i.e. the angle as measured from the radius vector to the
pericentre. Eqn. 2.1 can also be written as
r =a (1 − e2)
1 + e cos (f)(2.2)
In one orbital period, T , the true longitude increases from θ to θ+2π (or θ− 2π for
retrograde motion). The angular velocity, f , of the radius vector from the ‘filled’
24
CHAPTER 2. ORBITS AND ORBIT DETERMINATION 25
apocentrepericentre
‘empty’ focus ‘filled’ focus
position of object
reference direction
ae f
x
yr
a
b
ϖ
θ = 0
θ = ƒ + ϖ
Figure 2.1: The geometry of an ellipse of semi-major axis a, semi-minor axis b,eccentricity e and longitude of pericentre .
focus to the object is not constant (Kepler’s second law). However, we can define a
‘mean’ angular velocity, called the mean motion, n, as
n =2π
T(2.3)
. A more general form of which is
n =
√
µ
a3(2.4)
which is a form of Kepler’s third law of planetary motion, where µ = G×Mass . G
is the Universal gravitational constant and Mass is the mass of the object at the
centre of motion i.e. the body being orbited.
Some way is needed to uniquely locate the position of an object on the ellipse
at a specific time. The only variable in Eqn. 2.1 that is a function of time is θ.
Unfortunately since the angular velocity, θ, is not constant θ does not vary linearly
with time. What is needed is an angle that is not only 2π periodic but also varies
CHAPTER 2. ORBITS AND ORBIT DETERMINATION 26
linearly with time. The angle which we use is the mean anomaly,M , which is defined
in terms of the mean motion, n, and the time when the object passes through the
pericentre, τ , by
M = n (t− τ) (2.5)
The mean anomaly, M , has no simple geometrical representation but it can be
related to an angle that does. This angle is the eccentric anomaly, E. Consider
E f
ellipse
circumscribed circle
centre of circle focus of ellipse
a
r
x
y
object
Figure 2.2: Definition of the eccentric anomaly E.
an ellipse with semi-major axis a and eccentricity e. Place a circumscribed circle
of radius a concentric with the ellipse. A line is extended from the major axis
of the ellipse through the point occupied by the orbiting object, perpendicular to
the major axis of the ellipse, intersecting the circle. The eccentric anomaly, E, is
defined as the angle between the major axis of the ellipse and the radius vector
from the centre of the circle to the intersection point on the circle’s circumference.
The geometry is illustrated in Fig 2.2. Converting the polar coordinates (r, θ) into
cartesian coordinates (x, y) is accomplished using
x = a (cosE − e) (2.6)
CHAPTER 2. ORBITS AND ORBIT DETERMINATION 27
y = a√
1 − e2 sinE (2.7)
The mean and eccentric anomalies are related through Eqn. 2.8, Kepler’s equation.
M = E − e sinE (2.8)
At this stage a final angle is introduced, the mean longitude, λ, which is defined by
λ = M + (2.9)
Since λ depends only on M and the constant it is a linear function of time which
has no geometrical interpretation. Combining Eqns. 2.5 and 2.9 gives
λ = n (t− τ) + (2.10)
if at time t = t0, λ = λ0 then using Eqn. 2.10
λ0 = n (t0 − τ) + (2.11)
subtracting Eqn. 2.11 from Eqn. 2.10 gives
λ = n (t− t0) + λ0 (2.12)
In celestial mechanics a fixed time, called an epoch, is used when defining orbits and
reference frames. The value of an orbital variable, usually called an orbital element,
at the chosen epoch will be denoted by the subscript 0. So the mean longitude at
epoch is λ0, the mean anomaly at epoch M0 and so on.
In a system consisting of only two bodies, which always orbit each other in the
same plane, there is no need to extend a 2-dimensional representation of an elliptical
orbit as illustrated in Fig 2.1 to a 3-dimensional one. When a system contains three
or more bodies, the orbits are usually not confined to a single plane and a fixed
reference plane therefore needs to be defined. The introduction of a reference plane
and the move from a 2- to a 3-dimensional representation of elliptical orbits requires
the introduction of an additional 3 orbital variables. The angle of inclination of the
orbital plane to the reference plane is the inclination of the orbit, i. The line of
intersection of the orbital plane and the reference plane is the line of nodes. The
point in both planes where the orbit crosses from below the reference plane to above
CHAPTER 2. ORBITS AND ORBIT DETERMINATION 28
it is the ascending node. The angle between the reference direction, which lies in
the reference plane, and the radius vector to the ascending node is the longitude of
ascending node, Ω. The angle between the radius vector to the ascending node and
the radius vector to the pericentre is the argument of pericentre, ω. The geometry
of an elliptical orbit in 3-dimensions is illustrated in Fig 2.3.
planereference
orbit plane
referencedirection
pericentre
ascending node
object
Ωω
f
i
direction oforbitalmotion
Figure 2.3: The geometry of an orbit in 3-dimensions.
The inclination always lies in the range 0 ≤ i ≤ 180. The motion is prograde if
i < 90 and retrograde if 90 ≤ i ≤ 180. The longitude of pericentre, , is defined
as
= Ω + ω. (2.13)
When i = 0, is the angle between the reference direction and the radius vector
to the pericentre.
It is usually stated in the literature that the unperturbed elliptical orbit of a
body is defined by six orbital elements (e.g. Danby 1992 and Roy 1988). For these
six orbital elements we will use a, λ, e, , i and Ω. Knowledge of these six orbital
elements at a particular epoch does indeed uniquely define the position of the body
at that epoch but it does not allow the position of the body to be calculated at
any time. A seventh orbital parameter is required in order to determine the body’s
CHAPTER 2. ORBITS AND ORBIT DETERMINATION 29
position at any time. Examination of Eqn. 2.12 clearly shows that the calculation of
the mean longitude, λ, at any time given the mean longitude at epoch, λ0, requires
the mean motion, n. We will use n as the seventh orbital parameter. The mean
motion as the seventh orbital parameter is usually overlooked. Kepler’s third law of
planetary motion is often used to calculate the orbital period, T , and mean motion,
n, directly from the semi-major axis, a using Eqns. 2.3 and 2.4. Use of Kepler’s third
law in this way requires that the exact orbital period and semi-major axis already
be known for another body orbiting the same central object as the body of interest.
This criterion is not always satisfied e.g. when Dactyl was discovered orbiting the
asteroid Ida by the Galileo spacecraft in 1993 there were no other satellites, let
alone ones with accurately determined a and T . Knowing the mass of the central
body would also allow Kepler’s third law to be utilised, Eqn. 2.4, but again this
is not always known a priori. Therefore in the most general case an unperturbed
elliptical orbit has six orbital elements and a seventh orbital parameter.
2.1.1 The position of an object in an unperturbed Keplerian orbit
Given the 6 orbital elements (or the six orbital elements and the seventh orbital
parameter in the most general case) of an object at epoch its location at any time
in the past or future can be calculated. Assuming that the mass of the object being
orbited is known, the 6 orbital elements at epoch are a0, λ0, e0, 0, i0 and Ω0. The
epoch is t0 and the time at which the location of the orbiting object is required is t.
The mean motion, n, is calculated using Kepler’s third law, Eqns. 2.3 or 2.4. Since
we are considering an unperturbed Keplerian ellipse the only one of the orbital
elements which is time dependent is the mean longitude, λ. All the other elements
retain their epoch values.
Using Eqn. 2.12 the value of λ at time t is calculated,
λ = n (t− t0) + λ0
then Eqn. 2.9 is used to determined the mean anomaly, M ,
M = λ+0
CHAPTER 2. ORBITS AND ORBIT DETERMINATION 30
Kepler’s equation, Eqn. 2.8, is then solved for the eccentric anomaly, E,
E − e0 sinE = M
Kepler’s equation is transcendental in E and in general it is impossible to express
E as a simple function of M . Kepler’s equation can be solved numerically, e.g. the
Newton-Raphson method, or by using a series solution. The solution of Kepler’s
equation is well covered in the literature (e.g. see Murray and Dermott 1999 or
Danby 1992) and we shall not detail it here.
With E known the x and y coordinates in the cartesian system illustrated in
Fig. 2.2 are calculated using Eqns. 2.6 and 2.7.
x = a0 (cosE − e0)
y = a0
√
1 − e20 sinE
The (x,y) coordinates of the object must then be transformed into a 3 dimensional,
inertial Cartesian reference frame with axes X, Y and Z which is illustrated in
Fig. 2.4. The chosen reference direction lies along the X axis of the reference frame.
Orbital plane coordinates are transformed into reference plane coordinates using
(Murray and Dermott 1999)
X
Y
Z
= r
cos Ω cos(ω + f) − sin Ω sin(ω + f) cos i
sin Ω cos(ω + f) − cos Ω sin(ω + f) cos i
sin(ω + f) sin i
(2.14)
where we have assumed that the object of interest lies in the orbital plane and
r =√x2 + y2.
The pointing direction to the object from an observer at reference frame coor-
dinates Xobserver, Yobserver, Zobserver is P where
Px
Py
Pz
=
X
Y
Z
−
Xobserver
Yobserver
Zobserver
(2.15)
and the unit pointing vector from the observer to the object is P. We are ignoring
the effects of the finite speed of light, c, and planetary aberration at this stage.
CHAPTER 2. ORBITS AND ORBIT DETERMINATION 31
X
Y
Z
x
y
z
pericentre of orbit
orbit plane
referenceplane
i
Ω
ω
Ascending node
reference direction
x
X reference plane
orbit plane axes
axes
directionof motion
f
object
Figure 2.4: Illustration of relationship between orbit plane coordinates x, y, z andreference frame coordinates X, Y, Z.
Following the described procedure allows the position of an object in an unper-
turbed Keplerian orbit to be calculated at any time. The pointing vector to the
object from an observer can then be calculated providing the coordinates of the ob-
server is known in reference frame coordinates. If the observer is also in a Keplerian
orbit its position in reference frame coordinates must also be determined.
2.1.2 Orbital and reference frames
An object’s orbital frame has its origin at the centre of the object’s motion. In the
Solar System it is usually the Sun, heliocentric, for planets, asteroids and comets.
For planetary satellites it is the parent planet, planetocentric. In certain cases using
the centre of mass, the barycentre, of a system of objects as the origin provides a
better representation of the orbit. For Atlas, Prometheus and Pandora a planeto-
CHAPTER 2. ORBITS AND ORBIT DETERMINATION 32
centric origin provides a much better fit to the data.
The object’s velocity vector, along with the centre of its motion, defines the
orbital plane. The x- and y-axes are in the orbital plane with the x-axis point-
ing towards some pre-chosen reference direction, we use the radius vector to the
pericentre (after Murray and Dermott 1999). The z-axes is perpendicular to the
orbital plane, parallel to the angular momentum vector of the object. The y-axis is
orthogonal to both x and z as defined by the vector cross-product y = z × x. The
orbital frame is shown in Fig. 2.5 As long as , Ω and i remain constant the orbital
x
y
z
orbital plane
pericentredirection of motion
centre of motion
Figure 2.5: The orbital frame for a body.
frame will remain fixed in inertial space. When a system contains more than two
CHAPTER 2. ORBITS AND ORBIT DETERMINATION 33
bodies, as do all real world systems, mutual interactions between the bodies cause
the orbital elements to change, rotating the orbit and its orbital frame through in-
ertial space. This change in the orbital frame due to the effects of the perturbations
of additional objects is called precession.
The z coordinate of a body in its own orbital frame is always z = 0 but other
bodies may have z 6= 0 in that particular frame. The orbital frame of one body
usually will not coincide with that of another, although the orbital planes of bodies
within a system are often nearly co-planar.
Reference planes are chosen such that the X-axis is parallel to a fixed vector,
the reference direction. The Z-axis is parallel to another fixed vector which is
orthogonal to the reference direction with the Y-axis defined by the vector cross-
product Y = Z×X. The J2000 reference frame has the X-axis parallel to the vector
from the centre of the Earth pointing towards the intersection of the ascending
node of the mean Earth equator and the ecliptic at J2000 ( 2000 JAN 01 12:00:00.0
TDB). The Z-axis is parallel to the spin angular momentum vector of the Earth
at J2000 (i.e. parallel to the North polar axis) as shown in Fig. 2.6. The origin of
x
yz
ecliptic
Earth equator plane
North pole
Figure 2.6: The J2000 reference frame.
CHAPTER 2. ORBITS AND ORBIT DETERMINATION 34
the J2000 frame can be translated through space without changing the orientation
of the cartesian axes. For example, observations in the J2000 frame are centered
on the observer. The position and velocity of an object at a specified time with
respect to an origin is called a state, or state vector. States in the J2000 frame need
to have the origin specified, usually chosen to be the Sun, a planet or the system
barycentre. The J2000 frame is the current standard frame for observations, star
positions and states.
Previous standard frames include the B1950.0 frame, which is similar to the
J2000 frame but uses the mean Earth equator and North polar axis at 1949 DEC
31 22.09:07.2 TDB. All the Voyager data has observations and states in the B1950
reference frame. Transforming from one reference frame to another involves a simple
rotation.
2.2 Osculating and geometric elements
A body moving in a Keplerian orbit, or Keplerian ellipse, obeys Kepler’s three laws
of planetary motion exactly.
1. Objects move in ellipses with the centre of motion at one focus.
2. A radius vector from the centre of motion to the orbiting object sweeps out
equal areas in equal times.
3. The square of an object’s orbital period is proportional to the cube of its
semi-major axis.
For a Keplerian orbit, the conversion of orbital elements to states and vice versa
is straightforward (e.g. see Taylor 1998 or Danby 1992). The osculating elements of
an orbit are those calculated from an instantaneous state assuming perfect Keplerian
motion. For a system consisting of only two point masses the osculating elements
describe the exact trajectory followed by one mass with respect to the other. This
was demonstrated by Newton (1687).
When there are three or more masses in the system, the masses do not follow
perfect Keplerian curves due to the perturbations of the other objects as proved by
Poincare (1892-1899). In the n-body case (n > 2) the osculating elements do not
CHAPTER 2. ORBITS AND ORBIT DETERMINATION 35
describe the exact trajectory followed by the objects. The exact trajectory followed
by an object is described by its geometric elements. The geometric elements take
into account the perturbations of other masses in the system or the effects of an
oblate mass (Greenberg 1981). The major differences are usually in n and a while
and Ω are no longer constant but precess.
For the planets the osculating elements describe the actual trajectories very
well. For planetary satellites the effects of oblate primaries make the osculating
elements an inadequate description of the trajectories. Throughout this work all
orbital elements will be the geometric elements unless specifically stated otherwise.
2.3 Orbit determination
The problem of determining the orbit of an object divides naturally into two parts.
1. The determination of a preliminary, approximate orbit from a minimal number
of observations.
2. Improving the preliminary orbit by using as many observations as possible.
A very common problem in celestial mechanics is finding the orbit of a newly discov-
ered asteroid or comet using a very small number of observations. The preliminary
orbit is used to predict the object’s position in the future so that additional obser-
vations can be made thus allowing the orbit to be improved.
The two main methods used for preliminary orbit determination were developed
from the techniques of Laplace and Gauss. These approaches, called Laplace’s
method and Gauss’ method after their originators, are extensively described in the
literature (e.g. Escobal 1965, Roy 1988, Marsden 1985 and Danby 1992) and so
will only be summarised here. In both Laplace’s and Gauss’ method of initial orbit
determination the position, velocity and acceleration (R, R and R) of the observer
is known at all times.
CHAPTER 2. ORBITS AND ORBIT DETERMINATION 36
2.4 Preliminary orbit determination
2.4.1 Laplace’s method
Consider an observer and an object with position vectors R and r with respect to
the centre of motion of the object. The position vector of the object with respect
to the observer is p as in Fig. 2.7. The relationship between the vectors R, r and
R
r
Observer
Object
Central body e.g. the Sun
p
Figure 2.7: Laplace’s method
p is
p = r −R (2.16)
or
pp = r − R (2.17)
where the position of the observer, R, and the unit pointing vector to the object,
p are known. The two unknowns are the radial vector to the object, r, and the
observer-object separation, p. Squaring Eqn. 2.17 leads to
p2 = r2 − 2r · R +R2 (2.18)
CHAPTER 2. ORBITS AND ORBIT DETERMINATION 37
differentiating Eqn. 2.17 with respect to time gives
p ˙p + pp = r − R (2.19)
and differentiating again gives
p¨p + 2p ˙p + pp = r − R (2.20)
A minimum of three closely space observations are required, i.e. three values of
p. From three observations the values of ˙p and ¨p can be calculated numerically
using Lagrangian interpolation (Escobal 1965), where t2 is taken to be 0.
p(t) =t(t− t3)
t1(t1 − t3)p1 +
(t− t1)(t− t3)
t1t3p2 +
t(t− t1)
t3(t3 − t1)p3 (2.21)
with
˙p =(2t− t3)
t1(t1 − t3)p1 +
2t− t3 − t1t1t3
p2 +2t− t1
t3(t3 − t1)p3 (2.22)
and
¨p =2
t1(t1 − t3)p1 +
2
t1t3p2 +
2
t3(t3 − t1)p3 (2.23)
where pi and ti (i = 1, 2, 3) are the unit pointing vectors and times of the three
observations and p(t) is the pointing vector at time t. Eqn. 2.20 is multiplied by
(p× ˙p) and utilising the fact that
(p × ˙p) · p = 0 (2.24)
(p × ˙p) · ˙p = 0 (2.25)
becomes
p¨p · (p× ˙p) = (p × ˙p) · r − (p× ˙p) · R (2.26)
Using Newtonian gravitation
r = −GMr3
r (2.27)
so
(p× ˙p) · r = −GMr3
(p× ˙p) · r (2.28)
which using Eqn. 2.17 becomes
(p× ˙p) · r = −GMr3
(p × ˙p) · (pp + R) (2.29)
CHAPTER 2. ORBITS AND ORBIT DETERMINATION 38
or
(p× ˙p) · r = −GMr3
(p(p × ˙p) · p + (p× ˙p) · R) (2.30)
which since (p× ˙p) · ˙p = 0 is
(p× ˙p) · r = −GMr3
(p× ˙p) · R (2.31)
Substituting Eqn. 2.31 back in Eqn. 2.26 gives
p¨p · (p × ˙p) = −GMr3
(p× ˙p) · R − (p × ˙p) · R (2.32)
Eqns. 2.32 and 2.18 are a coupled set of equations with two unknowns, r and p,
which are usually solved iteratively. Once p is known r immediately follows from
Eqn. 2.17 and r from Eqn. 2.19.
2.4.2 Gauss’ method
Again considering the geometry of object, observer and central body as illustrated
in Fig. 2.7. Let the radius vector of the object at three times be r1, r2 and r3. Since
the plane of the object’s orbit remains constant (assuming no perturbations) there
are scalars c1 and c3 such that
r2 = c1r1 + c3r3 (2.33)
The constants c1 and c3 can be calculated using Gauss’ sector-triangle ratios (the
following is from Danby 1992). Let [ri, rj] represent the area of the triangle formed
by ri and rj. Let the area swept out by the radius vector in moving between ri and
rj be (ri, rj). The constants c1 and c3 are
c1 =|r2 × r3||r1 × r3|
(2.34)
=[r2, r3]
[r1, r3](2.35)
c3 =|r1 × r2||r1 × r3|
(2.36)
=[r1, r2]
[r1, r3](2.37)
the sector-triangle ratios y1, y2 and y3 are defined such that
y1 =(r2, r3)
[r2, r3](2.38)
CHAPTER 2. ORBITS AND ORBIT DETERMINATION 39
y2 =(r1, r3)
[r1, r3](2.39)
y3 =(r1, r2)
[r1, r2](2.40)
Eqn. 2.36 can be written
c1 =[r2, r3]
[r1, r3]
(r2, r3)
(r2, r3)
(r1, r3)
(r1, r3)(2.41)
which using Eqns. 2.39 and 2.40 becomes
c1 =(r2, r3)
(r1, r3)
y2
y1
(2.42)
Using Kepler’s second law of planetary motion, a radius vector sweeps out equal
areas in equal times i.e. area = kt where k is a constant. Eqn. 2.42 becomes
c1 =k(t3 − t2)
k(t3 − t1)
y2
y1
or
c1 =(t3 − t2)
(t3 − t1)
y2
y1(2.43)
and in a similar way
c3 =(t2 − t1)
(t3 − t1)
y2
y3(2.44)
Using Eqn. 2.17 in Eqn. 2.33 we obtain
c1p1p1 − p2p2 + c3p3p3 = c1R1 −R2 + c3R3 (2.45)
Eqn. 2.45 is a coupled set of three equations with three unknowns and can be solved
iteratively. Initial values of c1 and c3 are calculated assuming that the sector-triangle
ratios are exactly equal to one so
c1 =t3 − t2t3 − t1
c3 =t2 − t1t3 − t1
and p1, p2 and p3 solved for. The corresponding values of ri are obtained from
Eqn. 2.17. The values of c1 and c3 at the next iteration are calculated using
Eqns. 2.43 and 2.44 with y1, y2 and y3 determined using ri from the previous
iteration. The determination of the sector triangle ratios, yi, is covered in great
detail in Danby (1992). Once the solution has converged r1, r2 and r3 are known.
CHAPTER 2. ORBITS AND ORBIT DETERMINATION 40
2.4.3 The f and g functions
Laplace’s and Gauss’ method only calculate the position of the object, not its
velocity. The velocity is obtained from the position using the f and g functions.
For an object in a Keplerian orbit the radius vector at time t can be expressed in
terms of the radius and velocity vectors at time t0
r(t) = fr0 + gv0 (2.46)
Given the f and g functions and the radius vectors at two times, the velocity at one
of those times can be calculated. The f and g functions can be calculated using
the radius vectors, the sector-triangle ratios and µ as part of Gauss’ method as
described by Danby (1992). The less accurate truncated f and g series have to be
used to obtain the velocities using Laplace’s method
f = 1 − 1
2
µ
r30
(t− t20) + . . . (2.47)
g = (t− t0) −1
6
µ
r30
(t− t0)3 + . . . (2.48)
The additional terms in the two series include the velocities so have to be discarded.
When both the position and velocity of the object are known its osculating orbital
elements can be calculated (section 2.2).
2.4.4 The suitability of Laplace’s and Gauss’ methods for planetary satellites
Laplace’s method may be used for any number of observations, with the minimum
number being three. In its original form it could only be used for three observations,
Herget (1948) suggested that the inclusion of a fourth observation would improve the
accuracy of the orbit determination. Lagrangian interpolation allows any number
(≥ 3) of observations to be utilised (Escobal 1965). The greater the number of
observations used the more accurate the determination of the orbit. Laplace’s
method requires that the observations be made very close together, each observation
occurring a short time after the previous one. This is not a problem for observations
of asteroids or planets which have orbital periods of years. Observations separated
by days are typically ∼ 0.01 of an orbital period apart, close enough for Laplace’s
method to be valid. When a satellite with an orbital period of ∼ 0.5 days is being
CHAPTER 2. ORBITS AND ORBIT DETERMINATION 41
observed from a spacecraft it is rarely possible to take a series of observations say
every 12 minutes for an entire orbital period. Successive observations of a particular
satellite may be separated by several orbital periods. In this case Laplace’s method
is not valid.
The use of the truncated f and g series to calculate the velocities introduces
additional errors into the derived orbital elements. For asteroids, comets and planets
the effects of an oblate primary (section 2.2) are not a consideration. For planetary
satellites, particularly those of the giant planets, the effects of the oblate planet on
the orbit of a satellite are significant.
Gauss’ method may only be used for exactly three observations. If a large data
set is available different combinations of three observations would have to be used
in turn and a mean set of elements determined. It has the advantage that the f
and g functions are used to determine the velocity not the truncated f and g series.
Successive observations do not have to be closely spaced, they can be separated by
several orbital periods. Again the effects of an oblate primary are not accounted
for.
Only states and osculating elements can be obtained from Laplace’s or Gauss’
method. As discussed in section 2.2 the actual geometric shape of a satellite’s orbit
around an oblate primary is not accurately described by osculating elements.
There is no guarantee that Laplace’s or Gauss’ method will work for a particular
set of observations. Sometimes the solutions just will not converge. We programmed
Gauss’ method to determine orbits from three observations. In tests the determined
orbits of objects were fairly accurate unless one or more of the observations was
made from a position further away from the primary than the object. In this case
the solution failed to converge. We were unable to determine whether this is an
intrinsic flaw in the method or a fault with the algorithm used. Since spacecraft
observations of satellites are generally made from positions exterior to the satellite’s
orbit this difficulty is a great limitation.
Laplace’s and Gauss’ method are very useful where there is a minimal set of ob-
servations available. Where there are large data sets they are incapable of achieving
accuracies commensurate with the number of observations. These methods are use-
CHAPTER 2. ORBITS AND ORBIT DETERMINATION 42
ful for making an initial orbit determination. When there are more than a minimal
number of observations other techniques model the actual dynamics with greater
accuracy.
2.4.5 Determining the orbits of planetary satellites from spacecraft
For most planetary satellites assuming a circular, prograde, equatorial orbit pro-
vides sufficiently accurate starting conditions for improving an orbit. To date all
planetary probes have approached a planet in a trajectory inclined to the planet’s
equatorial plane. In this situation one observation is sufficient to determine a cir-
cular, prograde, equatorial orbit.
A difficulty with many spacecraft (e.g. the Voyager probes) is that they do not
orbit a planet but simply fly by. This limits the time over which observations can be
made. The Voyager observations of Prometheus only span ∼ 30 days. Fortunately
this will not be a difficulty for Cassini with a four year tour planned.
2.5 Improving the orbit
Once an approximate orbit is known for an object the only way to improve the
orbit, so that it more accurately represents the actual dynamics of the system
under consideration, is to obtain additional observations, the more the better. The
available techniques for orbit improvement are of two basic types
1. numerical integration of the equations of motion
2. fitting a precessing elliptical model
and both types are in use. Papers which publish orbital elements for satellites of
the outer planets from Voyager data using a precessing elliptical model fit to obser-
vations include Smith et al. (1981), Smith et al. (1982), Synnott et al. (1983), Owen
and Synnott (1987) and Owen et al. (1991). Others which numerically integrate
the equations of motion to fit to the observational data are Jacobson et al. (1991),
Jacobson et al. (1992) and Jacobson (1998a,1998b).
CHAPTER 2. ORBITS AND ORBIT DETERMINATION 43
2.5.1 Integration of the full equations of motion for an entire system
The gravitational force, Fij , acting on a point mass mi due to a point mass mj is
given by Newton’s universal law of gravitation (Newton 1687),
Fij =Gmimj(rj − ri)
|(rj − ri)|3(2.49)
where ri and rj are the position vectors of mi and mj with respect to some non-
accelerating origin in inertial space. The order of the indices is important. We are
modeling the effect on the object denoted by the first index caused by the object
denoted by the second index. Eqn. 2.49 can be extended to include the effects of
more than one mass acting on mi,
Fij =n∑
j=1,j 6=i
Gmimj(rj − ri)
|(rj − ri)|3(2.50)
where n is the total number of masses in the system. Dealing with accelerations
instead of forces Eqn. 2.50 is,
ri =n∑
j=1,j 6=i
Gmj(rj − ri)
|(rj − ri)|3(2.51)
Eqn. 2.51 is the n-body problem which is not generally soluble analytically. It is
however soluble numerically. Given the states (positions and velocities) and masses
of the n bodies at a particular epoch it is possible to calculate their states at
any time in the past or future. Planets and satellites are not point masses and
the effects of their shapes on their gravitational fields has to be accounted for. A
common method is to model them as oblate spheroids which adds the terms Aij
and Aji into Eqn. 2.51 (e.g. Peters 1981).
ri =n∑
j=1,j 6=i
Gmj(rj − ri)
|(rj − ri)|3−Aji + Aij (2.52)
The set of equations made up from Eqn. 2.52 for the n bodies are the full equations
of motion of the system.
Eqn. 2.52 can be solved for the entire n-body system using iterative numerical
integration techniques. The values of ri and ri (i = 1, n) are calculated at each
iteration using the values from the previous step. A traditional scheme for integrat-
ing the equations of motion is the Runge-Kutta 4th order scheme and is extensively
described in the literature (e.g. Danby 1992).
CHAPTER 2. ORBITS AND ORBIT DETERMINATION 44
In recent years much more accurate and powerful techniques have come into use.
The scheme that has been used at QMW is the Runge-Kutta-Nystrom 12th order
scheme of Dormand et al. (1987a, 1987b), specifically written to solve Eqn. 2.52.
Analysis has shown the Runge-Kutta-Nystrom 12th order technique to be extremely
accurate (Hadjifotinou and Harper 1995).
2.5.2 The precessing ellipse model
The orbit of an object is modeled as a simple precessing ellipse. The following
treatment of a precessing elliptical model for a satellite orbit is taken from Taylor
(1998). When inclinations and eccentricities are small the pericentres and nodes of
an orbit are difficult to determine. In such cases in place of the eccentricity, e and
the longitude of pericentre, it is easier to use h and k where
h = e sin (2.53)
k = e cos (2.54)
and p and q instead of the inclination, i and the longitude of ascending node, Ω
where
p = sin i sin Ω (2.55)
q = sin i cos Ω (2.56)
The six orbital elements at anytime are given by
a = a0 (2.57)
λ = λ0 + nt (2.58)
h = h0 cos βt+ k0 sin βt (2.59)
k = k0 cosβt− h0 sin βt (2.60)
p = p0 cos γt+ q0 sin γt (2.61)
q = q0 cos γt− p0 sin γt (2.62)
where a is the orbital semi-major axis, λ is the mean longitude, n is the sidereal
mean motion and β, γ are the apsidal and nodal precession rates respectively. The
terms a0, λ0, h0, k0, p0 and q0 are the values of the orbital elements at epoch.
CHAPTER 2. ORBITS AND ORBIT DETERMINATION 45
Where there are resonant relationships between the mean motions of two or more
satellites additional periodic terms have to be included in the mean longitude. We
neglected these periodic terms for Atlas and Prometheus as they have no significant
resonances with any other detected saturnian satellite. Pandora however, is very
close to a resonance with Mimas and this is taken into account (Chapter 7). It
must be stressed that these are not osculating but ‘geometric’ orbital elements
which include the effects of the primary’s oblateness (Greenberg 1981).
The apsidal and nodal precession rates are calculated directly from a, n the
equatorial radius of the primary, R, and the first two even zonal gravitational
harmonics J2 and J4 which (for small e and i) gives (Murray and Dermott 1999),
β = n
[
3
2J2
(
R
a
)2
− 9
8J2
2
(
R
a
)4
− 15
4J4
(
R
a
)4]
(2.63)
γ = −n[
3
2J2
(
R
a
)2
− 27
8J2
2
(
R
a
)4
− 15
4J4
(
R
a
)4]
(2.64)
The semi-major axis is determined in two ways. In the differential correction process
(see section 2.6) a, λ, n, h, k, p and q are free parameters. The a from the fitting
process is afitted and can be considered to be a scale factor. The a that is actually
used is acalc and is derived from the sidereal mean motion n using (Murray and
Dermott 1999)
n2 =GM
a3calc
(
1 +3
2J2
(
R
acalc
)2
− 15
8J4
(
R
acalc
)4)
+ . . . (2.65)
It is acalc that is used to calculate β and γ in Eqns. 2.63 and 2.64.
Let the position vector of the satellite in the planetocentric frame be O where
(to O(e2) and O(sin2 i)) from Taylor (1998)
OX = a
[
−3k
2+(
1 − 3
8k2 − 5
8h2 − 1
2p2)
cosλ
+(
1
4hk +
1
2pq)
sinλ+1
2k cos 2λ+
1
2h sin 2λ
+3
8
(
k2 − h2)
cos 3λ+3
4hk sin 3λ
]
(2.66)
OY = a[
−3
2h +
(
1
4hk +
1
2pq)
cosλ
+(
1 − 5
8k2 − 3
8h2 − 1
2q2)
sin λ− 1
2h cos 2λ
+1
2k sin 2λ− 3
4hk cos 3λ+
3
8
(
k2 − h2)
sin 3λ]
(2.67)
CHAPTER 2. ORBITS AND ORBIT DETERMINATION 46
OZ = a[
−3
2hq +
3
2kp− p cosλ+ q sin λ
−1
2(hq + kp) cos 2λ+
1
2(kq − hp) sin 2λ
]
(2.68)
where OX points along the intersection of the ascending node of the primary’s
equator at epoch with the Earth mean equator at J2000 and OZ lies parallel to the
primary’s spin angular momentum vector at epoch. Finally, OY is in the plane of
the primary’s equator such that X, Y , Z form a right handed triad. This frame is
illustrated in Fig. 2.8.
planetary equator plane at epoch
mean Earth equatorplane at J2000
ascending node of planetary equator
planet
on mean Earth equator
centre of planet
sense of rotation
O
OO
x
yz
Figure 2.8: Reference frame for precessing elliptical orbit.
The apparent pointing direction at time t is
P (t) = O (t− τ) − S (t− τ) (2.69)
where
τ =| O(t− τ) − S(t) |
c(2.70)
τ is the light travel time, c is the speed of light and S is the planetocentric position
vector of the spacecraft. Eqn. 2.70 has to be solved iteratively to obtain a value
for τ . The apparent pointing direction calculated using Eqn. 2.69 allows for both
light travel time and classical stellar aberration effects (e.g. Hohenkerk et al. 1992).
CHAPTER 2. ORBITS AND ORBIT DETERMINATION 47
We have ignored the effects of gravitational light bending and relativistic stellar
aberration as these are much smaller than the accuracy of our observations.
2.5.3 Generation of an ephemeris
Both techniques for the improvement of orbits have similarities. A set of starting
parameters is needed. These are the seven orbital elements at epoch for the pre-
cessing ellipse and the state vector at epoch for all the bodies in the system along
with their masses (usually G × Mass) for the numerical integration. Additional
parameters can be added to each model. The precessing ellipse can have β and γ
as variables instead of calculating them from Eqns. 2.63 and 2.64. The masses and
J2 and J4 values (J2 and J4 from Campbell and Anderson 1989) can be allowed to
vary for any or all of the bodies under consideration in the numerical model.
Whatever model is being used, a trajectory for an object is calculated using that
model and then the pointing vector to the object from the observer’s location at
the observation times is determined. The calculated trajectory, positions at given
times for the precessing ellipse and state vectors at given times for the numerical
model, constitutes an ephemeris. The calculated pointing vectors are compared to
the actual observations of the object and the residuals calculated. The residual of
an observation is given by
residual = |Pobserved − Pcalculated| (2.71)
where P is the unit pointing vector from the observer to the object. The starting
parameters of the model are changed to reduce the size of the residual. Once the
residual has been minimised the modeled trajectory is the ‘best’ dynamical fit to the
observations given the restrictions and limitations of the model used. The residual
is minimised using a least-squares differential correction technique (section 2.6).
2.6 The differential correction of elements
An orbit is fitted to the observations using a differential correction process. The
following treatment of the differential correction of orbits is from Danby (1992).
It is assumed that the trajectory of a body can be described by some model e.g
CHAPTER 2. ORBITS AND ORBIT DETERMINATION 48
Newtonian Gravitation, Keplerian ellipses etc. All the parameters which are used
in this model are contained in the vector X. Using the model and a given set of
parameters X, the values of a set of observations can be calculated. This set of
observations calculated using X and the model are contained in the vector Y. So
Y is some function of X
Yc = f (X) (2.72)
with f () representing the model. When X is input into the model Y is output. The
subscript c means a value calculated using the model whereas the subscript o means
the actual observed value. By initially assuming that there are no observational
errors, to find Yo given X we need to solve
Yo = f (X) (2.73)
taking an initial estimate for X, call it Xe Eqn. 2.72 gives
Yc = f (Xe) (2.74)
the residual in the observations, y is then
y = Yo −Yc (2.75)
If the model used is accurate and there are no observational errors a correct value
of X exists where
X = Xe + x (2.76)
where x is the difference between the estimated and actual value of X. Using
Eqn. 2.76 in Eqn. 2.73 gives
Yo = f (Xe + x) (2.77)
Now letting the Jacobian matrix J be
J =dY
dX(2.78)
where J, with elements Ji,j, is determined by varying the input parameters X by a
small amount δXj
dYi
dXj=f (Xj + δXj) − f (Xj − δXj)
2δXj(2.79)
CHAPTER 2. ORBITS AND ORBIT DETERMINATION 49
with i being the number in the observations set and j the number of parameters.
Expanding Eqn. 2.77 using Newton’s method and neglecting terms of order x2 or
higher gives
f (Xe + x) ≃ f (Xe) + Jx (2.80)
or
Jx ≃ f (Xe + x) − f (Xe) (2.81)
using Eqns. 2.75, 2.77 and 2.74 and taking Eqn. 2.81 to be precise gives
Jx = y (2.82)
Eqn. 2.82 is solved to give x and x is added to Xe to give an improved estimate for
X as per Eqn. 2.76. This improved value for X is inserted back into Eqn. 2.81 as
Xe and the procedure repeated iteratively until convergence in X achieved. At this
point X = Xe and the differential correction process is completed.
The precessing ellipse, for satellite orbits, is described in a planetocentric carte-
sian reference frame. This frame is defined in a very similar way to the J2000
reference frame (Section 2.1.2). The X-axis is parallel to the vector from the centre
of the planet pointing towards the intersection of the ascending node of the plane-
tary equator at epoch on the mean Earth equator at J2000 ( 2000 JAN 01 12:00:00.0
TDB). The Z-axis is parallel to the spin angular momentum vector of the planet
at epoch. The Y-axis being defined by the vector cross-product Y = Z × X. This
is illustrated in Fig. 2.9. The orbital elements are defined as in Fig. 2.5.
The actual observations have to be converted into the reference frame that is
used for the precessing elliptical model for the differential correction process to
work. If α and δ are the right ascension and declination of the primary’s pole in the
co-ordinate system of the observations (J2000 reference frame for our data) then
PX
PY
PZ
=
cosN − sinN cos J sinN sin J
sinN cosN cos J − cosN sin J
0 sin J cos J
T
px
py
pz
(2.83)
where
N = α + 90
J = 90 − δ
CHAPTER 2. ORBITS AND ORBIT DETERMINATION 50
mean Earth equatorplane at J2000
ascending node of planetary equator
planet
on mean Earth equator
centre of planet
sense of rotation
α
J2000z
J2000 y
J2000x
planetary equatorplane at J2000
X
YZ
90 − δ
Figure 2.9: Reference frame for orbital elements of a satellite.
2.7 Discussion
2.7.1 Integration of the full equations of motion
Provided all the masses that influence a system, along with their state vectors
at epoch, are included and accurate a numerical integration gives a dynamically
correct representation of any system under study. The trajectory of an object from
a numerical integration is as close to the actual trajectory of that object as it is
possible to calculate.
A correctly executed numerical integration models the dynamical characteristics
of an entire system at once. However this means that a lot of information is required
to begin with, accurate masses and states for all the objects at epoch.
It is very difficult to get an accurate feel for an orbit just from a state vector.
It is of course possible to convert the state vector into a set of osculating elements
(section 2.2) but as previously discussed osculating elements are not necessarily a
CHAPTER 2. ORBITS AND ORBIT DETERMINATION 51
good representation of an orbit’s actual shape. The subtle effects on the orbit of a
satellite due to the perturbations of another, while accurately modeled, are difficult
to determine from state vectors.
An approach used by a number of authors is to fit an orbit to observations using
a numerical integration technique and then generate an ephemeris. A precessing
elliptical model is then fitted to this generated ephemeris (e.g. Harper 1987, Harper
and Taylor 1993 and Taylor 1998). The elliptical model can easily be modified to
include terms representing the effects of other bodies in the system. In this the way
subtle, long term effects (e.g. of resonances) can be extracted from the numerically
generated ephemeris.
2.7.2 The precessing ellipse model
The precessing ellipse model is very simple to implement, it requires no great pro-
gramming skill and little computing time. The best fit orbit can be easily visualised
from the seven orbital elements at epoch corresponding to the ‘best’ fit solution. It
can provide a highly accurate fit to an orbit (e.g. see the fits to the JPL ephemerides
in Chapters 5, 6 and 7). Its great disadvantage is that it does not include the ef-
fects of perturbations due to other objects. In many cases these perturbations are
small and average out over a long period of time. Where this is the case the use
of a precessing ellipse model is perfectly valid. Where a resonance occurs however,
the effects of perturbations are significant and periodic, and the precessing ellipse
model can produce inaccurate results. Modifications can be made to account for
perturbations (section 7.8) but only when the magnitudes of the relevant terms are
known beforehand.
2.8 Conclusions
Fitting an orbit to observations using a numerical integration technique provides
the most accurate dynamical representation of the actual trajectory of an object.
It includes subtle effects due to perturbations from other objects in the system.
The precessing ellipse model is easy to program and fit to observations. It does
CHAPTER 2. ORBITS AND ORBIT DETERMINATION 52
not include the effects due to other objects in a system. Where these effects are
small the inaccuracies introduced are also small. In such cases a precessing ellipse
provides an accurate model for the actual dynamics of a system.
For Prometheus, which has no low order resonances with any object in the
saturnian system, a precessing ellipse model is an accurate method of orbit deter-
mination. Since a precessing ellipse is an uncomplicated, valid and accurate way
for modeling the orbit of Prometheus it is used in this work.
The same model is also used for determining the orbits of Atlas and Pandora
as an extension to the primary work on Prometheus. While valid for Atlas the
precessing ellipse requires some modification to accurately model Pandora’s motion
(section 7.8).
Chapter 3
Voyager Images
3.1 Introduction
The majority of the work presented in this thesis uses Voyager images, therefore
a discussion of the Voyager spacecraft and the quirks of Voyager images would be
useful. In this chapter information regarding the Voyager spacecraft in general, and
the Imaging Science Subsystem (ISS) in particular, is presented and discussed. The
techniques for the processing of Voyager ISS images are described in detail.
3.2 Voyager spacecraft and instruments
The two Voyager spacecraft were launched in 1977, Voyager 1 bound for Jupiter
and Saturn and Voyager 2 for Jupiter, Saturn, Uranus and Neptune in the “Grand
Tour”. To date the Voyager project, which is still on-going, is the most ambitious
program of outer Solar System exploration ever undertaken.
Each spacecraft carries an array of instruments which are described in detail in
the published literature (e.g. Danielson et al. 1981, Smith et al 1977 and Thompson
1990). The only instruments of interest for this work are the two cameras of the
Imaging Science Subsystem (ISS).
A Voyager spacecraft carries two cameras, a narrow angle camera (NAC) and
a wide angle camera (WAC). The NAC has a focal length of ∼ 1500 mm and the
WAC a focal length of ∼ 200mm. The corresponding fields of view are 7.4 × 7.4
53
CHAPTER 3. VOYAGER IMAGES 54
mrad for the NACs and 56 × 56 mrad for the WACs. The cameras and other ISS
instrumentation are mounted on a scan platform which allowed them to be pointed
in directions independent of the orientation of the spacecraft bus during planetary
encounters. The cameras could be shuttered simultaneously (called a BOTSIM) or
separately. Since the cameras are boresighted with each other, in BOTSIM mode
the NAC image is a high resolution blowup of the central part of the WAC image.
Each Voyager camera consists of an eight position, off-axis filter wheel, a lens
system, a shutter and a magnetically focused slow-scan vidicon tube with its sup-
plementary electronics. The Voyager camera optics were state of the art in 1977
and indeed compare favourably with modern systems. In fact the optics from one
of the flight spare WACs are now on their way to Saturn as part of the Cassini
wide angle camera. The seventies vintage vidicon tube however does not compare
favourably with modern CCDs.
The lens system brings the image to a focus on the face plate of the vidicon
tube. The face plate is a layer of photosensitive selenium sulphide one inch square.
Charge builds up where photons hit the face plate and the two dimensional image is
stored on the plate as a pattern of electrostatic charge. The vidicon electron beam
then scans the plate line by line, converting the image into a sequence of electrical
signals. In many ways the vidicon system is very similar to a standard photocopier.
The face plate of the vidicon tube is divided into a 800×800 square grid for readout
purposes. However, it is important to remember that the face plate is not actually
physically divided into such a grid in the way that a CCD is. Each element of this
800 × 800 grid is square in shape and is referred to as a pixel.
All Voyager images are subject to radiometric responsiveness induced errors,
geometric distortions and the introduction of artifacts. The artifacts include, but
are not limited to, “hot pixels” and the famous dust “donuts”. To make matters
more complicated the geometric distortions introduced vary from one image to the
next. The errors introduced are due to a combination of the camera optics and
the processes of building up the image and then reading it off the face plate of the
vidicon tube.
CHAPTER 3. VOYAGER IMAGES 55
3.3 Voyager images
Each image as it is read from the vidicon tube is an 800×800 array, giving 640,000
square pixels. Each pixel has a brightness level, called a DN (Digital number),
between 0-255 i.e. 256 brightness (DN) levels. Since 256 is 28 or an eight digit
binary number the images are often referred to as eight bit images. Such 800× 800
images which haven’t been processed in any way to remove radiometric or geometric
distortions are called ‘raw’, or ‘image-space’, images and will be referred to as such
from now on.
The coordinate system used for Voyager images has its origin in the upper
left corner of the image. The positive x direction, traditionally referred to as the
sample direction, is to the right. The positive y direction, traditionally called the
line direction, is downwards. The Voyager image coordinate system is illustrated
in Fig. 3.1. It is usual to give coordinates as (line,sample) not (sample,line) and
so we will follow common usage and give coordinates in the (line,sample) format.
Each pixel can have integer line and sample coordinates in the range 0-799.
Therefore, the pixel in the lower left hand corner has coordinates (799,0); the upper
right hand corner (0,799); the lower right (799,799) etc.
The fact that each pixel has integer coordinates can cause problems when co-
ordinates to sub-pixel accuracy are required. The integer coordinates of a pixel,
which varies from 0-799 in both sample and line, are the coordinates of the exact
centre of the square pixel. In this ‘pixel coordinate’ system the coordinates of the
top left, top right, bottom left and bottom right corners of the raw image are (-0.5,-
0.5), (-0.5,799.5), (799.5,-0.5) and (799.5,799.5) respectively. Whenever coordinates
are given to sub-pixel accuracy in the ‘pixel coordinate’ system, this is the frame
of reference used. We also used another coordinate system we called ‘continuous
coordinates’, which can take all values in the range 0.0-800.0 for raw images. The
coordinates in the ‘pixel’ system are converted to continuous coordinates by
linecontinuous = linepixel + 0.5 (3.1)
samplecontinuous = samplepixel + 0.5 (3.2)
The very top left of the image is (0.0,0.0); top right (0.0,800.0); top left pixel
CHAPTER 3. VOYAGER IMAGES 56
(0,0) (0,799)
(799,799)(799,0)
sample
line
Figure 3.1: Raw Voyager image orientation.
(0.5,0.5); bottom right pixel (799.5,799.5) etc. With continuous coordinates it is
clear that a sample value of 456.7 means that the raw image location is 456.7/800.0
of the way along the length of the x axis. Unless it is specifically stated otherwise
the ‘pixel coordinate’ system is used.
3.4 The geometrical correction of Voyager im-
ages
As previously discussed, the images from the Voyager NAC and WAC cameras are
subject to geometrical distortions. These distortions have to be removed thus pro-
ducing a geometrically corrected image. During the geometrical correction process
CHAPTER 3. VOYAGER IMAGES 57
1000x1000 pixel object-space image
Transformed 800x800 pixelimage-space image
Figure 3.2: Typical shape of 800× 800 image-space pixel grid mapped onto 1000×1000 pixel grid of object-space image.
the 800×800 pixel raw image is mapped onto a 1000×1000 pixel grid. The 800×800
grid of the raw image is called ‘image-space’, while the 1000× 1000 pixel grid onto
which the geometrically corrected raw image is mapped is referred to as ‘object-
space’. Of course the 640,000 pixels of the raw image cannot fill the 1,000,000 pixel
grid of the object-space image. The geometrical distortion introduced into Voyager
images by the imaging system is of ‘barrel’ type. Imagine the raw image as being
a square piece of rubber. Take it and stretch the corners diagonally away from
the centre. The resulting shape is roughly what the geometrically corrected raw,
‘image-space’, image looks like when mapped onto the 1000 × 1000 pixel object-
CHAPTER 3. VOYAGER IMAGES 58
space image as illustrated by Fig. 3.2. The distortion not only varies from one
image to the next but also from one region of the image to another, making a single
global correction for the image impossible. Since the introduced distortion varies
even within a single image it is unfeasible to calculate a single set of transformation
parameters that will geometrically correct more than a small region of a single raw
image. A different set of parameters must be calculated for every image shuttered
and they can only be determined after the image has been taken.
3.4.1 Overview of the geometrical correction process
The process of transforming each 800×800 raw Voyager image into a geometrically
corrected 1000 × 1000 image has the following steps:
• Locate the reseau marks on the raw image.
• Generate the 85 pseudo-reseaus.
• Calculate the transformation parameters for each of the 506 triangular areas an
image is divided into.
• For each of the 1,000,000 pixels in the object-space image determine which trian-
gular area it falls into. Use the appropriate transformation parameters to transform
the object-space pixel into image-space.
• Determine the DN value of the object-space pixel by either the nearest neighbour
or pixel interpolation technique.
3.4.2 Reseau marks
The exact geometrical distortion in each image must be determined separately,
using only information contained within the image itself and knowledge of the char-
acteristics of each individual vidicon tube. The necessary information is contained
in a grid of points called ‘reseau’ marks. Reseau is taken from the French for ‘web’
and the reseau marks are a grid of points actually etched onto the face of the vidi-
con tube itself. Where a reseau has been etched the selenium sulphide layer is
destroyed and the face plate becomes insensitive to incident light i.e. there is no
electrostatic charge build up in that location. A pixel in the location of a reseau
mark has a DN of 0. The reseau marks can be clearly seen in raw Voyager im-
CHAPTER 3. VOYAGER IMAGES 59
ages, see Fig. 3.3. Published images usually have these reseau marks removed for
Figure 3.3: A raw Voyager image showing reseau marks
an improved aesthetic effect, although they contain no more information and can
indeed be misleading. Reseau marks are removed by averaging the DN values of
the surrounding pixels and replacing the ∼ 0 DN pixels within the reseau by this
averaged value. We retained the reseau marks in all the images used.
Each vidicon tube has a slightly different pattern of reseau marks etched onto
its face-plate. The actual physical locations of the reseau marks on the face-plates
are known to sub-pixel accuracy. After manufacture each vidicon tube was put
through an exhaustive series of laboratory tests. The location of the reseau marks
CHAPTER 3. VOYAGER IMAGES 60
on the vidicon face-plates was measured using a theodolite, in conjunction with
accurate surveying techniques, to accuracies of ±0.002 mm. When these positions
are converted to line and sample coordinates the locations of the reseaus in the
object-space image is known to sub-pixel accuracy. These object-space reseau lo-
cations served to define a geometrically correct space which would result if, say
a grid target, were imaged through a geometrically perfect camera system. The
object-space reseau locations for a particular vidicon tube remain constant. Com-
parison of the actual positions of the reseaus in image-space on the raw image with
their object-space locations enables the determination of a set of ‘distortion pa-
rameters’. These distortion parameters map the object-space reseau locations onto
their image-space locations and vice versa. By taking large numbers of test images
in the laboratory a set of nominal image-space reseau coordinates was obtained for
each vidicon tube. A reseau’s nominal location is the average of its actual locations
in the test images. For every image taken using that vidicon tube the actual image-
space location of the reseau marks in the raw image should be near their nominal
location, say within 10 pixels or so. Both the nominal image-space reseau locations
and their corresponding constant object-space locations are used in the geometrical
correction of the raw image.
3.4.3 Location of reseau marks
Each of the vidicon tubes manufactured for the Voyager project has 202 reseau
marks etched onto the face-plate, Fig. 3.4 shows the object-space reseau mark lo-
cations for the Voyager 2 narrow angle camera. The object-space reseau locations
for all the other cameras are similar. Fig. 3.4 clearly shows that the reseaus are
concentrated around the edge of the image with fewer in the middle. Presumably
this is because the centre of the image is the region of greatest importance and
reseaus here might obscure items of interest.
Clearly the greater the number of reseau marks the better any distortion correc-
tions will be. However the more reseaus the greater the area of the image obscured,
202 reseaus seemed a reasonable compromise.
CHAPTER 3. VOYAGER IMAGES 61
1 2 3 4 5 6 7 8 9 10 11 12
1314 15 16 17 18 19 20 21 22
23
24 25 26 27 28 29 30 31 32 33 34 35
36 37 38 39 40 41 42 43 44 45 46
47 48 49 50
51 52 53 54 55 56 57 58 59 60 61
62 63 64 65
66 67 68 69 70 71 72 73 74 75 76
77 78 79 80
81 82 83 84 85 86 87 88 89 90 91
92 93 94 95
96 97 98 99 100 101 102 103 104 105 106
107 108 109 110
111 112 113 114 115 116 117 118 119 120 121
122 123 124 125
126 127 128 129 130 131 132 133 134 135 136
137 138 139 140
141 142 143 144 145 146 147 148 149 150 151
152 153 154 155
156 157 158 159 160 161 162 163 164 165 166
167 168 169 170 171 172 173 174 175 176 177 178
179180 181 182 183 184 185 186 187 188
189
190 191 192 193 194 195 196 197 198 199 200 201
202
Figure 3.4: Object-space image reseau mark locations for vidicon tube number 5,the tube used in the Voyager 2 narrow angle camera (NAC). The reseau ID numbersare included above the corresponding reseau mark.
CHAPTER 3. VOYAGER IMAGES 62
3.4.4 The detection of reseaus in a raw image
The region of an image around the nominal image-space reseau location of each
reseau mark is searched. If an actual physical reseau mark is located its position is
used, if a physical mark isn’t located within a fixed distance of its nominal image-
space location then the nominal image-space location is used. In this way the actual
image-space space location of the 202 reseau marks on an image are determined.
3.4.5 The generation of ‘pseudo-reseau’ marks
For the purposes of geometrical correction Voyager images are divided up into 506
triangular areas. The reseau grid illustrated in Fig. 3.4 clearly cannot be divided
into 506 triangular areas with any great ease. The solution is to generate extra
reseau marks. These extra reseaus, called ‘pseudo-reseau’ marks, are generated
from already existing ‘real’ reseau mark locations. In this way 85 pseudo-reseau
marks are created making a total of 287 marks in total, both real and pseudo. The
positions of the extra pseudo-reseau marks are illustrated in Fig. 3.5.
The image is then divided into triangular areas with reseau marks, real and
pseudo-, at the vertices as illustrated in Fig. 3.6. The extra pseudo-reseau marks
are used to ensure that the triangular areas the image is divided into are of roughly
equal size. This ensures that the transformation parameters are reasonably contin-
uous between one triangle and its immediate neighbours.
The image-space locations of the reseau marks and triangular areas for the
Voyager 2 NAC image FDS43686.55 are shown in Fig. 3.7 for comparison with the
object-space locations of the same features in Fig. 3.6.
3.4.6 Mapping image-space locations into object-space
As described previously a raw image is divided into 506 triangular areas, with
the vertices of the triangles denoted by the position of reseau marks. Similarly
the geometrically correct image in object-space is also divided into 506 triangular
areas with the vertices being co-incident with the object-space locations of the same
reseau marks that denote the vertices of the triangles in image-space.
The locations of reseau marks in image-space transform exactly to the reseau
CHAPTER 3. VOYAGER IMAGES 63
1 2 3 4 5 6 7 8 9 10 11 12
1314 15 16 17 18 19 20 21 22
23
24 25 26 27 28 29 30 31 32 33 34 35
36 37 38 39 40 41 42 43 44 45 46
47 48 49 50
51 52 53 54 55 56 57 58 59 60 61
62 63 64 65
66 67 68 69 70 71 72 73 74 75 76
77 78 79 80
81 82 83 84 85 86 87 88 89 90 91
92 93 94 95
96 97 98 99 100 101 102 103 104 105 106
107 108 109 110
111 112 113 114 115 116 117 118 119 120 121
122 123 124 125
126 127 128 129 130 131 132 133 134 135 136
137 138 139 140
141 142 143 144 145 146 147 148 149 150 151
152 153 154 155
156 157 158 159 160 161 162 163 164 165 166
167 168 169 170 171 172 173 174 175 176 177 178
179180 181 182 183 184 185 186 187 188
189
190 191 192 193 194 195 196 197 198 199 200 201
202
Figure 3.5: Generated object-space pseudo-reseau marks locations using the object-space image reseau mark locations for vidicon tube 5. The pseudo-reseaus are thered dots.
mark locations in object-space. This of course means that the vertices of a particular
triangular area in image-space transform exactly to the vertices of the corresponding
triangle in object-space. We make the assumption that every point that lies within
the boundary of a image-space triangle transforms into object-space in exactly the
same way. This of course is not necessarily true but since the area of the triangle is
small when compared to the total area of the image any variations in transformation
parameters across the triangle are also likely to be small.
For each triangle there are six variables, the coordinates of the vertices, in image-
CHAPTER 3. VOYAGER IMAGES 64
1 2 3 4 5 6 7 8 9 10 11 12
1314 15 16 17 18 19 20 21 22
23
24 25 26 27 28 29 30 31 32 33 34 35
36 37 38 39 40 41 42 43 44 45 46
47 48 49 50
51 52 53 54 55 56 57 58 59 60 61
62 63 64 65
66 67 68 69 70 71 72 73 74 75 76
77 78 79 80
81 82 83 84 85 86 87 88 89 90 91
92 93 94 95
96 97 98 99 100 101 102 103 104 105 106
107 108 109 110
111 112 113 114 115 116 117 118 119 120 121
122 123 124 125
126 127 128 129 130 131 132 133 134 135 136
137 138 139 140
141 142 143 144 145 146 147 148 149 150 151
152 153 154 155
156 157 158 159 160 161 162 163 164 165 166
167 168 169 170 171 172 173 174 175 176 177 178
179180 181 182 183 184 185 186 187 188
189
190 191 192 193 194 195 196 197 198 199 200 201
202
Figure 3.6: Object-space image divided into grid of 506 triangles using reseau marklocations for vidicon tube 5, the tube in the Voyager 2 narrow angle camera (NAC).
CHAPTER 3. VOYAGER IMAGES 65
12 3 4 5 6 7 8 9 10 11
12
13 14 15 16 17 18 19 20 21 2223
24 25 26 27 28 29 30 31 32 33 34 35
36 37 38 39 40 41 42 43 44 45 46
47 4849 50
51 52 53 54 55 56 57 58 59 60 61
62 6364 65
66 67 68 69 70 71 72 73 74 75 76
77 7879 80
81 82 83 84 85 86 87 88 89 90 91
92 9394 95
96 97 98 99 100 101 102 103 104 105 106
107 108109 110
111 112 113 114 115 116 117 118 119 120 121
122 123124 125
126 127 128 129 130 131 132 133 134 135 136
137 138139 140
141 142 143 144 145 146 147 148 149 150 151
152 153154 155
156 157 158 159 160 161 162 163 164 165 166
167 168 169 170 171 172 173 174 175 176 177 178
179180 181 182 183 184 185 186 187 188 189
190191 192 193 194 195 196 197 198 199 200
201
202
Figure 3.7: Image-space locations of reseau and pseudo-reseau marks and triangulargrid for the Voyager 2 NAC image FDS43686.55.
CHAPTER 3. VOYAGER IMAGES 66
space that transform into six variables in object-space. Since all twelve variables are
known the simultaneous linear equations describing the transformation from image-
space to object-space for the triangle are soluble. The simultaneous equations for
transforming from object-space to image-space are
limage = a + b× lobject + c× lobjectsobject (3.3)
simage = d+ e× lobject + f × lobjectsobject (3.4)
and those for transforming from image-space to object-space are
lobject = g + h× limage + i× limagesimage (3.5)
sobject = j + k × limage +m× limagesimage (3.6)
For each triangle the simultaneous equations are solved giving the transformation
parameters a, b, c, d, e, f , g, h, i, j, k and m. Any point within the boundary
of the triangle can be transformed from an image-space to object-space location or
vice versa.
The transformation parameters for all 506 triangles are obtained in turn. For
those parts of the image that lie outside a defined triangular area, basically around
the very edge of the image, the transformation parameters of the closest triangular
area are utilised.
Each pixel in a image is checked to see which triangular area it falls into, or
which area is closest. The pixel is then mapped from its image-space location
to its object-space location using the transformation parameters for the identified
triangular area. Of course pixels can also be mapped from object-space to image-
space in the same way.
It is highly unlikely that the image-space position of a pixel will map exactly
onto a valid pixel location in the 1000 × 1000 object-space grid. It is much more
likely that the image-space location of a pixel will map across four object-space
pixels as illustrated in Fig. 3.8, although in this figure an object-space pixel is
being mapped onto an image-space grid the principle is the same. In Fig. 3.8 what
should the intensity levels, i.e. DN values, of the image-space pixels 1, 2, 3 and 4
be? The situation becomes even more complicated when additional object-space
pixels are mapped onto the same area in image-space. To avoid this problem when
CHAPTER 3. VOYAGER IMAGES 67
a raw image is geometrically corrected the image-space pixels are not transformed
into object-space. A slightly different technique is used.
A 1000 × 1000 pixel grid is generated, each pixel in the grid being initially
assigned a DN of 0. This blank grid will become the geometrically corrected image in
object-space. The transformation parameters for each of the triangular areas in the
image-space image are calculated as previously described. Each pixel in the object-
space grid is then stepped through in turn and transformed into its corresponding
location on the image-space raw image leading to a situation illustrated in Fig. 3.8
1 2
3 4
P
Figure 3.8: Object-space pixel, P, mapped onto image-space pixel grid.
In Voyager image processing there are two main ways that the DN value of
an object-space pixel are determined, both involve transforming the pixel into its
corresponding image-space position. If the object-space pixel is mapped to an
invalid image-space location the DN of the object-space pixel is set equal to 0. An
invalid image-space location is one that is not on the raw image e.g. image-space
location (801,200).
When a raw image-space image is transformed to a geometrically corrected
object-space image the technique used involves mapping object-space pixels to
CHAPTER 3. VOYAGER IMAGES 68
image-space locations. The DN value of each object-space pixel is then calculated
using the nearest image-space pixels to its mapped image-space location.
3.4.7 The nearest neighbour pixel mapping technique
The nearest neighbour technique involves mapping an object-space pixel onto its
image-space location and then determining which image-space pixel is closest to
that mapped position. In Fig. 3.8 the largest fraction of the mapped object-space
pixel lies within image-space pixel 1. Therefore, image-space pixel 1 is closest to
the mapped position of the object-space pixel. The DN of the object-space pixel is
set equal to the value of image-space pixel 1. This procedure is performed on all
1,000,000 object-space pixels.
3.4.8 Pixel interpolation method
Again an object-space pixel is mapped onto its image-space location as in Fig. 3.9.
In this case the intensity level of all the pixels 1-4 is used to find an average for the
DN value of the object-space pixel,
DNobject = (x1y1DN1) + (x2y1DN2) + (x1y2DN3) + (x2y2DN4) (3.7)
where DNi denotes the intensity of image-space pixel i.
3.4.9 Comparison of the two methods
The nearest neighbour method is very simple to use and any loss of fine detail in an
image due to complicated interpolation of DN values is kept to a minimum. With
pixel interpolation it is possible to get DN values in the geometrically corrected
image that don’t occur in the raw image. Averaging over four image-space pixels
can lead to loss of fine detail. Due to the nearest neighbour method’s superior
ability to retain fine detail this was use throughout this work.
One slight drawback of the nearest neighbour method is that due to the trans-
formation technique, spatial features can be offset by up to half a pixel in the
geometrically corrected object-space image. This offset can lead to to a ‘wavy’
appearance of a straight feature in the corrected image.
CHAPTER 3. VOYAGER IMAGES 69
xx
y
y
1 2
1
2
1 2
3 4
P
Figure 3.9: Object-space pixel mapped onto image-space pixel grid for pixel inter-polation.
3.5 VICAR
Traditionally, Voyager images have been analysed using a software package called
VICAR (Video Image Communication and Retrieval). The VICAR software was
originally written in the early 1970’s, to run on hardware that is long since obsolete
and under operating systems that have not been in widespread use for many years.
The software had very little associated documentation, either within the code itself
or in the users manual, the VICAR Users Guide (circa. 1977). Images are processed
by VICAR mainly in an interactive mode, requiring a great deal of input by the
operator. Batch processing of multiple images is difficult. Generally VICAR is
still only in use by scientists who learned its use during the active portion of the
Voyager project, which effectively ended with Voyager 2’s encounter with Neptune
in 1987. Because of its age, and various shortcomings that became apparent to the
community during its use, VICAR has been modified many times. These modifica-
tions were made in no apparent coherent, centralised or controlled manner. Each
institution using VICAR has made its own modifications to suit its needs at the
CHAPTER 3. VOYAGER IMAGES 70
time. In some cases the modifications are quite major. Imperial College has a mod-
ified version of VICAR, called BISHOP, which was designed to run on a DEC/VAX
11/780 (Thompson 1990). The image format used by BISHOP is incompatible with
the standard VICAR format. It is unknown whether BISHOP is still functioning.
In many cases institutions have lost the ability to use VICAR as the machines it
was tuned to run on have been scrapped. Indeed the machine the OPNAV team
at the Jet Propulsion Laboratory (JPL) used to run VICAR has been retired and
they have lost the ability to process raw Voyager images. The machine used to run
VICAR at the the Lunar and Planetary Laboratory at the University of Arizona
is nine years old. It is probable that any copies of VICAR still running are totally
incompatible with each other. The shortcomings of VICAR are a direct result of it
being designed for a specific set of hardware and operating system over twenty years
ago. The limitations of computer hardware and software in the 1970’s, portability
across systems being totally unknown, meant that software had to be tuned for each
individual machine it was used on. Because of the great difficulty of using VICAR
at an institution where it wasn’t originally installed in the 1980’s for the Voyager
project, some authors have abandoned it entirely for Voyager image processing. A
combination of a single global transformation for geometrical correction and SPICE
FORTRAN routines for image navigation was used by Gordon (1994) and Gordon
et al. (1995). The only part of VICAR that has survived intact, and is still in
widespread use, is the VICAR image format. The images from the Cassini Imaging
Science Subsystem (ISS) cameras are in the VICAR image format.
3.6 MINAS
To avoid the problems that have been described with VICAR, and to take advantage
of the advances in computer hardware and software since the 1970’s, the Cassini
ISS team decided that a completely new software package should be written for
use with Cassini ISS images. One of the criteria was that the software should
be easily installable across a wide variety of hardware and operating systems. This
portability was achieved by programming in the commercial language IDL, which is
CHAPTER 3. VOYAGER IMAGES 71
widely used for graphical applications. IDL commands are identical no matter what
computer or operating system is used. The extensive task of writing IDL compilers
for various systems, and ensuring compatibility across them, is the responsibility of
IDL’s commercial suppliers. Since IDL is in general use in the scientific, medical
and business communities there is a wide range of documentation available. This
widespread commercial use is likely to ensure that IDL will be supported for many
years to come, with compilers being written for new hardware/operating systems as
they are released. Backwards compatibility with earlier versions of IDL is assured
by the supplier.
The image processing and navigation package being developed at CICLOPS
(Cassini Imaging Central Laboratory for Operations) at the University of Arizona,
under the direction of Carolyn Porco, for the Cassini ISS team is called MINAS
(Modular Image Navigation and Analysis Software). Since MINAS is written in
IDL, the package is system independent, the same code is installed on Sun ma-
chines running Solaris and PCs running Linux for example. Although specifically
written for Cassini, MINAS was designed so that it could be used with images taken
by any imaging system be it ground based telescope or as yet undesigned spacecraft.
This of course means that MINAS can be used to process, geometrically correct and
navigate Voyager images. MINAS wasn’t specifically written for Voyager images
and so lacks the routines necessary for reseau detection and geometrical correction.
However, the modular nature of MINAS means that IDL routines for the processing
of Voyager images can be easily integrated. Indeed, one of MINAS’ design philoso-
phies is that individual members of the ISS team should write MINAS compatible
IDL procedures to perform specific tasks and then CICLOPS will include these
procedures in MINAS itself.
3.7 Routines for reseau location and geometrical
correction
The FORTRAN77 subroutines within VICAR that carry out reseau location and
geometrical correction are called RESLOC and GEOMA respectively. We wrote
CHAPTER 3. VOYAGER IMAGES 72
MINAS compatible IDL procedures that perform the same functions as VICAR’s
RESLOC and GEOMA. The only available guides to exactly how RESLOC and
GEOMA perform their tasks is the undocumented source code for the subroutines
and brief descriptions of the subroutines in the VICAR Users Guide (circa 1977).
The source code was available as a legacy from an unsuccessful attempt to install
VICAR at Queen Mary in the early 1990’s. The source code was written in the
1970’s for the computers of the day and various tortuous programming tricks are
used because of the limited memory available to such computers. The lack of
documentation, and the programming style used, makes the source code for the
subroutines very difficult to follow.
The two main MINAS compatible IDL procedures written were resloc.pro and
geoma.pro. They perform identical tasks to the FORTRAN77 VICAR subroutines
RESLOC and GEOMA. Other lower level procedures were written but are not
discussed individually since they are called by resloc.pro and geoma.pro. They
are covered in the descriptions of those top-level procedures.
All the IDL procedures written for the geometrical correction of Voyager images
were written solely by the author, based on the FORTRAN77 VICAR subrou-
tines RESLOC and GEOMA. Where appropriate pre-existing MINAS procedures
were utilised instead of writing brand new code. The procedures written are fully
compatible with MINAS and will be included in future MINAS distributions by
CICLOPS.
3.7.1 resloc.pro: locating reseau marks
Each raw image should contain 202 reseau marks. In real images the geometrical
distortion introduced by the imaging system pushes some of the reseaus off the
800 × 800 pixel grid of the raw image. This results in each raw image having a
variable number of reseaus with a maximum of 202.
The nominal image-space reseau locations for each camera are known. Each
reseau mark has a physical size of 0.040×0.040 mm on the face-plate of the vidicon
tube. Fig. 3.10 shows a 9×9 pixel region of a raw image around the actual location
of a reseau mark. Within resloc.pro a reseau is modeled by a two dimensional
CHAPTER 3. VOYAGER IMAGES 73
Figure 3.10: 9 × 9 pixel region around the location of a reseau mark.
shape function. This function is constant for all reseau marks and is taken to be a
Gaussian function in two dimensions with the form
F(x, y) = 255(1 − e(x2+y2)/2) (3.8)
where x and y are sample and line coordinates respectively. The line and sample
coordinates take integer values from −4 to +4. F(x, y) takes integer values between
0-255 so 255(1 − e(x2+y2)/2) is rounded to the nearest integer. The reseau is thus
modeled as a 9 × 9 pixel square region.
The reseau shape function, F(x, y), is then scanned over the raw image area
with line coordinates between 4-795 and sample coordinates between 4-795. At
each pixel location within the defined image area the cross-correlation term of the
image data within the 9 × 9 area centred on the pixel location and the Gaussian
shape reseau profile, F(x, y), is calculated. The locations of local maxima in the
grid of correlation coefficients are then identified. If the value of the correlation
coefficient at an identified local maxima is less than some minimum set value, ccmin,
then the point is discarded. The default value of ccmin is 0.75 and we used this value
throughout our image processing. The correlation maxima can only be located to
the nearest line and sample at this stage.
Each of the 202 reseau marks is stepped through in turn. The closest local
CHAPTER 3. VOYAGER IMAGES 74
maximum in the correlation coefficient to the nominal image-space location of the
reseau mark is assumed to be the actual location of the reseau mark in the raw
image. If this assumed location is more then a certain number of pixels from the
nominal image-space location then the reseau is flagged as NOT LOCATED. If the
assumed location is within the specified number of pixels than the reseau is flagged
as LOCATED. The maximum distance an assumed pixel location can be from its
nominal image-space location is an input variable to resloc.pro. The default value
is 5.0 pixels, and this has been used throughout this work.
The next step is to determine the image-space locations of all LOCATED reseaus
to sub-pixel accuracy. If the assumed location of a LOCATED reseau is (i, j) then
the value of the correlation coefficient at that point is ρ(i, j). Sub-pixel accuracy
is obtained by fitting a paraboloid to the 2-dimensional correlation function in the
vicinity of pixel (i, j). We used a 3 × 3 pixel area centred on pixel (i, j).
The image-space locations of the LOCATED reseau marks are known at this
point. The locations of the NOT LOCATED reseaus are determined next. Each
of the 202 reseau marks has a number of near-neighbour reseau marks, varying
between 3 and 6 depending on the location of the reseau mark on the grid. For
example reseau mark 1 has reseaus 2, 13 and 24 as near-neighbours while reseau 120
has reseaus 109, 119, 121 and 124 see Fig. 3.4. There is a list of near-neighbours for
each reseau mark embedded within VICAR. This is not a straightforward list of the
adjacent reseau marks, sometimes a reseau is listed as a near-neighbour twice or an
adjacent reseau isn’t listed at all. In VICAR the list is generated using an involved
algorithm, we ran the algorithm to generate a simple list of near-neighbours for
each reseau.
Each NOT LOCATED reseau is stepped through in turn and its near-neighbour
reseaus examined. If all the near-neighbour reseaus are also NOT LOCATED then
the program moves on the next NOT LOCATED reseau. If one or more of the
near-neighbour reseaus are LOCATED the average displacement of the LOCATED
near-neighbour reseaus from their nominal image-space locations is calculated. The
NOT LOCATED reseau is then displaced from its nominal image-space location by
this averaged amount and then marked as LOCATED. However, this reseau remains
CHAPTER 3. VOYAGER IMAGES 75
NOT LOCATED when it is used as a near-neighbour. All the NOT LOCATED
reseaus are stepped through, at the end of the iteration all the newly located re-
seaus are also marked as LOCATED when used as near-neighbours. The process is
repeated until all 202 reseaus are LOCATED.
Once all 202 ‘real’ reseau marks have been located, the 85 ‘pseudo-reseau’ marks
are generated from specific real reseaus. The list of exactly which ‘real’ reseaus are
used in the generation of each pseudo-reseau was recreated from VICAR via software
archeology. In general, the sample coordinate of a pseudo-reseau is the average of
the sample coordinates of the two nearest real reseau marks on the previous line
of real reseaus. The line coordinate being the average of the line coordinates of
the nearest two real reseau marks on the same line. This averaging technique
is only used for the object-space pseudo-reseau coordinates. The corresponding
coordinates in image-space are calculated using a bilinear mapping technique using
nearest neighbour real reseau marks, four in the centre of the image and three
around the edges. For a particular pseudo-reseau mark, near the image edge, its
image-space coordinates are given by
lineimage = d+ e× sampleobject + f × lineobject (3.9)
sampleimage = a + b× sampleobject + c× lineobject (3.10)
where (lineimage, sampleimage) and (lineobject, sampleobject) are the coordinates of the
pseudo-reseau mark in image-space and object-space respectively. The constants a,
b, c, d, e and f are obtained by solving
sampleimageX
sampleimageY
sampleimageZ
=
1.0 sampleobjectX lineobjectX
1.0 sampleobjectY lineobjectY
1.0 sampleobjectZ lineobjectZ
a
b
c
(3.11)
and
lineimageX
lineimageY
lineimageZ
=
1.0 sampleobjectX lineobjectX
1.0 sampleobjectY lineobjectY
1.0 sampleobjectZ lineobjectZ
d
e
f
(3.12)
where (lineimageI, sampleimageI) and (lineobjectI, sampleobjectI) are the coordinates
are the near-neighbour reseau marks in image-space and object-space respectively.
CHAPTER 3. VOYAGER IMAGES 76
Here the variable I takes the values X, Y and Z which are the ID numbers of the
three nearest neighbour real reseau marks.
A similar scheme is followed for pseudo-reseau creation in the centre of the image
where four instead of three near-neighbour real reseaus are used
line image = e+ f × sampleobject + g × lineobject
+h× sampleobject × lineobject (3.13)
sample image = a+ b× sampleobject + c× lineobject
+d× sampleobject × lineobject (3.14)
with a, b, c, d, e, f , g and h being the solutions to a set of simultaneous linear
equations similar to Eqns. 3.11 and 3.12 but for four points instead of three.
The coordinates of all 287 reseau marks, both real and pseudo, in both image-
space and object-space are now known. The vertices of each of the 506 triangu-
lar areas correspond to the locations of specific reseau marks. The output from
resloc.pro is a set of parameters, called the geoma parameters, which contains
the coordinates of the vertices of the 506 triangular areas in both image-space and
object-space.
3.7.2 geoma.pro: Performing the Geometrical Correction
The actual transformation of a raw image into geometrically correct object-space
image is performed by the routine geoma.pro. Input variables are the raw image
and the geoma parameters for a particular raw image output by resloc.pro. The
output variable is the 1000×1000 object-space image. The pixels are mapped using
the nearest neighbour technique as the default although pixel interpolation can also
be used.
The transformation parameters are calculated for each of the 506 triangular
areas that the image is divided into. For each area the transformation parameters
a, b, c, d, e, f , g, h, i, j, k and m are the solutions to
limage = a+ b× lobject + c× sobject
simage = d+ e× lobject + f × sobject (3.15)
CHAPTER 3. VOYAGER IMAGES 77
and
lobject = g + h× limage + i× simage
sobject = j + k × limage +m× simage (3.16)
where l and s are the line and sample coordinates of the vertices of the triangular
area. Since there are three vertices, and thus 12 equations, they are soluble for the
12 transformation parameters. The sets of Eqns. 3.15 and 3.16 are solved for each
of the 506 areas in turn using a using a ‘back-substitution’ with Singular Value
Decomposition technique (the IDL procedure svsol).
Each of the 1,000,000 pixels in the object-space image initially has its DN set to
0. The 506 triangular areas are then stepped through and the index numbers of the
pixels contained within the area determined. The index numbers run sequentially
from 0 to 999,999. Pixel 0 is the pixel at the extreme left of the top row in the
object-space image. The index number then increases to the right along the row,
when the end of a row is reached the next index number is given to the extreme
left pixel of the next row down and the index again increases to the right along the
row. Pixel 999,999 is at the extreme right of the bottom row. Fig 3.11 illustrates
the scheme for allocating index numbers for a 4× 4 pixel image. An index number
1 2 3 45 6 7 89 10111213141516
Figure 3.11: Index numbers for a 4 × 4 pixel image.
CHAPTER 3. VOYAGER IMAGES 78
is converted into integer pixel coordinates using
l = int
(
index
Nline
)
s = index− l ×Nline (3.17)
where Nline is the number of pixels in each line of the image, index is the index
number and int() gives the nearest integer rounded down. As already discussed we
assumed that the transformation parameters for all the pixels within a particular
triangular area are the same as the transformation parameters calculated for the
vertices of the area.
Taking the 506 areas in turn, each of the identified pixels within the area is
transformed into its raw-space location. The DN value of the object space pixel
under consideration is determined from its 4 nearest neighbour raw-space pixels
using the nearest neighbour or pixel interpolation technique. The actual DN of that
object-space pixel in the 1000 × 1000 geometrically corrected image is then set to
this determined value. For pixels lying outside a triangular area the transformation
parameters for the nearest area are used.
In this way a geometrically corrected version of the raw image is built up on the
1000 × 1000 grid of the object-space image. The output from geoma.pro is this
1000 × 1000 pixel geometrically corrected object-space image.
3.8 Comparison of VICAR and MINAS results
Images corrected using MINAS were compared to the same images corrected using
VICAR. The VICAR corrected images were provided by Vance Haemmerle at CI-
CLOPS. Also provided were the reseau locations in each image as determined by
VICAR. The MINAS and VICAR images were compared by subtracting one from
the other. Identical images resulting in each pixel in the 1000× 1000 grid having a
DN of 0.
When the reseau locations, as determined by MINAS and VICAR, were com-
pared the mean residual in the line and sample identified positions of the reseaus
was on the order of a few hundredths of a pixel. Occasional pixels had residuals of
up to 0.5 pixels or so. These slight differences in the identified reseau mark locations
CHAPTER 3. VOYAGER IMAGES 79
are consistent with slight differences in the code used to locate them. The pixels
with the large residuals being those where the position had to be interpolated.
When the VICAR reseau locations were used in the MINAS geoma.pro routine
the MINAS corrected image was essentially identical to the VICAR version. There
were occasional limited areas of the image where the DN levels differed by 1 DN
or so. When an image processed entirely with MINAS was compared to a VICAR
image, again they were practically identical. In some small areas of the image the
DN values differed by up to 3 DN levels. All features, ring edges, satellites were
however in the same location in both images.
The differences between the VICAR and MINAS geometrically corrected images
can be explained by slight differences in some of the parameters used in the cor-
rection process along with minor differences in algorithms. VICAR itself does not
produce 100% accurate corrected images and can only realistically locate reseaus to
within ∼ 0.1 pixels (VICAR users guide circa. 1977). Differences in located reseau
positions of ∼0.03 pixels or so between VICAR and MINAS and the differences in
the corrected image resulting from them are insignificant. We were therefore confi-
dent that the routines written for MINAS were producing geometrically corrected
images that were as accurate as those produced by VICAR.
Chapter 4
Image Navigation
4.1 Introduction to image navigation
As discussed in the description of methods of orbit determination (Chapter 2) the
data used, the observations, consists of sets of unit pointing vectors along with
their corresponding observation times. The raw data, images taken by telescopes or
spacecraft cameras, are simply pictures of areas of the sky. While the observation
time associated with an image is known, the pointing vector to an object in an
image is not immediately apparent. The process of determining the actual pointing
direction of the centre of an image, along with the orientation of the image, is called
image navigation. Once an image has been navigated the pointing vector to any
object in the image can be easily calculated.
Image navigation is necessary because while the approximate pointing of the
telescope or camera when the image was shuttered is known, the information gen-
erally contains inaccuracies. For example, the camera pointing information in the
Voyager SEDRs includes errors due to inaccuracies in the positions of the Voyager
spacecraft (on the order of 10km) and uncertainties in the orientation of the scan
platform on which the cameras are mounted. These combine to give an uncertainty
in the pointing direction of the cameras of up to ∼ 2.5 × 10−3 radians (Showalter
1991), which is approximately one third of the field of view of the Narrow Angle
Camera (NAC).
80
CHAPTER 4. IMAGE NAVIGATION 81
4.2 Methods of image navigation
While not strictly an image navigation technique, angular separation of satellite
pairs is used for orbit determination purposes. Large numbers of observations of
the satellites of the outer planets are in the form of angular separations between
satellite pairs (see for example Harper and Taylor 1993). Measuring such angular
separations with micrometers was the only practical accurate method before the
widespread use of photographic plates. Even with ground based photographs it is
often very difficult to locate a single satellite accurately in terms of right ascension
and declination. The difficulty stems from the need for two or more reference stars
(with accurately known positions) to navigate an image. Two such relative positions
enables the absolute inertial position of a satellite to be determined. Often there
are not two such reference stars in an image and other methods have to be used.
Often the positions of a satellite relative to the known, and assumed accurate,
ephemeris positions of other satellites or the centre of the primary planet are used
(e.g. Nicholson et al. 1996, Poulet et al. 2000) in image navigation. The disadvan-
tage of these methods is that the positions of other useful reference satellites are
not always known to the required accuracy and that the centre of the image of an
oblate planet, possibly obscured by a ring system, is difficult to fix. Synnott et
al. (1983) used the positions of reference stars and the satellites Mimas, Enceladus,
Tethys, Dione and Rhea as fiducial points (Smith et al. 1981). This requirement for
two or more fiducial points limited the number of images that Synnott et al. (1983)
could navigate and therefore use.
The advantage of using the angular separation of satellite pairs is that it is easy
to accurately measure and requires no a priori knowledge about the actual inertial
position of either satellite. All the information necessary for orbit determination
purposes can be obtained directly from an image. Often sets of angular separation
measurements of various combinations of satellite pairs are obtained from the same
image. The great disadvantage is that the orbits of both satellites (or a whole
system if a combination of pairs is used) has to be solved simultaneously. Also
you need to have the same two satellites in every image, this criterion is usually
satisfied for ground based and HST images but is rarely the case for most of the
CHAPTER 4. IMAGE NAVIGATION 82
higher resolution Voyager images of Atlas, Prometheus and Pandora. Because of
this the satellite pair separation method is unsuitable for determining the orbits of
these satellites.
For the planet Saturn, the locations of circular, or near circular, features in
the ring system are known to high accuracy, ∼ 100 metres or so (Porco et al. 1984,
Nicholson et al. 1990 and French et al. 1993). Such features can be used to navigate
images and thus determine the pointing direction to an object, such as a satellite,
providing there is sufficient curvature of any feature used in a particular image
(Gordon 1994, Gordon et al. 1996 and Murray et al. 1997). We used circular
features in the outer ring system of Saturn along with the position of the planetary
limb to navigate the Atlas, Prometheus and Pandora images used in this work.
4.3 Software for image navigation, MINAS
We used the software package MINAS (see section 3.6) for the purposes of image
navigation. MINAS has routines that allow geometrically correct images (see Chap-
ter 3) to be navigated interactively using planetary limbs and circular features in
ring systems.
For a particular image the MINAS routines require information on the charac-
teristics of the planet and ring system, the pixel scale of the image, the state of the
observer with respect to the planet, and the approximate orientation of the image
at the observation time. Since the required data is general, MINAS can navigate
images taken by any camera or telescope.
For the purpose of navigating Voyager images we wrote routines that extracted
the state of the spacecraft with respect to Saturn from a NAIF SP kernel (Ac-
ton 1990) and the approximate pointing of the camera from a NAIF C kernel.
NAIF is the Navigation and Ancillary Information Facility node of the Planetary
Data System (PDS). It is responsible for the implementation and operation of the
SPICE information system. SPICE stands for Spacecraft, Planet, Instrument,
“C-matrix” and Events and it is a way of providing ancillary observation geome-
try data and related tools used in the interpretation and planning of observations
CHAPTER 4. IMAGE NAVIGATION 83
made by spacecraft. NAIF provides a toolkit of FORTRAN 77 and C subroutines
along with ‘kernels’ containing information on spacecraft trajectories, planetary
ephemerides, pointing directions for instruments and so forth. The SP kernels used
were vg1 sat.bsp and vg2 sat.bsp for Voyager 1 and Voyager 2 respectively. These
kernels are available via ftp at
ftp:naif.jpl.nasa.gov:/pub/naif/voyager/spk.
The C kernels were vg1 sat qmw na.bc, vg1 sat qmw wa.bc, vg2 sat qmw na.bc and
vg2 sat qmw wa.bc which were created from Voyager SEDRs at QMW for use with
Voyager Saturn images (Gordon 1994 and Gordon et al. 1996). SP kernels con-
tain ephemerides for spacecraft, planets and satellites etc. while C kernels contain
pointing information for instruments. SEDRs are Supplementary Experiment Data
Records and contain ancillary information for an image like the state of the space-
craft, temperature of the cameras, pointing direction and so on. The extracted
information is passed directly to the MINAS navigation procedures.
4.4 Navigating an image
From the information supplied, the MINAS routines present the locations of two
ring features and the planetary limb, calculated from the supplied geometry, su-
perimposed on the image. MINAS draws in the position of these features based on
the information it has been given. Due to inaccuracies in the pointing direction the
drawn in locations are unlikely to coincide with the actual features in the image.
Fig. 4.1 illustrates a Voyager image with drawn in positions for ring features based
on the initially supplied information.
The position and orientation of the dotted lines superimposed on the image are
then interactively modified. When the lines indicating the calculated positions of
the features exactly coincide with the actual positions of the features in the image,
the pointing direction assumed by the MINAS procedure at that point is the actual
pointing direction of the instrument (see Fig. 4.2). The process of matching the
calculated and actual positions of features can be performed automatically, but we
found from experience that visually matching the lines with the corresponding fea-
CHAPTER 4. IMAGE NAVIGATION 84
Figure 4.1: Example of initial image displayed by MINAS. The white dotted linesindicate the locations of the outer edge of the A ring and the inner edge of theEncke gap based on the information supplied to the program. The image is ageometrically corrected FDS43686.55 which has been stretched to bring out thedotted lines against the background of the rings. The pointing information suppliedis clearly inaccurate.
CHAPTER 4. IMAGE NAVIGATION 85
tures provided better results. The now accurate pointing direction used to calculate
Figure 4.2: Example of final image displayed by MINAS. The white dotted linesindicating the locations of the outer edge of the A ring and the inner edge of theEncke gap now coincide with the features’ actual location in the image. The pointingdirection assumed for the instrument is now accurate.
the feature locations is out-putted, along with the image ID number and image time
for future use. The pointing direction of the camera/telescope at the image time
as determined by MINAS is out-putted as a SPICE C-matrix. Each image must
be navigated separately using this software, batch processing is not possible at the
current stage in the MINAS’ development.
CHAPTER 4. IMAGE NAVIGATION 86
x
y
z
image plane
origin of imagecoordinate system
y
x
centre ofimage
x,y,z instrument coordinate
x,y image coordinate frame
frameinstrument boresightpointing to target
Figure 4.3: Instrument frame for Voyager cameras
4.5 Calculating the pointing vector to an object
in an image
The line,sample (l, s) coordinates of an object in a navigated image have to be
converted into a pointing vector in inertial space. The first step is to convert the line,
sample coordinates into a pointing vector in the instrument frame, the frame that is
fixed with respect to the camera/telescope. The origin of the instrument frame is at
the centre of the image plane with the z-axis pointing along the instrument boresight
vector. The x- and y-axes are orthogonal to each other in the plane perpendicular
to the z-axis. The orientation of the x- and y-axes may be defined differently for
different instruments, the instrument frame for the Voyager cameras is illustrated
in Fig. 4.3. This figure illustrates the relationship between the instrument frame
and image coordinate system, it is not an accurate representation of the actual path
followed by a light ray traveling through the imaging system.
The origin of the image frame has to be mapped onto the origin of the instrument
frame. If the origin of the instrument frame has coordinates lcentre, scentre in the
CHAPTER 4. IMAGE NAVIGATION 87
image frame then
linstrument = limage − lcentre (4.1)
sinstrument = simage − scentre (4.2)
gives the line(l) and sample(s) coordinates of a point in the image in instrument
coordinates. For space probe images it is usual to refer to coordinates in the x and
y directions as line and sample respectively and we will follow convention and refer
to line and sample coordinates. The pointing vector to an object in the instrument
frame, Q, is defined by (VICAR users guide circa 1977)
Qx =1
σsinstrument (4.3)
Qy =1
σlinstrument (4.4)
Qz = f (4.5)
where f is the focal length of the camera in mm and σ is a scale factor measured in
pixels per mm. The values of f and σ for the Voyager cameras is given in Danielson
et al (1981) and Table 5.1.
The unit pointing vector in the instrument frame, Q, is related to the unit
pointing vector, P, in the reference frame co-moving with the instrument, the J2000
frame (Earth mean equator, dynamical equinox of J2000) throughout this work,
through the transpose of C, the SPICE C-matrix for that image.
P = CTQ (4.6)
In this way, P, the unit vector pointing from the instrument towards the position
of the target object in the J2000 frame co-moving with the instrument, can be
calculated.
Chapter 5
Atlas
5.1 Introduction
The small saturnian satellite Atlas orbits just exterior to the outer edge of the A
ring (see Fig. 5.1). It was discovered during the Voyager 1 encounter in 1980 and
its orbital elements determined using only 8 images (Smith et al. 1981). Atlas was
also imaged by the Voyager 2 cameras in 1981 but no orbital elements have been
published that utilise these Voyager 2 images. The Voyager 2 science report simply
reproduces the Voyager 1 elements (Smith et al. 1982).
The pre-1995 orbits of Prometheus and Pandora calculated by Synnott et al. -
(1983) have been checked using the original data (Jacobson private communication)
but no independent check has been made on Atlas’ orbit. In light of the changes
in Prometheus’ orbit (Nicholson et al. 1996, Bosh and Rivkin 1996), indications
of undetected mass in the region between the outer edge of the A ring and the F
ring and the fact that Atlas’ orbit is unchecked and only uses Voyager 1 data an
investigation of the orbit of Atlas is warranted.
If significant changes in the orbit of Atlas are detectable between the Voyager
1 and 2 epochs, constraints could then be placed on possible mechanisms for the
observed changes in Prometheus’ motion.
88
CHAPTER 5. ATLAS 89
Figure 5.1: Atlas in FDS43917.52. The F ring is in the bottom right of the imageand the outer edge of the A ring at the top left. Atlas is the circled object. Onlya small section of the complete image is shown. This is the raw image with theintensity enhanced somewhat
CHAPTER 5. ATLAS 90
5.2 Search methodology
Software was written by the author in FORTRAN77 using the SPICE library (Ac-
ton 1990). The program, findimage.f, searched through the Voyager dataset for
candidate images that could include the object of interest, in this case Atlas.
In order to locate candidate images, an orbit had to be assumed for the object.
We used the orbit within the NAIF SP kernels vg1 sat.bsp and vg2 sat.bsp for
the Voyager 1 and Voyager 2 images respectively. Since no great accuracy in the
assumed orbit is required a circular orbit with the elements from Smith et al. (1981)
would have sufficed.
Using the ephemerides of the Voyager spacecraft and Atlas contained within
the utilised SP kernels, the pointing direction from the spacecraft to Atlas at the
exposure time of each image, P, is easily calculated. The effects of the light time,
and planetary aberration effects were allowed for.
Each of the images in the Voyager dataset for Saturn was stepped through in
turn. The pointing direction to Atlas from the spacecraft was calculated at the
image mid-time. The image mid-time is simply the shutter opening time plus half
the length of the exposure. This information is given in the headers of the Voyager
images.
The assumed pointing direction of the camera is then obtained from a SPICE
C-kernel. We used the kernels vg1 sat qmw na.bc, vg1 sat qmw wa.bc,
vg2 sat qmw na.bc and vg2 sat qmw wa.bc for the Voyager 1 NAC and WAC and
Voyager 2 NAC and WAC respectively. As stated in Chapter 4 these kernels were
prepared at QMW from Voyager SEDRs (Gordon 1994). A rotation matrix, called a
C-matrix , is obtained from SPICE C-kernels. Using Eqn. 4.6 the pointing direction
of the camera, BJ2000, is then simply the transpose of the C-matrix times the
boresight vector of the camera in the instrument frame, Binstrument
BJ2000 = CTBinstrument (5.1)
where BJ2000 is in the J2000 reference frame (section 2.1.2).
Due to Atlas’ small size, 18.5 × 17.2 × 13.5 km (Davies et al. 1994), and corre-
sponding faintness when compared to the much larger Prometheus, images where
CHAPTER 5. ATLAS 91
the calculated resolution was worse than 150 km per pixel were instantly rejected.
These images were excluded from the next stage in the process.
The angular separation between the calculated pointing direction to the obj-
ect, P, and the boresight vector of the camera, BJ2000, is then determined. If this
angular separation is greater than a set value it is assumed that Atlas is not in the
image and the image is then rejected. The image is marked for further examination
if the angular separation is less than the set value, as Atlas could possible be in
the image. The set value used in this work was 12(FOV of the camera)+ 2.5× 10−3
radians. The error in the pointing direction of the camera can be up to 2.5 × 10−3
radians (Showalter 1991).
The output from findimage.f is a list of images that have been marked for
further examination. Each of these images is then viewed using the XV image tool
and Atlas searched for. If an image includes an object that could be Atlas it is
marked as an identified image. False identifications of Atlas are removed at the
orbit determination stage 5.6.
5.3 Geometrical correction
All identified images were geometrically corrected using MINAS and the procedures
resloc.pro and geoma.pro as described in Chapter 3. The 800 × 800 pixel raw
image-space images being transformed into 1000 × 1000 pixel object-space images.
5.4 Image navigation
After geometrical correction, as described in Chapter 3, all the identified images
were navigated using MINAS, as described in Chapter 4. The images were navigated
using the outer edge of the A Ring, the inner edge of the Encke gap and the limb
of Saturn. The inner edge of the Encke Gap is circular (Showalter 1991), but the
outer edge of the A Ring is not. Its shape is consistent with the seven lobed pattern
expected due to the 7:6 inner Linblad resonance with the co-orbital satellites Janus
and Epimetheus (Porco et al. 1984). However, the amplitude of the radial distortion
in the A Ring edge is only 6.7 ± 1.5 km (Porco et al. 1984), so for our purposes
CHAPTER 5. ATLAS 92
treating the ring edge as circular is a reasonable approximation. The parameters
used for Saturn, the ring features, and the Voyager cameras in the image navigation
and orbit fitting processes are listed in Table 5.1 Any images for which navigation
Table 5.1: Constants used in the image navigation and orbit determination process.
Where f and σ are the focal length and scale factor of the camera as used in
Eqns. 4.3-4.5
Parameter Value
Radius Outer Edge A-Ring 136774.4kma
Radius Inner Edge Encke Gap 133423.53kmb
Equatorial Radius of Saturnc 60330.0kmd
Radii of Saturne 60268.0 × 60268.0 × 54364.0kmf
J2 0.016298d
J4 −0.000915d
f (Vgr 1 NAC) 1500.19mmg
f (Vgr 1 WAC) 200.29mmg
f (Vgr 2 NAC) 1503.49mmg
f (Vgr 2 WAC) 200.77mmg
σ 84.8214 pixels/mmg
aPorco et al. (1984)bNicholson et al. (1990)cused to calculate spherical harmonicsdCampbell and Anderson (1989)eused to calculate location of limbfDavies et al. (1994)gDanielson et al. (1981)
was not possible, mainly due to the absence of ring features, were discarded. The
SPICE C-matrices for the navigated images, along with the image FDS (Flight
Data System) ID numbers are stored in a text file.
CHAPTER 5. ATLAS 93
5.5 Generating a set of observations
The centroided line,sample (l, s) coordinates of Atlas on the raw image-space images
were transformed into object-space line,sample coordinates using the appropriate
transformation parameters and geoma.pro (section 3.7.2). For each image the
object space line,sample (l, s) coordinates were mapped to instrument frame coor-
dinates using Eqns. 4.1 and 4.2
linstrument = lobject−space − 499.5
sinstrument = sobject−space − 499.5
where (499.5,499.5) are the pixel-coordinates of the centre of the image in object-
space and the origin of the line and sample directions in the instrument frame. The
pointing direction to Atlas, Q, in the instrument frame using Eqns. 4.3, 4.4 and 4.5
is
Qx =1
σsinstrument
Qy =1
σlinstrument
Qz = f
with σ and f from Table 5.1. The unit pointing vector to Atlas in the instrument
frame is simply Q. This is then transformed into a unit pointing vector, P, in
the J2000 reference frame with the origin at the Voyager spacecraft at the image
mid-time using Eqn. 4.6
P = CT Q
where C is the C-matrix determined for the image in the image navigation process
(section 5.4).
Once P had been calculated for each image the final result of the image naviga-
tion process was a text file containing the image mid-time and unit pointing vector
to Atlas for each image. This text file was the set of observations used for the
process of orbit determination.
CHAPTER 5. ATLAS 94
5.6 Orbit determination
A FORTRAN77 program was written by the author for the purpose of orbit deter-
mination utilising the SPICE library (Acton 1990) and SPICE library extensions
written by Gordon (1994). This program, diffcorr.f, fits a set of observations to
the precessing elliptical model of Taylor (1998) which is described in Section 2.5.2.
The program takes as input an ephemeris for the observer and a set of initial
orbital elements at epoch. Seven parameters are fitted a0, λ0, n0, e0, 0, i0 and Ω0.
It is assumed that a, n, e and i have no time dependency. The effects of any body
other than Saturn are ignored
Based on the initially assumed orbital elements a set of expected pointing vec-
tors to Atlas at the observations mid-times are calculated. Full allowance is made
for light time and planetary aberration effects. Differences between the expected
pointing vectors and the actual observations are used, along with a differential cor-
rection technique, to make a ‘better’ estimate of the initial orbital elements (section
2.6). A ‘better’ estimate of the initial orbital elements is one where the root mean
square (rms) of the difference between the expected pointing vector and the actual
observations is reduced.
The initial orbital elements used to calculate the expected pointing vectors are
then updated to this ‘better’ value, the differences recalculated and the differential
correction algorithm run again.
The process is repeated iteratively, with the rms of the differences being reduced
at each iteration, until the program has converged. When convergence occurs the
difference between the initial orbital elements and the ‘better’ elements is smaller
then the formal errors from the fitting process associated with each parameter.
The final orbital elements at epoch after convergence provide the least squares
best fit to the supplied observations.
A complementary program to diffcorr.f was also written by the author. Again
written in FORTRAN 77 using the SPICE libraries the program, findlinesample.f,
calculates line and sample co-ordinates of an object in an image. The input variables
are the 7 elements of the object’s orbit and the navigated C-matrix for the image.
In some cases the identification of a satellite is ambiguous, there may be several
CHAPTER 5. ATLAS 95
objects in approximately the right location which could be the satellite. An orbit
is first determined using the unambiguous detections and then findlinesample.f
used to calculate the co-ordinates the object should be at in all the other images.
The other candidate images can then be searched near the identified co-ordinates.
Experience has shown that if the identified co-ordinates are on the image there is
usually a satellite candidate object within 2-3 pixels. This technique allows the
identification of faint satellite detections that are missed during the initial visual
examination of an image.
During the early stages of the research the orbits were fitted using a method
involving the numerical integration of the full equations of motion (section 2.5.1).
The Runge-Kutta-Nystrom 12th order scheme of Dormand et al. (1987a, 1987b) was
used to integrate the equations. As detailed in Chapter 2, the end result from a
numerical integration is the state at epoch for all the bodies in the system. A state is
of little utility when a determined orbit is to be compared with the orbital elements
obtained by other authors. At the time I was unable to fit a precessing elliptical
model to an ephemeris generated by numerical integration. All the elliptical models
used failed to converge during the differential correction process.
The precessing elliptical model of Taylor (1998) was first used in the final year
of the research, and was the first such model to successfully converge during the
differential correction process. Time constraints prevented its use to fit an orbit to
a numerically generated ephemeris.
5.7 Identified images
The 7 Voyager 1 and 18 Voyager 2 navigable images of Atlas that we identified
are listed in Tables 5.2 and 5.3 respectively. Of these images the 4 Voyager 1 and
7 Voyager 2 which were marked in the data of Synnott et al. (1983) as used by
Jacobson (private communication) as including Atlas are indicated.
CHAPTER 5. ATLAS 96
Table 5.2: Voyager 1 Images of Atlas
FDS Image Mid-time Used by Line Sample Phase Res. a
No. UTC Synnott lcont. scont. Angle() km
34667.32 Nov. 3 18:34:58.7 • 687.5 588.4 12.4 99.2
34775.24 Nov. 7 08:52:36.9 • 558.7 111.5 12.4 61.4
34785.21 Nov. 7 16:50:12.9 • 167.6 784.5 14.5 57.9
34785.25 Nov. 7 16:53:24.9 • 627.4 303.6 14.5 57.8
34830.20 Nov. 9 04:49:22.5 235.6 232.5 12.5 42.4
34930.48 Nov. 12 13:11:46.7 0.5 592.4 39.7 5.7
34930.59 Nov. 12 13:20:34.7 388.6 165.6 40.0 55.3
aThe year of all images is 1980. All images are taken with the Narrow Angle Camera (NAC)
except 34930.59 which is a Wide Angle Camera (WAC) image. The resolution and sun phase
angle information is calculated using the derived Voyager 1 orbit in Table 5.4 .
CHAPTER 5. ATLAS 97
Table 5.3: Voyager 2 Images of Atlas.
FDS Image Mid-time Used by Line Sample Phase Res.a
No. UTC Synnott lcont. scont. Angle() km
43655.12 Aug. 14 11:04:00.5 • 303.5 257.5 7.9 92.5
43655.28 Aug. 14 11:16:48.5 • 413.5 175.6 7.9 92.3
43655.44 Aug. 14 11:29:36.5 • 342.5 112.5 7.9 92.1
43656.00 Aug. 14 11:42:24.5 • 391.2 290.4 7.9 92.0
43656.16 Aug. 14 11:55:12.5 • 465.4 153.9 7.9 91.8
43684.35 Aug. 15 10:34:24.7 201.5 297.5 6.8 85.6
43703.18 Aug. 16 01:32:48.5 164.5 418.8 6.9 81.1
43737.47 Aug. 17 05:08:00.5 677.4 533.5 6.9 72.0
43752.38 Aug. 17 17:00:49.4 425.8 98.6 7.3 67.2
43779.59 Aug. 18 14:53:35.5 505.5 643.7 8.6 61.8
43810.56 Aug. 19 15:39:11.5 12.4 273.5 7.5 54.0
43818.37 Aug. 19 21:47:59.5 175.5 31.6 9.6 51.2
43835.06 Aug. 20 10:59:11.5 380.6 604.5 9.8 47.6
43843.48 Aug. 20 17:56:47.5 614.3 133.5 7.8 44.5
43854.20 Aug. 21 02:22:23.5 100.9 47.1 10.5 42.3
43861.50 Aug. 21 08:22:23.5 500.2 362.3 8.1 39.9
43917.52 Aug. 23 05:11:59.5 • 626.2 431.7 9.2 26.0
43938.11 Aug. 23 21:27:13.9 • 263.2 6.4 10.6 21.3
aThe year of all images is 1981. All images are taken with the Narrow Angle Camera (NAC).
The resolution and sun phase angle information is calculated using the derived Voyager 2 orbit in
Table 5.5 .
CHAPTER 5. ATLAS 98
An additional Voyager 1 image was identified in the Synnott et al. (1983) data,
FDS 34667.36 taken at 1980 Nov. 3 18:38:08.5 UTC, which we were unable to
confirm actually existed let alone locate. Smith et al. (1981) state that 8 Voyager
1 images were used, we are currently unable to account for the extra 3 images.
Possibly they are the additional 3 images that we identified.
5.8 Results
We determined orbits at the Voyager 1 and Voyager 2 epochs separately using
the 7 Voyager 1 and 18 Voyager 2 images respectively. A determination using the
combined data was also calculated. The orbital elements are shown in Table 5.4.
The Voyager 1 elements of Smith et al. (1981) are shown for comparison.
Table 5.4: The orbital elements of Atlas. The epoch for the elements from thiswork is 2444839.6682. The Smith et al. (1981) elements are epoch 2444513.5 withλ advanced forwards to 2444839.6682 using n = 598.08/day.
para- Smith Voyager 1 Voyager 2 Combined
metera et al. (1981) only only Voyager 1 & 2
a 137670 137621± 62 137714 ± 14 137704 ± 12
acalc - 137669 ± 7 137667.0± 0.5 137666.37 ± 0.03
λ 247 ± 16 312 ± 13 317.43 ± 0.02 317.45 ± 0.01
n 598.08 ± 0.05 598.29 ± 0.05 598.302 ± 0.003 598.3069± 0.0002
e (2 ± 3) × 10−3 (1.2 ± 1.4) × 10−3 (9 ± 2) × 10−4 (6 ± 1) × 10−4
i 0.0 ± 0.2 0.17 ± 0.06 0.02 ± 0.01 0.017 ± 0.007
- 198 ± 18 243 ± 8 233 ± 8
Ω - 82 ± 5 318 ± 16 317 ± 19
aDistances are in km, longitudes in degrees and rates in degrees/day. Errors for elements
from this work are the formal errors from the fitting process. All longitudes measured from the
ascending node of Saturn’s equator at epoch on the Earth mean equator at J2000, i measured
from Saturn’s equatorial plane at epoch.
CHAPTER 5. ATLAS 99
5.9 Discussion
The mean motions derived from the Voyager 1 & 2 data separately are within 1σ
of each other. These results may be misleading due to the large error associated
with the Voyager 1 mean motion. This large error is a direct consequence of the
low number (7) of images utilised.
The difference between the Voyager 2 only and combined mean motions may be
a better value to use, they differ by 1.7σ. A difference which is not large enough to
be significant.
The fit to the Voyager 1 data is clearly much worse than the fit to the Voyager
2 data. The error in the Voyager 1 value of e is actually larger than the derived
value.
It is reasonable to conclude that there was no significant change in the orbit of
Atlas between the Voyager encounters. This being the case, fitting a single orbit to
the combined dataset gives the most accurate determination. As a result we adopt
the combined orbit fit for the orbital elements of Atlas (see Table 5.8).
The adopted combined fit differs significantly from that of Smith et al. (1981),
the mean motion being ∼ 0.22/day higher. Taking the mean longitude at JD
2444513.5 of Smith et al. (1981) and precessing it forwards to JD 244839.6682
using the n from our combined fit, 598.3069± 0.0002/day, gives a mean longitude
at epoch, λ, of 321, close to the combined fit value of 317. It appears that the
mean motion of Smith et al. (1981) is too low by ∼ 0.22/day. To allow a direct
comparison to be made between our results and those of Smith et al. (1981) we
derived a fit to the Voyager 1 dataset at the Voyager 1 epoch, JD 2444513.5, see
Table 5.5.
Table 5.5 clearly shows that the mean longitude at JD 2444513.5 of Smith et
al. (1981) is comparable with that of our Voyager 1 fit at JD 2444513.5. The large
difference in λs in table is therefore primarily a result of the low mean motion of
Smith et al. (1981).
CHAPTER 5. ATLAS 100
Table 5.5: Voyager 1 orbital elements of Atlas. The epoch is JD 2444513.5. TheSmith et al. (1981) elements have been changed into the reference frame used forthis work.
para- Smith et al. (1981) Voyager 1
metera only
a 137670 137621 ± 62
acalc - 137669 ± 7
λ 292 ± 1.5 289 ± 2
n 598.08 ± 0.05 598.29 ± 0.05
e (2 ± 3) × 10−3 (1.2 ± 1.1) × 10−3
i 0.0 ± 0.2 0.17 ± 0.05
- 339 ± 50
Ω - 296 ± 10
aDistances are in km, longitudes in degrees and rates in degrees/day. Errors for elements
from this work are the formal errors from the fitting process. All longitudes measured from the
ascending node of Saturn’s equator at epoch on the Earth mean equator at J2000, i measured
from Saturn’s equatorial plane at epoch.
5.10 The distribution of longitudes at observa-
tion mid-times
Figs. 5.2 and 5.3 show the x-y coordinates of Atlas, during the Voyager 1 and
Voyager 2 encounters, in the planetary reference frame used for Saturn, at the
observation mid-times. The x-y plane is the equator plane of Saturn at epoch. The
positions are numbered in ascending chronological order, the same order as they
are presented in Tables 5.2 and 5.3.
CHAPTER 5. ATLAS 101
1
2
34
5
6
7
mean direction to Sun
Figure 5.2: The x-y plane coordinates of Atlas at the Voyager 1 observation mid-times calculated from the combined Voyager fit in Table 5.4
CHAPTER 5. ATLAS 102
1
2
3
4
5
6
7
8
9
10
1112
13
14
15
16
17
18
mean direction to Sun
Figure 5.3: The x-y plane coordinates of Atlas at the Voyager 2 observation mid-times calculated from the combined Voyager fit in Table 5.4
CHAPTER 5. ATLAS 103
Examination of Figs. 5.2 and 5.3 clearly shows that none of the images used has
Atlas either transiting Saturn or in the shadow of the planet. This lack of observa-
tions in these longitudes ranges is unsurprising, Atlas being effectively invisible in
the visible light range in the shadow and swamped by light from the planetary disc
when transiting.
Locations which are marked in the same colour (except black) indicate observa-
tions which are close together in time, all occurring within 1/2 an orbital period.
Ideally the entire longitude range for the orbit should be sampled, this is especially
important for accurate determination of e (Synnott et al. 1983). The Voyager 1
data does not sample the available longitude range very well at all, not unsurprising
considering there are only 7 images available.
The Voyager 2 observations (Fig. 5.2) sample the longitude range reasonably
well. Although there is some clustering of observations, it is not enough to cause
concern over the accuracy of the orbital fit.
5.11 The orbit of Atlas in JPL Ephemerides
Since there is only one published orbit for Atlas, that of Smith et al. (1981), an
analysis was performed on the JPL Ephemerides to investigate the orbit of Atlas
as presented in them.
A program was written which fits the freely precessing ellipse model of Taylor
(1998), detailed in section 2.5.2, to a set of planetocentric position vectors obtained
from an ephemeris. The perturbations due other bodies in the system, resonances
etc. were ignored.
The reference frame used is a Cartesian coordinate system frame. The z-axis
is parallel to the spin angular momentum vector of Saturn at epoch. The x-axis
points along the direction to the intersection at epoch of the ascending node of the
saturnian equator with the mean Earth Equator at J2000. The y-axis lies in the
plane of the saturnian equator orthogonal to both the x- and z-axes. This is the
planetary reference frame (for Saturn) as illustrated in Fig. 2.9.
The fitting program, FitOrb2Ephm.f, generates a specified number of plan-
CHAPTER 5. ATLAS 104
etocentric position vectors for the object at given times. The precessing ellipse is
then fitted to the set of position vectors using a least squares differential correction
technique. The process is identical to that described in sections 2.5.2 and 2.6 except
that the observations are now positions not pointing vectors and the rms error is a
distance in km not an angle.
For Atlas, 160 position vectors at 3600.0 second intervals were obtained from an
ephemeris. A precessing ellipse was fitted to the data with the starting values for
the parameters of the ellipse being taken from the adopted combined orbit fit from
Table 5.5. A second fit was also performed with the apsidal and nodal precession
rates, β and γ set equal to 0.0/day. Position vectors were obtained from the SP-
kernels vg2 sat.bsp (the Voyager 2 ephemeris) and sat081.4.bsp (the current Cassini
small satellites ephemeris). The results obtained are presented in Tables 5.6 and
5.7. In both cases the North Pole orientation of Saturn is taken from French et
al. (1993).
The fitting process was very robust, a wide range of starting values for the
parameters of the ellipse converged on the exact values given in Tables 5.6 and 5.7.
The only strong correlations were between the mean longitude at epoch (λ) and the
mean motion (n).
Examination of the data presented in Tables 5.6 and 5.7 indicates that a simple
ellipse is a very good model for the ephemerides of Atlas, at least over several tens
of days. Somewhat surprisingly the non-precessing ellipse is a better fit to the
ephemerides than the precessing model. The very low eccentricies suggest that it
was assumed that Atlas was in a circular orbit for the generation of the ephemerides.
The JPL ephemerides are generated by numerical integration of the full equa-
tions of motion for an object in a many body system (Standish 1990). Our analysis
of the ephemerides of Atlas in vg2 sat.bsp and sat081.4.bsp suggests that they are
so consistent with a non-precessing elliptical model that such a model may well
have been used in their generation and not a numerical integration.
It is possible that the very limited available data (12 images in the Synnott et
al. (1983) data supplied to Jacobson (private communication)) prevented reliable
numerical determination of Atlas’ orbit and forced the use of ellipses.
CHAPTER 5. ATLAS 105
Table 5.6: Orbital elements for Atlas from a fit to the SP kernel vg2 sat.bsp: Epoch2444839.6682 JED. 160 positions at 3600.0 sec. intervals starting at 244824.6682JED.
para- Precessing Ellipse Model Non-Precessing Ellipse Model
metera β = γ = 0.0
a 137640.00± 0.10 137639.99998± 0.00003
acalc 137666.512± 0.003 137666.5118296± 0.0000009
λ 317.6214 ± 0.0002 317.62142221± 0.00000007
n 598.30600 ± 0.00002 598.306000000± 0.000000006
e (0.0 ± 4.5) × 10−7 (2.6 ± 0.1) × 10−10
i (1.2752 ± 0.006) × 10−2 (1.279067 ± 0.000002) × 10−2
224 ± 134 191 ± 3
Ω 322.9 ± 0.2 356 ± 7
rms 0.17 0.00005
aDistances are in km, longitudes in degrees and rates in degrees/day. Errors for elements
from this work are the formal errors from the fitting process. All longitudes measured from the
ascending node of Saturn’s equator at epoch on the Earth mean equator at J2000, i measured
from Saturn’s equatorial plane at epoch.
The fits to vg2 sat.bsp and sat081.4.1.bsp differ slightly. Some of the difference
is almost certainly due to the fact the the ephemerides cover epochs 20 years apart.
Since the only observations of Atlas is the small Voyager set, none of the differences
can be due to additional data on Atlas. Other factors must be responsible.
In numerical integrations of the full equations of motion of a body the positions,
masses and velocities of other bodies are factors. Sat081.4.bsp is more recent than
vg2 sat.bsp and presumably uses more reliable data on the positions, masses and
velocities of Saturn, the saturnian satellites, Jupiter, Uranus, the Sun etc. As such
the orbital elements derived from sat081.4.1.bsp have a higher degree of confidence
than those from vg2 sat.bsp.
The mean motion derived from sat081.4.bsp, n = 598.30666 ± 0.00002/day,
is within 1σ of our derived value from the combined Voyager 1 & 2 dataset of
n = 598.3069 ± 0.0002/day. Interestingly the mean motions derived from both
CHAPTER 5. ATLAS 106
Table 5.7: Orbital elements for Atlas from a fit to the SP kernel sat081.4.bsp: Epoch2453371.00 JED. 160 positions at 3600.0 sec. intervals starting at 2453360.00 JED.
para- Precessing Ellipse Model Non-Precessing Ellipse Model
metera β = γ = 0.0
a 137666.47± 0.09 137666.47 ± 0.02
acalc 137666.412± 0.003 137666.4117± 0.0006
λ 230.0435 ± 0.0002 230.04348± 0.00003
n 598.30666 ± 0.00002 598.306656± 0.00004
e (1.3 ± 0.4) × 10−6 (1.29 ± 0.09) × 10−6
i (1.141 ± 0.005) × 10−2 (1.147 ± 0.001) × 10−2
143 ± 16 121 ± 2
Ω 338.8 ± 0.2 0.97 ± 0.06
rms 0.15 0.03
aDistances are in km, longitudes in degrees and rates in degrees/day. Errors for elements
from this work are the formal errors from the fitting process. All longitudes measured from the
ascending node of Saturn’s equator at epoch on the Earth mean equator at J2000, i measured
from Saturn’s equatorial plane at epoch.
ephemerides are much higher than the published value of n = 598.08 ± 0.05/day
(Smith et al. 1981) but much closer to all our derived values (see Table 5.4).
5.12 Conclusions
There is no evidence that the orbit of Atlas changed between the Voyager 1 and
Voyager 2 encounters. As such Atlas appears to be behaving in a completely pre-
dictable manner. The elements of this moon’s orbit have been determined to much
greater accuracy then previously, enabling predictions of its position when first de-
tected by Cassini to be made with higher confidence. We have also been able to
place much tighter constraints on Atlas’ eccentricity and inclination. The orbit we
have adopted is from the combined Voyager data and summarised in Table 5.8.
Our adopted value for the mean motion, n = 598.3069 ± 0.0002/day is ∼0.22/day higher than the published value (Smith et al. 1980), but within 1σ of the
CHAPTER 5. ATLAS 107
Table 5.8: The adopted orbital elements of Atlas: epoch2444839.6682
parametersa Adopted Orbit
a 137704 ± 12
acalc 137666.37± 0.03
λ 317.45 ± 0.01
n 598.3069 ± 0.0002
e (6 ± 1) × 10−4
i 0.017 ± 0.007
233 ± 8
Ω 317 ± 19
aDistances are in km, longitudes in degrees and rates in degrees/day.
Errors for elements from this work are the formal errors from the fitting
process. All longitudes measured from the ascending node of Saturn’s
equator at epoch on the Earth mean equator at J2000, i measured from
Saturn’s equatorial plane at epoch.
value derived from the JPL Ephemeris sat081.4.bsp. The similarity between n from
this work and from the JPL Ephemerides suggests that unpublished data may be
included in the JPL Ephemerides.
Chapter 6
Prometheus
6.1 Introduction
The saturnian satellite Prometheus, provisionally named S1980S27, was discovered
during the Voyager 1 encounter with Saturn in 1980 (Smith et al. 1981). Like Atlas
(see Chapter 5), Prometheus orbits between the outer edge of the A ring and the
F ring. The first published orbit for Prometheus was derived using 18 Voyager 1
images (Smith et al. 1981). Additional images of Prometheus were obtained during
the Voyager 2 encounter in 1981 and 27 of them were used by Synnott et al. (1983)
to derive a new orbit. Fig. 6.1 shows the highest resolution image of Prometheus
currently available. Fig. 6.2 is a WAC image shuttered at the same time as the
NAC image in Fig. 6.1, clearly illustrating Prometheus’ location with respect to the
F ring and the outer edge of the A ring.
Synnott et al. (1983) derived separate values for Prometheus’ eccentricity, e, and
longitude of pericentre, , at each encounter using the 18 Voyager 1 and 27 Voyager
2 images, while the mean motion, n, and semi-major axis, a, were derived using
combined images from both encounters. The value for the inclination, i, was derived
from Voyager 2 data only. Synnott et al. (1983) stated that derived solutions for
e, i and using the combined data set were consistent with the Voyager 2 values.
The full combined solution and a separate value for the Voyager 2 mean motion
were not published.
108
CHAPTER 6. PROMETHEUS 109
Figure 6.1: Prometheus in FDS43998.29. This is a region of the highest resolution,∼ 6km per pixel, narrow angle camera image of Prometheus. This is a raw imagewith the intensity slightly enhanced.
CHAPTER 6. PROMETHEUS 110
Figure 6.2: Prometheus in FDS43998.32. This is a raw wide angle camera imageshuttered at the same time as FDS43998.29. The Encke and Keeler Gaps in theA ring are clearly visible as is the narrow F ring running diagonally through thecentre of the image. Prometheus is the circled object.
CHAPTER 6. PROMETHEUS 111
The commonly quoted values for Prometheus’ orbital elements (see, for example,
Burns 1986) are those from Synnott et al. (1983), with e, and i derived only from
the 27 Voyager 2 images. Only the determination of n, and therefore a, also includes
data from the 18 Voyager 1 images.
In determining an orbit it is standard practice to combine data sets obtained at
widely separate epochs to produce a single solution (see, for example, Harper and
Taylor 1994). This combination of observations, often separated by time intervals
as long as decades, usually leads to highly accurate determinations. There was no
reason at the time to think that combining Voyager 1 and Voyager 2 results would
not lead to a more accurate determination of Prometheus’ orbit.
As described in section 1.1, in 1995 Prometheus was observed to be lagging
behind its expected location based on the Voyager ephemerides. We discussed
possible explanations for this lag in section 1.3. We re-examined the Voyager dataset
in an attempt to locate additional images to those used by Synnott et al. (1983).
All the located images were used to determine an orbit for Prometheus. Three
separate orbit determinations were performed
1. Using only the Voyager 1 images
2. Using only the Voyager 2 images
3. Using the combined Voyager 1 and Voyager 2 images
in an attempt to ascertain if the orbit of Prometheus’ changed significantly in the
9 months between the two Voyager encounters. At the very least, use of images
in addition to those of Synnott et al. (1983) would allow an improvement in the
determined elements. In addition, an independent check on Prometheus’ published
orbital elements (Jacobson used Synnott et al.’s (1983) data) would immediately
indicate if the elements of Synnott et al. (1983) are grossly incorrect.
6.2 Search methodology and orbit determination
The search methodology, geometrical correction, image navigation and orbit deter-
mination techniques used for Prometheus were almost identical to those already
CHAPTER 6. PROMETHEUS 112
detailed for Atlas (sections 5.2, 5.3, 5.4 and 5.6 respectively). The only differences
being that Prometheus not Atlas was being searched for and images where the
resolution was worse then 320 km per pixel (not 150 km per pixel) were instantly
rejected. This figure was chosen because the worst resolution image used by Synnott
et al. (1983) has a resolution of 320 km per pixel.
Since these techniques have already been described in Chapter 5 we shall not
describe them again here.
6.3 Identified images
The 56 Voyager 1 and 66 Voyager 2 navigable images of Prometheus that we iden-
tified are listed in Tables 6.1 and 6.2 respectively. Also indicated in the tables is
whether a particular image was used by Synnott et al. (1983) for their Prometheus
orbit determination. Synnott et al. ’s (1983) data was also used by Jacobson (private
communication) for an independent determination of Prometheus’ orbital elements.
The line and sample coordinates are given in continuous and not integer pixel coor-
dinates (see section 3.3). The resolutions and solar phase angle information included
for each image were calculated using the individually determined orbits from this
work for Prometheus at the Voyager 1 and Voyager 2 epochs (see Tables 6.3 and
6.4 respectively).
CHAPTER 6. PROMETHEUS 113
Table 6.1: Voyager 1 Images.
FDS Image Mid-time Used by Line Sample Phase Res.a
No. UTC Synnott lcont. scont. Angle() km
34151.07 Oct. 17 13:26:60 • 293.6 200.4 13.3 278.5
34271.15 Oct. 21 13:33:23 539.6 742.4 12.9 235.8
34388.24 Oct. 25 11:16:35 146.8 463.5 13.0 196.9
34388.30 Oct. 25 11:21:23 100.6 366.4 13.0 196.9
34388.36 Oct. 25 11:26:11 100.6 385.5 13.1 196.9
34388.42 Oct. 25 11:30:59 92.6 451.5 13.1 196.8
34388.48 Oct. 25 11:35:47 115.6 438.5 13.1 196.7
34388.54 Oct. 25 11:40:35 189.3 278.2 13.1 196.7
34389.00 Oct. 25 11:45:23 217.9 274.3 13.1 196.6
34389.06 Oct. 25 11:50:11 181.6 445.7 13.1 196.6
34389.12 Oct. 25 11:54:59 202.7 394.3 13.1 196.5
34389.18 Oct. 25 11:59:47 192.4 286.4 13.1 196.5
34389.24 Oct. 25 12:04:35 47.6 397.4 13.1 196.4
34389.30 Oct. 25 12:09:23 127.9 317.7 13.1 196.4
34389.36 Oct. 25 12:14:11 • 245.8 274.3 13.1 196.3
34389.42 Oct. 25 12:18:59 120.7 274.3 13.2 196.3
34389.48 Oct. 25 12:23:47 90.9 254.9 13.2 196.2
34389.54 Oct. 25 12:28:35 51.8 374.6 13.2 196.1
34390.00 Oct. 25 12:33:23 95.8 303.4 13.2 196.1
34390.06 Oct. 25 12:38:11 122.6 292.9 13.2 196.0
34390.12 Oct. 25 12:42:59 132.1 371.7 13.2 196.0
34390.18 Oct. 25 12:47:47 • 190.5 261.5 13.2 195.9
34390.24 Oct. 25 12:52:35 • 220.8 240.6 13.2 195.8
34390.30 Oct. 25 12:57:23 221.7 316.5 13.2 195.8
aThe year of all images is 1980. All images are taken with the Narrow Angle Camera (NAC).
The resolution and sun phase angle information is calculated using the derived Voyager 1 orbit in
Table 6.3
CHAPTER 6. PROMETHEUS 114
FDS Image Mid-time Used by Line Sample Phase Res.a
No. UTC Synnott lcont. scont. Angle() km
34390.36 Oct. 25 13:02:11 215.5 407.3 13.2 195.7
34390.42 Oct. 25 13:06:59 197.7 318.4 13.2 195.6
34390.54 Oct. 25 13:16:35 83.2 375.0 13.2 195.5
34391.00 Oct. 25 13:21:23 150.6 282.5 13.2 195.4
34391.06 Oct. 25 13:26:11 • 178.6 325.7 13.2 195.4
34391.12 Oct. 25 13:30:59 • 201.7 370.8 13.3 195.3
34391.18 Oct. 25 13:35:47 • 256.5 289.6 13.3 195.2
34391.24 Oct. 25 13:40:35 • 257.5 392.7 13.3 195.2
34391.30 Oct. 25 13:45:23 • 313.6 280.8 13.3 195.1
34391.36 Oct. 25 13:50:11 287.6 454.7 13.3 195.0
34400.10 Oct. 25 20:41:24 762.2 752.5 12.7 191.3
34444.00 Oct. 27 07:45:22 • 700.7 392.5 13.1 177.6
34637.32 Nov. 02 18:34:59 • 552.6 599.5 12.8 108.7
34637.36 Nov. 02 18:38:11 571.7 530.3 12.7 108.6
34637.40 Nov. 02 18:41:23 576.7 488.5 12.7 108.6
34667.16 Nov. 03 18:22:10 • 224.7 260.4 13.6 99.6
34667.20 Nov. 03 18:25:23 • 221.6 350.2 13.6 99.6
34667.24 Nov. 03 18:28:35 • 251.2 337.4 13.6 99.5
34701.59 Nov. 04 22:08:35 458.5 19.5 13.4 88.1
34702.07 Nov. 04 22:14:59 404.6 48.4 13.4 88.0
34702.11 Nov. 04 22:18:11 171.9 352.4 13.5 88.0
34702.15 Nov. 04 22:21:23 216.2 268.5 13.5 88.0
34702.19 Nov. 04 22:24:35 141.1 431.3 13.5 87.9
34727.25 Nov. 05 18:29:23 • 33.4 155.8 13.3 77.3
34727.35 Nov. 05 18:37:23 • 104.5 22.3 13.3 77.2
34757.08 Nov. 06 18:15:47 113.6 269.4 13.7 68.9
aThe year of all images is 1980. All images are taken with the Narrow Angle Camera (NAC).
The resolution and sun phase angle information is calculated using the derived Voyager 1 orbit in
Table 6.3
CHAPTER 6. PROMETHEUS 115
FDS Image Mid-time Used by Line Sample Phase Res.a
No. UTC Synnott lcont. scont. Angle() km
34779.22 Nov. 07 12:02:59 98.2 76.6 14.4 59.9
34785.36 Nov. 07 17:02:13 • 203.5 621.6 12.7 57.3
34795.50 Nov. 08 01:13:25 685.7 478.4 14.5 54.8
34802.52 Nov. 08 06:50:59 • 758.6 139.8 13.1 51.0
34832.01 Nov. 09 06:10:13 681.9 376.9 15.1 42.3
34832.05 Nov. 09 06:13:25 83.6 712.9 15.2 42.2
aThe year of all images is 1980. All images are taken with the Narrow Angle Camera (NAC).
The resolution and sun phase angle information is calculated using the derived Voyager 1 orbit in
Table 6.3
CHAPTER 6. PROMETHEUS 116
Table 6.2: Voyager 2 Images
FDS Image Mid-time Used by Line Sample Phase Res. a
No. UTC Synnott lcont. scont. Angle() km
42728.56 Jul. 14 14:03:11 • 665.5 720.5 6.9 319.4
42893.36 Jul. 20 01:47:11 • 45.5 38.4 6.8 278.9
43023.52 Jul. 24 09:59:59 • 50.5 113.7 6.7 247.7
43393.15 Aug. 05 17:30:29 149.8 489.6 6.6 157.7
43398.41 Aug. 05 21:51:16 • 704.9 567.4 7.2 156.0
43401.17 Aug. 05 23:56:01 • 316.6 102.6 7.2 154.4
43479.29 Aug. 08 14:29:40 290.3 748.7 6.7 134.8
43491.09 Aug. 08 23:49:38 • 778.3 720.8 7.3 133.1
43494.02 Aug. 09 02:08:01 • 444.2 184.5 7.3 131.4
43536.26 Aug. 10 12:03:13 • 442.6 344.5 6.6 121.2
43544.23 Aug. 10 18:24:49 • 485.8 638.4 7.3 120.6
43585.25 Aug. 12 03:14:25 356.7 267.7 7.6 109.0
43585.29 Aug. 12 03:17:37 403.6 213.3 7.5 109.0
43626.58 Aug. 13 12:28:48 • 628.5 193.3 6.9 98.4
43627.30 Aug. 13 12:54:24 • 538.3 349.5 6.8 98.4
43628.34 Aug. 13 13:45:36 • 548.7 576.5 6.7 98.5
43629.38 Aug. 13 14:36:48 • 500.0 595.5 6.6 98.6
43630.42 Aug. 13 15:28:00 • 316.6 593.5 6.6 98.7
43635.50 Aug. 13 19:34:24 • 271.2 687.6 7.5 98.0
43646.06 Aug. 14 03:47:12 • 637.6 501.7 6.8 93.9
43655.12 Aug. 14 11:04:00 • 184.8 573.2 7.7 93.0
43655.28 Aug. 14 11:16:48 • 286.7 430.5 7.7 92.9
43655.44 Aug. 14 11:29:36 • 209.5 303.6 7.8 92.7
43656.00 Aug. 14 11:42:24 • 253.6 416.6 7.8 92.6
43656.16 Aug. 14 11:55:12 • 324.6 212.3 7.8 92.4
aThe year of all images is 1981. All images are taken with the Narrow Angle Camera (NAC).
The resolution and sun phase angle information is calculated using the derived Voyager 2 orbit in
Table 6.4
CHAPTER 6. PROMETHEUS 117
FDS Image Mid-time Used by Line Sample Phase Res. a
No. UTC Synnott lcont. scont. Angle() km
43657.20 Aug. 14 12:46:24 • 437.2 236.9 7.9 91.8
43657.52 Aug. 14 13:12:00 • 431.8 54.3 7.9 91.5
43663.24 Aug. 14 17:37:36 • 741.6 90.6 7.1 89.3
43663.56 Aug. 14 18:03:12 • 644.4 244.6 7.0 89.3
43684.29 Aug. 15 10:29:37 559.9 377.9 6.7 84.9
43684.35 Aug. 15 10:34:25 552.3 273.0 6.7 84.9
43684.41 Aug. 15 10:39:13 467.9 259.0 6.7 84.9
43686.55 Aug. 15 12:26:25 385.6 416.6 6.8 85.1
43686.59 Aug. 15 12:29:37 376.4 517.9 6.9 85.1
43695.16 Aug. 15 19:07:12 253.4 115.5 8.1 81.9
43695.20 Aug. 15 19:10:25 260.3 187.6 8.1 81.9
43695.24 Aug. 15 19:13:37 318.6 123.2 8.1 81.9
43703.18 Aug. 16 01:32:49 752.9 504.0 6.8 80.3
43703.22 Aug. 16 01:36:01 732.3 537.1 6.8 80.3
43710.08 Aug. 16 07:00:48 • 252.8 140.7 8.0 79.4
43723.20 Aug. 16 17:34:25 • 158.6 216.5 6.9 75.9
43732.18 Aug. 17 00:44:49 253.4 125.3 8.4 72.6
43732.22 Aug. 17 00:48:01 250.6 193.5 8.4 72.6
43732.26 Aug. 17 00:51:13 335.4 98.7 8.4 72.5
43737.47 Aug. 17 05:08:01 503.0 58.7 7.2 71.0
43737.51 Aug. 17 05:11:13 512.2 136.7 7.2 71.0
43737.55 Aug. 17 05:14:26 519.1 219.6 7.1 71.0
43774.21 Aug. 18 10:23:12 480.0 57.5 7.1 74.1
43785.17 Aug. 18 19:08:00 313.7 346.6 7.4 61.7
43785.21 Aug. 18 19:11:12 124.5 145.8 9.0 60.1
43796.05 Aug. 19 03:46:25 782.5 575.3 9.0 60.1
aThe year of all images is 1981. All images are taken with the Narrow Angle Camera (NAC).
The resolution and sun phase angle information is calculated using the derived Voyager 2 orbit in
Table 6.4
CHAPTER 6. PROMETHEUS 118
FDS Image Mid-time Used by Line Sample Phase Res. a
No. UTC Synnott lcont. scont. Angle() km
43796.13 Aug. 19 03:52:48 366.5 311.3 7.2 57.4
43805.50 Aug. 19 11:34:24 502.4 84.4 7.2 57.4
43843.36 Aug. 20 17:47:12 749.2 170.5 9.3 54.2
43849.10 Aug. 20 22:14:24 700.7 241.9 9.9 44.3
43858.21 Aug. 21 05:35:12 237.3 604.6 7.9 43.2
43858.25 Aug. 21 05:38:24 68.9 411.1 10.3 41.8
43859.55 Aug. 21 06:50:24 631.1 98.8 10.4 41.7
43932.27 Aug. 23 16:52:00 110.5 754.6 10.6 40.8
43932.37 Aug. 23 17:00:00 180.1 558.8 14.7 22.4
43932.47 Aug. 23 17:08:00 80.8 544.5 14.8 22.2
43933.07 Aug. 23 17:24:07 86.7 110.4 15.0 22.0
43933.17 Aug. 23 17:32:07 133.6 35.2 15.0 21.9
43933.27 Aug. 23 17:40:07 100.6 38.6 15.0 21.8
44253.58 Sep. 03 10:04:48 114.5 326.5 92.9 67.8
44254.31 Sep. 03 10:31:12 8.6 624.5 92.7 67.9
aThe year of all images is 1981. All images are taken with the Narrow Angle Camera (NAC).
The resolution and sun phase angle information is calculated using the derived Voyager 2 orbit in
Table 6.4
CHAPTER 6. PROMETHEUS 119
6.4 Results
The derived orbit fits for Prometheus, using Voyager 1 and Voyager 2 data sepa-
rately are shown in Tables 6.3 and 6.4 respectively. All errors for our fits are the
formal errors from the fitting process.
For comparison, the fits of Synnott et al. (1983) and Jacobson (private com-
munication) are also shown. No errors have been included for Jacobson’s elements
since none are given in the source material. Jacobson’s elements are quoted to the
same number of significant figures as our own. The orbits were derived using 56
Voyager 1 and 66 Voyager 2 images. Synnott et al. (1983) used 18 Voyager 1 and 27
Voyager 2 images. Jacobson’s (private communication) fits were performed using
the data of Synnott et al. (1983). The longitudes of Synnott et al. (1983) have been
transformed into the same reference frame, and where necessary the same epoch,
as used in both this work and that of Jacobson (private communication). Synnott
et al. (1983) quote rms residuals of better than 0.5 pixels while we achieved rms
residuals of 0.20 NAC pixels for the Voyager 1 and 0.22 NAC pixels for the Voyager
2 fits.
In order to directly compare the methods of image navigation and orbit determi-
nation used in this work with those of Synnott et al. (1983) and Jacobson (private)
Tables 6.5 and 6.6 show derived fits using only the 18 Voyager 1 and 27 Voyager
2 images used by Synnott et al. (1983). We achieved rms residuals of 0.24 NAC
pixels (Voyager 1) and 0.33 NAC pixels (Voyager 2) fits.
Since all fits are at the Voyager 2 epoch, Julian Ephemeris Date 2444839.6682,
the error associated with the Voyager 1 mean longitude at epoch is an order of
magnitude higher than the Voyager 2 mean longitude at epoch error. Fits were
also performed for the Voyager 1 data at the Voyager 1 epoch, Julian Ephemeris
Date 2444513.5. The results for these orbit fits are shown in Tables 6.7 and 6.8, for
our 57 images and the 18 used by Synnott et al. (1983) respectively.
The errors in the Voyager 1 mean longitudes at the Voyager 1 epoch are com-
parable to the Voyager 2 mean longitude at epoch errors at the the Voyager 2
epoch. Although the aim of this work was to derive separate orbital elements for
Prometheus at the Voyager 1 and Voyager 2 epochs, we have included fits to the
CHAPTER 6. PROMETHEUS 120
Table 6.3: Voyager 1 Results: epoch 2444839.6682 JED.
element This Work Synnott et al. Jacobson Units
a 139393 ± 15 139353a 139377.4 km
acalc 139375.5± 0.4 – – km
λ 192.0 ± 0.8 181b ± 7 186 deg
e (1.4 ± 0.2) × 10−3 (3.0 ± 0.3) × 10−3 2.6 × 10−3 –
n 587.3010± 0.0028 587.28c ± 0.2 587.2833 deg/day
195 ± 5 218d ± 20 218 deg
i 0.028 ± 0.006 0.0 ± 0.15c 0.0 deg
Ω 13 ± 22 – – deg
avalue quoted in Smith et al. (1981) for a as no Voyager 1 value given by Synnott et al. (1983).bprecessed and transformed into the reference frame and epoch of this work from Smith et
al. (1981) as no Voyager 1 value given in Synnott et al. (1983)cfrom Smith et al. (1981) as no Voyager 1 value given in Synnott et al. (1983).dprecessed and transformed into the reference frame and epoch of this work.
Table 6.4: Voyager 2 Results: epoch 2444839.6682 JED.
element This Work Synnott et al. Jacobson Units
a 139287 ± 12 139353 139377.4 km
acalc 139377.5± 0.2 – km
λ 188.53 ± 0.01 – 188.54 deg
e (2.6 ± 0.1) × 10−3 (2.4 ± 0.2) × 10−3 2.3 × 10−3 –
n 587.2886± 0.0013 – 587.2896 deg/day
245 ± 1 213a ± 20 213 deg
i 0.025 ± 0.008 0.0 ± 0.1 0.0 deg
Ω 25 ± 13 – – deg
atransformed into the reference frame of this work.
CHAPTER 6. PROMETHEUS 121
Table 6.5: Voyager 1 Results using Synnott et al. ’s 18 images: epoch 2444839.6682JED
element This Work Synnott et al. Jacobson Units
a 139498 ± 21 139353a 139377.4 km
acalc 139376.9± 0.7 – – km
λ 189 ± 1 181b ± 7 186 deg
e (1.1 ± 0.3) × 10−3 (3.0 ± 0.3) × 10−3 2.6 × 10−3 –
n 587.2922± 0.0044 587.28c ± 0.2 587.2833 deg/day
170 ± 12 218d ± 20 218 deg
i 0.016 ± 0.013 0.0c ± 0.15 0.0 deg
Ω 296 ± 65 – – deg
avalue quoted in Smith et al. (1981) for a as no Voyager 1 value given by Synnott et al. (1983).bprecessed and transformed into the reference frame and epoch of this work from Smith et
al. (1981) as no Voyager 1 value given in Synnott et al. (1983)cfrom Smith et al. (1981) as no Voyager 1 value given in Synnott et al. (1983).dprecessed and transformed into the reference frame and epoch of this work.
Table 6.6: Voyager 2 Results using Synnott et al.’s 27 images: epoch 2444839.6682JED
element This Work Synnott et al. Jacobson Units
a 139362 ± 31 139353 139377.4 km
acalc 139378.7± 1.0 – km
λ 188.44 ± 0.06 – 188.54 deg
e (3.2 ± 0.3) × 10−3 (2.4 ± 0.2) × 10−3 2.3 × 10−3 –
n 587.2808± 0.0062 – 587.2896 deg/day
256 ± 2 213a ± 20 213 deg
i 0.033 ± 0.013 0.0 ± 0.1 0.0 deg
Ω 334 ± 34 – – deg
atransformed into the reference frame of this work.
CHAPTER 6. PROMETHEUS 122
Table 6.7: Voyager 1 Results: epoch 2444513.5 JED
element This Work Synnott et al. Jacobson Units
a 139393 ± 15 139353a 139377.4 km
acalc 139375.5 ± 0.4 – – km
λ 153.07 ± 0.10 149.0b ± 0.5 153 deg
e (1.4 ± 0.2) × 10−3 (3.0 ± 0.3) × 10−3 2.58 × 10−3 –
n 587.3010 ± 0.0028 587.28c ± 0.2 587.2833 deg/day
16 ± 5 – 8d deg
i 0.028 ± 0.007 0.0c ± 0.15 0.0 deg
Ω 187 ± 21 – – deg
avalue quoted in Smith et al. (1981) for a as no Voyager 1 value given by Synnott et al. (1983).btransformed into the reference frame of this work from Smith et al. (1981) as no Voyager 1
value given in Synnott et al. (1983)cfrom Smith et al. (1981) as no Voyager 1 value given in Synnott et al. (1983).dtransformed into the reference frame of this work.
Table 6.8: Voyager 1 Results using Synnott et al.’s 18 images: epoch 2444513.5JED
element This Work Synnott et al. Jacobson Units
a 139498 ± 21 139353a 139377.4 km
acalc 139376.9± 0.7 – – km
λ 153.4 ± 0.2 149.0b ± 0.5 153 deg
e (1.1 ± 0.3) × 10−3 (3.0 ± 0.3) × 10−3 2.6 × 10−3 –
n 587.2922± 0.0044 587.28c ± 0.2 587.2833 deg/day
352 ± 12 8d deg
i 0.016 ± 0.013 0.0c ± 0.15 0.0 deg
Ω 110 ± 60 – – deg
avalue quoted in Smith et al. (1981) for a as no Voyager 1 value given by Synnott et al. (1983).btransformed into the reference frame of this work from Smith et al. (1981) as no Voyager 1
value given in Synnott et al. (1983)cfrom Smith et al. (1981) as no Voyager 1 value given in Synnott et al. (1983).dtransformed into the reference frame of this work.
CHAPTER 6. PROMETHEUS 123
Table 6.9: Combined Voyager 1 & 2 Results: epoch 2444839.6682 JED
element This Work Synnott et al. Jacobson Units
a 139337 ± 10 139353 139377.43 km
acalc 139377.33± 0.01 −− km
λ 188.526 ± 0.009 – 188.54 deg
e (1.92 ± 0.09) × 10−3 – 2.3 × 10−3 –
n 587.28942 ± 0.00007 587.2890 ± 0.0005 587.28917 deg/day
228 ± 2 – 213 deg
i 0.030 ± 0.005 – 0.0 deg
Ω 53 ± 7 – – deg
Table 6.10: Combined Synnott et al.’s (1983) Voyager 1 & 2 Results: epoch2444839.6682 JED
element This Work Synnott et al. Jacobson Units
a 139394 ± 11 139353 139377.43 km
acalc 139377.31± 0.02 – km
λ 188.51 ± 0.02 – 188.54 deg
e (1.94 ± 0.23) × 10−3 – 2.3 × 10−3 –
n 587.28958 ± 0.00014 587.2890 ± 0.0005 587.28917 deg/day
237 ± 4 – 213 deg
i 0.016 ± 0.010 – 0.0 deg
Ω 337 ± 52 – – deg
CHAPTER 6. PROMETHEUS 124
combined Voyager 1 and Voyager 2 dataset for the sake of completeness. Table 6.9
has the fit for our 123 images while Table 6.10 has the fit just for the 45 images of
Synnott et al. (1983). The rms errors were 0.21 NAC pixels (our 123 images) and
0.31 NAC pixels (the 45 images of Synnott et al. (1983)).
6.5 The distribution of longitudes at observation
mid-times
Figs. 6.3 and 6.4 show the x-y coordinates of Prometheus during the Voyager 1
and Voyager 2 encounters, in the planetary reference frame used for Saturn at the
observation mid-times. The x-y plane is the equator plane of Saturn at epoch. The
positions are numbered in ascending chronological order, the same order as they
are presented in Tables 6.1 and 6.2.
CHAPTER 6. PROMETHEUS 125
1
23
45
67
89
1011
1213
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
3536
37
383940
4142
4344
45
46474849
50
51
52
53
54
5556
mean direction to Sun
Figure 6.3: The x-y plane coordinates of Prometheus at the observation mid-timescalculated from the Voyager 1 fit in Table 6.3
CHAPTER 6. PROMETHEUS 126
1
2
3
4
5
6
7
8
9
10
11
1213
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
3334
353637
3839
40
41
424344
454647
48
4950
51
5253
54
55
5657
58
59
60
61
62
63
64
65
66
mean direction to Sun
Figure 6.4: The x-y plane coordinates of Prometheus at the observation mid-timescalculated from the Voyager 2 fit in Table 6.4
CHAPTER 6. PROMETHEUS 127
Examination of Figs. 6.3 and 6.4 clearly shows that none of the images used has
Prometheus either transiting Saturn or in the shadow of the planet. This lack of
observations in these longitudes ranges is unsurprising, Prometheus being effectively
invisible in the visible light range in the shadow and swamped by light from the
planetary disc when transiting.
Locations which are marked in the same colour (except black) indicate observa-
tions which are close together in time, all occurring within 1/2 an orbital period.
Ideally the entire longitude range for the orbit should be sampled, this is especially
important for accurate determination of e (Synnott et al. 1983). The Voyager 1
data does not sample the available longitude range completely. There is a clustering
of observations around the West ring ansa in both longitude and observation times.
This clustering could effect the accuracy of the orbital fit. In Chapter 7 the effect
of clustering of observations is investigated. It is concluded that it does not affect
the orbit fit to any great extent. The Voyager 2 observations (Fig. 7.2) sample the
longitude range very well. Although there is some clustering of observations, it is
not enough to cause concern over the accuracy of the orbital fit.
6.6 Discussion
The derived orbital elements from this Chapter show some deviation from those
of Jacobson (private communication) and Synnott et al. (1983). Most noticeably
our mean motions at both the Voyager 1 and Voyager 2 epochs are higher than
the previously derived values. Importantly our rms residuals are only half as large
as those of Synnott et al. (1983). Prometheus’ inclination is in the range 0.026 −0.028 deg with a mean of 0.027 ± 0.006 deg, the value of Synnott et al. (1983) is
0.0 ± 0.1.
The new derivations of Prometheus’ orbit using a larger dataset than that of
Synnott et al. (1983) show a difference in mean motions between the Voyager 1 and
Voyager 2 epochs of 3σ. It therefore appears that Prometheus’ orbit underwent a
change between the two Voyager encounters.
Any explanation for possible causes of such a change in the orbit, and for the
CHAPTER 6. PROMETHEUS 128
∼ 19 lag seen in 1995, must account for the observed mean longitudes in 1980, 1981
and 1994-2000. Table 6.11 shows the calculated mean longitudes in 1995 based on
the derived 1980 and 1981 mean longitudes and mean motions from our Voyager 1,
Voyager 2 and combined fits in Tables 6.3, 6.4 and 6.9. Also included is the mean
longitude of Prometheus in 1995 from French et al. (2000) and the observed mean
longitude during the August 1995 ring plane crossing (Nicholson et al. 1996).
Table 6.11: Mean longitudes for Prometheus from vari-ous authors. Epoch: 2449940.0 JED
Fit Mean longitudea
Nicholson et al. (1996) 339.23 ± 0.10
French et al. (2000) 339.12 ± 0.05
Voyager 1 fit 62 ± 14
Voyager 2 fit 16 ± 6
Combined fit 359.0 ± 0.4
Jacobson (private communication) 358.32 ± 2.56
ain degrees
CHAPTER 6. PROMETHEUS 129
Clearly none of the derived mean longitudes at epoch and corresponding mean
motions give the observed mean longitude during the August 1995 ring plane cross-
ing event. The ∼ 19 longitude lag observed in 1995 is still present with our derived
values for the combined Voyager mean motion. The lag in 1995 based on the indi-
vidual Voyager 1 or Voyager 2 is much larger.
6.7 The orbit of Prometheus in JPL Ephemerides
The program FitOrb2Ephm.f, (section 5.6), was used to fit a precessing ellipse to
the ephemerides for Prometheus in the SP-kernels vg2 sat.bsp and sat081.4.bsp. For
Prometheus, 160 position vectors at 3600.0 second intervals were obtained from the
relevant ephemeris. A precessing ellipse was fitted to the data with starting values
for the parameters of the ellipse being taking from the combined orbit fit from Table
6.9. A second fit was also performed with the apsidal and nodal precession rates,
β and γ set equal to 0.0/day. Position vectors were obtained from the SP-kernels
vg2 sat.bsp (the Voyager 2 ephemeris) and sat081.4.bsp (the current Cassini small
satellites ephemeris). The results obtained are presented in Tables 6.12 and 6.13. In
both cases the North Pole orientation of Saturn is taken from French et al. (1993).
As with the fits for Atlas, see Chapter 5, the fitting process was very robust.
For a wide range of starting values for the parameters of the ellipse convergence to
the exact values given in Tables 6.12 and 6.13 was achieved. There were no strong
correlations between any of the fitted parameters.
Examination of the data presented in Tables 6.12 and 6.13 shows that a simple
ellipse is a very good model for the ephemerides of Prometheus, at least over several
tens of days. For the more recent ephemeris, sat081.4.bsp, a precessing ellipse
is a better fit to the ephemerides than the non-precessing model. While for the
older ephemeris, vg2 sat.bsp, the non-precessing ellipse fits the data better then
the precessing model.
The non-precessing ellipse fit to vg2 sat.bsp in Table 6.12 has identical mean
motion, n and eccentricity, e to that of Synnott et al. (1983), which has n =
587.2890±0.0005/day and e = (2.4±0.6)×10−3, while the other orbital parameters
CHAPTER 6. PROMETHEUS 130
Table 6.12: Orbital elements for Prometheus from a fit to the SP kernel vg2 sat.bsp:Epoch 2444839.6682 JED. 160 positions at 3600.0 sec. intervals starting at2444836.6682 JED.
para- Precessing Ellipse Model Non-Precessing Ellipse Model
metera β = γ = 0.0
a 139352 ± 2 139353.0000± 0.0001
acalc 139377.47± 0.08 139377.399474± 0.000004
λ 188.5080 ± 0.0009 188.50835287± 0.00000005
n 587.2886 ± 0.0005 587.28900046± 0.00000003
e (2.39 ± 0.01) × 10−3 (2.3999977 ± 0.0000006) × 10−3
i (1.3 ± 0.1) × 10−2 (1.27906 ± 0.00008) × 10−3
216.7 ± 0.2 217.52230± 0.00001
Ω 357 ± 6 356.3949± 0.0003
rms 3.84 0.00022
aDistances are in km (rms is a distance), longitudes in degrees and rates in degrees/day. Errors
for elements from this work are the formal errors from the fitting process. All longitudes measured
from the ascending node of Saturn’s equator at epoch on the Earth mean equator at J2000, i
measured from Saturn’s equatorial plane at epoch.
are well within the error bars. The non-precessing ellipse is such a good fit that
it suggests that the ephemeris of Prometheus in vg2 sat.bsp may well have been
generated using a non-precessing elliptical model with the parameters of Synnott
et al. (1983).
The precessing ellipse fit to sat081.4.bsp in Table 6.13 has identical mean mo-
tion, n and eccentricity, e to the combined Voyager 1 and Voyager 2 fit of Jacobson
(private communication), see Table 6.9, while the other orbital parameters are well
within the error bars. The precessing ellipse is such a good fit that it suggests that
the ephemeris of Prometheus in sat081.4.bsp may well have been generated using
a precessing elliptical model with the parameters of Jacobson (private communica-
tion).
CHAPTER 6. PROMETHEUS 131
Table 6.13: Orbital elements for Prometheus from a fit to the SP kernel sat081.4.bsp:Epoch 2453371.00 JED. 160 positions at 3600.0 sec. intervals starting at 2453368.00JED.
para- Precessing Ellipse Model Non-Precessing Ellipse Model
metera β = γ = 0.0
a 139377.42± 0.09 139377 ± 2
acalc 139377.372± 0.003 139377.38 ± 0.08
λ 67.31334± 0.00004 67.3122 ± 0.0009
n 587.28917 ± 0.00002 587.2891± 0.0005
e (2.2890 ± 0.0004) × 10−3 (2.277 ± 0.010) × 10−3
i (1.140 ± 0.005) × 10−2 (1.1 ± 0.1) × 10−3
336.762 ± 0.009 337.6 ± 0.2
Ω 1.9 ± 0.3 0 ± 6
rms 0.15 3.72
aDistances are in km (rms is a distance), longitudes in degrees and rates in degrees/day. Errors
for elements from this work are the formal errors from the fitting process. All longitudes measured
from the ascending node of Saturn’s equator at epoch on the Earth mean equator at J2000, i
measured from Saturn’s equatorial plane at epoch.
6.8 Comparison of the derived orbits with expla-
nations for the origins of Prometheus’ lag
We will now examine the possible explanations for the observed lag of Prometheus,
presented in section 1.3, in light of the obtained results.
6.8.1 A co-orbital companion
If an orbital switch between Prometheus and a hypothetical co-orbital occurred
between the two Voyager encounters, Prometheus’ two distinct mean motions are
587.3010 ± 0.0028 deg /year and 587.2927 ± 0.0011 deg /year. Since both of these
are higher than the Synnott et al. (1983) value of 587.2890 ± 0.0005 deg /year any
combination of them would lead to a lag greater than ∼ 19 deg in August 1995.
A companion satellite being maintained in a horseshoe orbit with an orbital
CHAPTER 6. PROMETHEUS 132
switch occurring between the two Voyager encounters can explain the mean motion
difference between the Voyager 1 and Voyager 2 epochs. However, such a companion
cannot explain the observed lag in 1995.
Of course, a co-orbital satellite, with the orbital switch occurring at a time other
than coincident with the Voyager encounters, can still explain the observed lag in
1995. Nicholson et al. (1996) considered an object the size of the saturnian satellite
Atlas with a libration period of ∼ 30 years. As Nicholson et al. (1996) point out
such a large object should have been seen by the Voyager spacecraft. However,
the difference between the derived Voyager 1 and Voyager 2 mean motions is not
explained by this hypothetical co-orbital of Nicholson et al. (1996).
6.8.2 Periodic encounters with the F ring
The problem is to account for both the observed lag and the observed mean motion.
Orbital fits to HST observations from 1994-2000 give a mean motion of 587.287555±0.000048 deg /day (French et al. 2000) with the lag increasing by 0.57 deg /year.
Even if Prometheus underwent a collision, or collisions, that reduced its mean mo-
tion from 587.2890 deg /day (Synnott et al. 1983) to 587.287555 deg /day immedi-
ately after the Voyager 2 encounter this would still only account for about 10 deg of
the ∼ 19 deg accumulated lag observed in 1995. Furthermore, it cannot account for
the differences in the mean motions between the Voyager 1 and Voyager 2 epochs.
6.8.3 Cometary impact
The probability of such an impact with a 0.2 km diameter object has been estimated
as 107±1year−1 (Nicholson et al. 1996). The argument is the same as for a collision
with an F ring object, a collision reducing the mean motion to that of French
et al. (2000) can only account for a 10 deg lag in 1995 and cannot explain the
difference in Voyager 1 and Voyager 2 mean longitudes. To be consistent with all
data would require at least three impacts. One occurring between the Voyager
1 and Voyager 2 encounters which reduces the mean motion, another between the
Voyager encounters and the 1995 ring plane crossing events which reduces the mean
motion even further and a final one just before the 1995 ring plane crossings which
CHAPTER 6. PROMETHEUS 133
raises the mean motion to that of French et al. (2000). Such a high frequency of
impacts with large objects seems highly implausible.
6.8.4 Gravitational interaction with the F ring
This theory requires the mass contained within the F ring to be quite high, of the
order of the mass of Prometheus itself. Due to the random, chaotic, nature of
the gravitational interactions this theory can explain the Voyager 1 and Voyager 2
mean motion differences, the 1995 lag and the 1994-2000 HST mean motion and
lags. The main problem is explaining how the F ring can maintain its well defined
structure under the influence of these random gravitational forces.
6.8.5 Other mechanisms
Gross errors in the Voyager ephemeris, the predicted lag due to back-reaction from
density wave torques and lag resulting from secular effects are all considered by
Nicholson et al. (1996) and rejected as being too small.
6.8.6 Comments
If the mean motion of Prometheus has indeed decreased from its Voyager 1 value of
587.3010±0.0028 deg /day (this work) to 587.287555±0.000048 deg /day (French et
al. 2000) then there is a corresponding increase in semi-major axis from 139375.5 km
to 139377.6 km respectively, an increase of ∼ 120 metres a year. If this rate of expan-
sion continued unchecked the semi-major axis of Prometheus’ orbit would double in
∼ 106 years. This would imply an unfeasibly short lifetime for the Prometheus, F
ring and Pandora system. Clearly this rate of change in the mean motion cannot be
maintained. If it was a one off event, occurring sometime between 1981 and 1995,
the probability that it would occur just as it was able to be observed is infinitesi-
mally small. This implies that it was not an isolated occurrence but simply one in
a sequence of such events. Every one of these events leading to a decrease in mean
motion leads to the afore-mention short lifetime for the Prometheus, F ring and
Pandora system. It is therefore reasonable to assume that for each event increasing
the mean motion there is, on average, one that decreases it by a similar amount.
CHAPTER 6. PROMETHEUS 134
Of course over a long enough period of time a slow increase or decrease due to a
net imbalance in mean motion increasing/decreasing events is perfectly possible.
So this change in Prometheus’ mean motion is unlikely to be an isolated event. It
would be interesting to know if the other objects in the region, Atlas and Pandora,
exhibit similar changes in mean motion.
6.9 Conclusion
Not only did Prometheus’ orbit demonstrably change between the Voyager encoun-
ters and the ring plane crossing events in 1995, but change is also apparent between
the Voyager 1 and Voyager 2 encounters. There is currently no single theory that
satisfactorily accounts for the changes in Prometheus’ orbit yet also reproduces the
observed mean longitudes at various epochs. The orbit of Prometheus should con-
tinue to be monitored at every opportunity in order to detect further changes and
thus provide more information enabling the exact dynamical mechanism behind
them to be explained.
Chapter 7
Pandora
7.1 Introduction
Like Atlas and Prometheus, the saturnian satellite Pandora was discovered during
the Voyager 1 encounter with Saturn in 1980 (Smith et al. 1981). Pandora orbits
in the region just exterior to the narrow F ring. The orbit initially published for
Pandora was derived using 32 Voyager 1 images (Smith et al. 1981). The Voyager 2
encounter with Saturn in 1981 provided another opportunity for imaging Pandora
and Synnott et al. (1983) used 39 images to improve Pandora’s orbit. Fig 7.1 shows
a typical late encounter image of Pandora shuttered by the Voyager 2 narrow angle
camera. Pandora’s location just exterior to the F ring is clearly illustrated.
7.2 Search methodology and orbit determination
The search methodology, geometrical correction, image navigation and orbit de-
termination techniques used for Pandora were almost identical to those already
detailed for Atlas (sections 5.2, 5.3, 5.4 and 5.6 respectively). The only differences
being that Pandora, not Atlas, was being searched for and images where the res-
olution was worse then 320 km per pixel (not 150 km per pixel) were instantly
rejected. This figure was chosen because the worst resolution image used by Syn-
nott et al. (1983) for the determination of the orbit of Prometheus has a resolution
of 320 km per pixel (Chapter 6). Pandora is smaller, and fainter, than Prometheus
135
CHAPTER 7. PANDORA 136
Figure 7.1: Pandora in FDS43854.11. The Encke Gap and F ring are clearly visible,Pandora is circled.
CHAPTER 7. PANDORA 137
so this figure for the minimum was deemed to be more than adequate.
Since these techniques have already been described in Chapter 5 we shall not
describe them again here.
7.3 Identified images
The 38 Voyager 1 and 49 Voyager 2 navigable images of Pandora that we identified
are listed in Tables 7.1 and 7.2 respectively. Also indicated is whether a particular
image was used by Synnott et al. (1983) for their Pandora orbit determination. The
data of Synnott et al. (1983) was also used by Jacobson (private communication) for
an independent determination of Pandora’s orbital elements. Of the images we used,
28 of the Voyager 1 and 37 of the Voyager 2 were also used by Synnott et al. (1983).
The line and sample coordinates are given in continuous and not integer pixel
coordinates (see section 3.3). The resolutions and solar phase angle information
included for each image were calculated using the individually determined orbits
for Pandora at the Voyager 1 and Voyager 2 epochs (see Tables 7.3 and 7.4).
CHAPTER 7. PANDORA 138
Table 7.1: Voyager 1 Images of Pandora
FDS Image Mid-time Used by Line Sample Phase Res.
No. UTC Synnott lcont. scont. Angle() km
34300.41 Oct. 22 13:06:10.5 573.5 418.2 13.2 225.4
34389.18 Oct. 25 11:59:47.0 432.6 510.5 13.1 196.5
34389.36 Oct. 25 12:14:11.0 446.6 480.3 13.1 196.4
34389.54 Oct. 25 12:28:35.0 208.7 558.4 13.2 196.2
34390.12 Oct. 25 12:42:59.0 246.4 532.5 13.2 196.0
34390.24 Oct. 25 12:52:35.0 • 304.6 385.6 13.2 195.9
34390.30 Oct. 25 12:57:23.0 • 289.6 453.5 13.2 195.9
34390.36 Oct. 25 13:02:11.0 • 267.7 535.4 13.2 195.8
34390.42 Oct. 25 13:06:59.0 • 234.6 437.4 13.2 195.7
34390.48 Oct. 25 13:11:47.0 126.6 321.4 13.2 195.7
34391.06 Oct. 25 13:26:11.0 • 151.8 408.6 13.2 195.5
34391.12 Oct. 25 13:30:59.0 • 159.5 443.0 13.2 195.4
34391.18 Oct. 25 13:35:47.0 • 197.6 352.5 13.2 195.3
34391.24 Oct. 25 13:40:35.0 • 182.5 446.2 13.3 195.3
34391.30 Oct. 25 13:45:23.0 • 223.4 323.5 13.3 195.2
34391.36 Oct. 25 13:50:11.0 • 181.6 487.6 13.3 195.1
34391.42 Oct. 25 13:54:59.0 • 214.2 369.5 13.3 195.1
34391.48 Oct. 25 13:59:47.0 • 202.6 396.4 13.3 195.0
34391.54 Oct. 25 14:04:35.0 • 163.5 375.8 13.3 194.9
34392.00 Oct. 25 14:09:23.0 97.5 339.3 13.3 194.9
34392.06 Oct. 25 14:14:11.0 • 17.0 363.2 13.3 194.8
34392.12 Oct. 25 14:18:59.0 • 79.4 259.4 13.3 194.7
34392.18 Oct. 25 14:23:47.0 • 81.1 400.3 13.3 194.6
34392.30 Oct. 25 14:33:23.0 • 177.7 306.4 13.3 194.5
34392.36 Oct. 25 14:38:11.0 212.0 293.9 13.3 194.4
34392.42 Oct. 25 14:42:59.0 • 131.6 426.7 13.3 194.4
CHAPTER 7. PANDORA 139
FDS Image Mid-time Used by Line Sample Phase Res. a
No. UTC Synnott lcont. scont. Angle() km
34392.48 Oct. 25 14:47:47.0 • 65.5 272.6 13.3 194.3
34392.54 Oct. 25 14:52:35.0 • 9.8 426.6 13.3 194.2
34393.12 Oct. 25 15:06:59.0 • 57.5 332.8 13.3 194.0
34404.34 Oct. 26 00:12:34.5 • 593.5 658.3 12.8 191.1
34449.06 Oct. 27 11:50:10.5 • 707.5 338.3 13.3 174.9
34457.52 Oct. 27 18:50:58.8 • 556.6 531.7 12.6 171.6
34467.01 Oct. 28 02:10:10.5 • 700.6 411.2 13.3 169.1
34701.45 Nov. 4 21:57:22.5 • 523.4 25.6 12.6 86.6
34704.23 Nov. 5 00:03:46.5 781.4 170.6 12.4 86.6
34704.35 Nov. 5 00:13:22.5 503.5 629.0 12.4 86.6
34817.30 Nov. 8 18:33:22.5 • 632.0 277.5 12.5 47.0
34835.25 Nov. 9 08:53:22.5 • 174.4 510.7 12.5 40.3
aThe year of all images is 1980. All images are taken with the Narrow Angle Camera (NAC).
The resolution and sun phase angle information is calculated using the derived Voyager 1 orbit in
Table .
CHAPTER 7. PANDORA 140
Table 7.2: Voyager 2 Images of Pandora
FDS Image Mid-time Used by Line Sample Phase Res.
No. UTC Synnott lcont. scont. Angle() km
42420.55 Jul. 4 07:38:23.5 • 41.5 50.4 7.2 394.1
42514.35 Jul. 7 10:34:23.5 • 209.5 219.3 7.1 371.1
42686.10 Jul. 13 03:50:23.5 • 100.5 129.8 6.9 330.0
42741.34 Jul. 15 00:09:35.5 • 239.6 9.6 6.9 316.0
43023.52 Jul. 24 09:59:59.5 • 90.5 77.3 6.7 247.0
43439.54 Aug. 7 06:49:37.5 • 595.2 654.9 6.5 145.6
43507.17 Aug. 9 12:44:01.5 • 311.9 318.3 7.3 128.2
43536.26 Aug. 10 12:03:13.5 • 110.6 412.4 6.7 122.5
43544.23 Aug. 10 18:24:49.5 • 346.1 268.7 7.5 119.3
43596.54 Aug. 12 12:25:37.4 428.5 722.8 7.4 107.6
43625.54 Aug. 13 11:37:36.5 • 670.8 393.0 6.8 98.8
43626.26 Aug. 13 12:03:12.5 • 563.8 456.9 6.7 98.8
43626.58 Aug. 13 12:28:48.5 • 538.2 631.2 6.7 98.9
43627.30 Aug. 13 12:54:24.5 • 432.2 681.6 6.6 98.9
43628.34 Aug. 13 13:45:36.5 • 423.8 673.2 6.6 99.0
43629.38 Aug. 13 14:36:48.5 • 371.2 451.6 6.7 99.2
43630.42 Aug. 13 15:28:00.5 • 201.9 231.6 6.8 99.2
43631.10 Aug. 13 15:50:24.5 • 189.0 483.4 6.9 99.2
43634.46 Aug. 13 18:43:12.5 • 219.5 471.8 7.5 98.2
43635.50 Aug. 13 19:34:24.5 • 339.4 208.1 7.7 97.7
43646.06 Aug. 14 03:47:12.5 • 555.5 761.6 6.7 94.2
43649.50 Aug. 14 06:46:24.5 • 317.5 238.5 6.9 94.5
43653.54 Aug. 14 10:01:36.5 • 280.5 427.7 7.7 93.4
43655.12 Aug. 14 11:04:00.5 • 252.6 302.5 7.8 92.7
43655.28 Aug. 14 11:16:48.5 • 359.4 194.2 7.9 92.5
43655.44 Aug. 14 11:29:36.5 • 286.6 104.6 7.9 92.4
CHAPTER 7. PANDORA 141
FDS Image Mid-time Used by Line Sample Phase Res.a
No. UTC Synnott lcont. scont. Angle() km
43656.00 Aug. 14 11:42:24.5 • 333.8 255.1 7.9 92.2
43656.16 Aug. 14 11:55:12.5 • 407.6 90.6 7.9 92.1
43657.20 Aug. 14 12:46:24.5 • 523.3 272.3 7.9 91.4
43657.52 Aug. 14 13:12:00.5 • 515.4 163.9 7.9 91.1
43662.52 Aug. 14 17:12:00.5 • 781.6 169.2 7.0 89.5
43663.24 Aug. 14 17:37:36.5 • 711.7 387.2 6.9 89.5
43663.56 Aug. 14 18:03:12.5 • 603.7 501.3 6.8 89.5
43684.35 Aug. 15 10:34:24.7 564.6 307.5 6.7 85.0
43686.55 Aug. 15 12:26:24.5 325.0 403.2 6.9 85.1
43695.24 Aug. 15 19:13:37.0 306.7 105.2 8.1 81.8
43703.18 Aug. 16 01:32:48.5 776.5 496.3 6.8 80.2
43710.08 Aug. 16 07:00:48.5 • 328.6 223.5 7.9 79.5
43715.52 Aug. 16 11:36:00.5 706.3 297.9 8.0 76.3
43723.20 Aug. 16 17:34:24.5 • 345.0 247.3 6.8 75.7
43724.41 Aug. 16 18:39:13.4 196.7 407.1 7.0 75.7
43732.22 Aug. 17 00:48:01.4 51.7 146.1 8.4 72.9
43752.26 Aug. 17 16:51:13.4 502.3 191.5 8.5 67.4
43780.11 Aug. 18 15:03:11.86 • 717.6 653.6 7.1 61.5
43796.09 Aug. 19 03:49:35.5 666.9 87.8 7.4 56.5
43805.50 Aug. 19 11:34:23.5 266.5 388.2 9.2 55.1
43818.49 Aug. 19 21:57:35.5 • 174.4 565.5 7.5 52.1
43854.11 Aug. 21 02:15:11.5 • 464.0 178.5 7.7 42.3
43854.36 Aug. 21 02:35:11.5 587.2 389.9 7.7 42.3
aThe year of all images is 1981. All images are taken with the Narrow Angle Camera (NAC).
The resolution and sun phase angle information is calculated using the derived Voyager 2 orbit in
Table .
CHAPTER 7. PANDORA 142
7.4 Results
The derived orbit fits for Pandora using Voyager 1 and Voyager 2 data separately
are shown in Table 7.3 and 7.4 respectively. All errors for our fits are the formal
errors from the fitting process.
For comparison, the fits of Synnott et al. (1983) and Jacobson (private com-
munication) are also shown. No errors have been included for Jacobson’s elements
since none are given in the source material. Jacobson’s elements are quoted to the
same number of significant figures as our own. The orbits were derived using 38
Voyager 1 and 49 Voyager 2 images. Synnott et al. (1983) used 32 Voyager 1 and
39 Voyager 2 images. Jacobson’s fits were performed using the observational data
of Synnott et al. (1983). The longitudes of Synnott et al. (1983) have been trans-
formed into the same reference frame, and where necessary the same epoch, as used
in both this work and by Jacobson (private communication). Synnott et al. (1983)
quote rms residuals of better than 0.5 pixels while we achieved rms residuals of 0.18
NAC pixels for the Voyager 1 and 0.24 NAC pixels for the Voyager 2 fits.
In order to directly compare the methods of image navigation and orbit determi-
nation used in this work with those of Synnott et al. (1983) and Jacobson (private
communication), Tables 7.5 and 7.6 show derived fits using the 28 of the 32 Voyager
1 and the 37 of the 39 Voyager 2 images used by Synnott et al. (1983) that we could
identify and navigate. We achieved rms residuals of 0.19 NAC pixels (Voyager 1)
and 0.29 NAC pixels (Voyager 2) fits.
A fit to the combined Voyager 1 and Voyager 2 dataset was also performed.
Table 7.7 has the fit for our 87 images while Table 7.8 has the fit just for the
65 images we could identify or navigate of Synnott et al.’s (1983) 71 images. We
achieved rms residuals of 0.21 NAC pixels and 0.24 NAC for our combined images
and the 65 images we used of the 71 of Synnott et al. (1983) respectively.
CHAPTER 7. PANDORA 143
Table 7.3: Voyager 1 Results: epoch 2444839.6682 JED.
element This Work Synnott et al. Jacobson Units
a 141869 ± 32 – 141712.61 km
acalc 141712.5 ± 0.8 141700a – km
λ 82 ± 1 71b ± 7 82 deg
e (5.7 ± 0.4) × 10−3 (4.4 ± 0.6) × 10−3 3.98 × 10−3
n 572.7895 ± 0.0046 572.77a ± 0.02 572.7877 deg/day
55 ± 3 70c ± 30 67 deg
i 0.09 ± 0.01 0.05a ± 0.15 0.0 deg
Ω 263 ± 1 – – deg
aThis is the value quoted in Smith et al. (1981) as there is no value in Synnott et al. (1983).bValue from Smith et al. (1981) precessed and transformed into the reference frame and epoch
used in this work.ctransformed into reference frame used in this work
Table 7.4: Voyager 2 Results: epoch 2444839.6682 JED.
element This Work Synnott et al. Jacobson Units
a 141670 ± 13 – 141712.61 km
acalc 141712.2 ± 0.5 141700 – km
λ 82.13 ± 0.02 – 82.19 deg
e (3.8 ± 0.1) × 10−3 (4.2 ± 0.6) × 10−3 4.73 × 10−3
n 572.7911 ± 0.0028 – 572.7927 deg/day
53 ± 1 62a ± 30 70 deg
i 0.050 ± 0.007 0.0 ± 0.1 0.0 deg
Ω 222 ± 6 – – deg
atransformed into reference frame used in this work
CHAPTER 7. PANDORA 144
Table 7.5: Voyager 1 Results using 28 of Synnott et al. ’s 32 images: epoch2444839.6682 JED.
element This Work Synnott et al. Jacobson Units
a 141863 ± 32 – 141712.61 km
acalc 141712.0 ± 0.9 141700a – km
λ 83 ± 2 71b ± 7 82 deg
e (5.7 ± 0.4) × 10−3 (4.4 ± 0.6) × 10−3 3.98 × 10−3
n 572.7924 ± 0.0053 572.77a ± 0.02 572.7877 deg/day
57 ± 3 70c ± 30 67 deg
i 0.08 ± 0.01 0.05a ± 0.15 0.0 deg
Ω 251 ± 4 – – deg
aThis is the value quoted in Smith et al. (1981) as there is no value in Synnott et al. (1983).bValue from Smith et al. (1981) precessed and transformed into the reference frame and epoch
used in this work.ctransformed into reference frame used in this work
Table 7.6: Voyager 2 Results using 37 of Synnott et al. ’s 39 images: epoch2444839.6682 JED.
element This Work Synnott et al. Jacobson Units
a 141713 ± 19 – 141712.61 km
acalc 141712.4 ± 0.6 141700 – km
λ 82.11 ± 0.03 – 82.19 deg
e (3.6 ± 0.2) × 10−3 (4.2 ± 0.6) × 10−3 4.73 × 10−3
n 572.7900 ± 0.0037 – 572.7927 deg/day
50 ± 2 62a ± 30 70 deg
i 0.06 ± 0.1 0.0 ± 0.1 0.0 deg
Ω 222 ± 8 – – deg
atransformed into reference frame used in this work
CHAPTER 7. PANDORA 145
Table 7.7: Combined Voyager 1 & 2 Results: epoch 2444839.6682 JED.
element This Work Synnott et al. Jacobson Units
a 141731 ± 13 – 141712.61 km
acalc 141712.63 ± 0.01 141700 km
λ 82.12 ± 0.01 – 82.15 deg
e (4.5 ± 0.1) × 10−3 – 4.37 × 10−3
n 572.78859± 0.00009 572.7891 ± 0.0005 572.78891 deg/day
57 ± 1 – 68 deg
i 0.053 ± 0.007 – 0.0 deg
Ω 245 ± 4 – – deg
Table 7.8: Combined 65 of Synnott et al. ’s(1983) Voyager 1 & 2 Results: epoch2444839.6682 JED.
element This Work Synnott et al. Jacobson Units
a 141744 ± 18 – 141712.61 km
acalc 141712.62± 0.02 141700 – km
λ 82.11 ± 0.01 – 82.15 deg
e (4.4 ± 0.2) × 10−3 – 4.37 × 10−3
n 572.78869 ± 0.00011 572.7891 ± 0.005 572.78891 deg/day
56 ± 1 – 68 deg
i 0.06 ± 0.01 – 0.0 deg
Ω 241 ± 5 – – deg
CHAPTER 7. PANDORA 146
7.5 The distribution of longitudes at observation
mid-times
Figs. 7.2 and 7.3 show the x-y coordinates of Pandora, during the Voyager 1 and
Voyager 2 encounters, in the planetary reference frame used for Saturn, at the
observation mid-times. The x-y plane is the equator plane of Saturn at epoch. The
positions are numbered in ascending chronological order, the same order as they
are presented in Tables 7.1 and 7.2.
1
234
56
78
910
1112
1314
1516
1718
1920
2122
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
mean direction to Sun
Figure 7.2: The x-y plane coordinates of Prometheus at the observation mid-timescalculated from the Voyager 1 fit in Table 7.3
CHAPTER 7. PANDORA 147
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
mean direction to Sun
Figure 7.3: The x-y plane coordinates of Prometheus at the observation mid-timescalculated from the Voyager 2 fit in Table 7.4
CHAPTER 7. PANDORA 148
Examination of Figs. 7.2 and 7.3 clearly shows that none of the images used
has Pandora either transiting Saturn or in the shadow of the planet. Position 18 in
Fig. 7.3 only appears to be in Saturn’s shadow because the mean region is shown
not the location of the shadow at the observation times. The lack of observations
in these longitudes ranges is unsurprising, Pandora being effectively invisible in the
visible light range in the shadow and swamped by light from the planetary disc
when transiting.
Locations which are marked in the same colour (except black) indicate observa-
tions which are close together in time, all occurring within 1/2 an orbital period.
Ideally the entire longitude range for the orbit should be sampled, this is especially
important for accurate determination of e (Synnott et al. 1983). The Voyager 1
data does not sample the available longitude range completely. In addition 28 of the
38 Voyager 1 observations were made within 3 hours of each other, when Pandora
was in the region of the West ring ansa. This closeness of the vast majority of the
observations, along with the limited sampling of the longitude range leads to the
fit to the Voyager 1 data being of lower confidence than would otherwise be the
case. A fit to the Voyager 1 data was performed with observations 3-14 and 16-28
inclusive removed, leaving just the first, middle and last of the cluster of images.
The determined elements were consistent with those obtained using the full 38 ob-
servations. So the cluster of 28 images does not seem to effect the accuracy of the
fit in any meaningful way.
The Voyager 2 observations (Fig. 7.2) sample the longitude range very well.
Although there is some clustering of observations, it is not enough to cause concern
over the accuracy of the orbital fit.
7.6 Discussion of the orbit fits
The mean motions from the three fits are all within 1σ of each other. The mean
longitudes and longitudes of pericentre are comparable. There is no indication that
Pandora’s orbit changed between the Voyager 1 and Voyager 2 encounters. The
number of images used (87) is not significantly greater than the 71 of Synnott et
CHAPTER 7. PANDORA 149
al. (1983). Our combined orbit (Table 7.7) is consistent with the combined orbit
of Synnott et al. (1983) and Jacobson (private communication), the mean motions
differing by only ∼ 1σ.
7.7 The orbit of Pandora in JPL Ephemerides
The program FitOrb2Ephm.f, described in Chapter 5, was used to fit a pre-
cessing ellipse to the ephemerides for Pandora in the SP-kernels vg2 sat.bsp and
sat081.4.bsp. For Pandora, 160 position vectors at 3600.0 second intervals were ob-
tained from the relevant ephemeris. A precessing ellipse was fitted to the data with
starting values for the parameters of the ellipse being taking from the combined or-
bit fit from Table 7.7. A second fit was also performed with the apsidal and nodal
precession rates, β and γ set equal to 0.0/day. Position vectors were obtained
from the SP-kernels vg2 sat.bsp (the Voyager 2 ephemeris) and sat081.4.bsp (the
current Cassini small satellites ephemeris). The results obtained are presented in
Tables 7.9 and 7.10. In both cases the North Pole orientation of Saturn is taken
from French et al. (1993).
As with the fits for Atlas, see Chapter 5, and Prometheus, Chapter 6, the fitting
process was very robust. For a wide range of starting values, the parameters of the
ellipse convergence to the exact values given in Tables 7.9 and 7.10 was achieved.
There were no strong correlations between any of the fitted parameters.
Examination of the data presented in Tables 7.9 and 7.10 shows that a simple
ellipse is a very good model for the ephemerides of Pandora, at least over several
tens of days. For the more recent ephemeris, sat081.4.bsp, a precessing ellipse
is a better fit to the ephemerides than the non-precessing model. While for the
older ephemeris, vg2 sat.bsp, the non-precessing ellipse fits the data better than
the precessing model.
The non-precessing ellipse fit to vg2 sat.bsp in Table 7.9 has identical mean
motion, n, and eccentricity, e, to that of Synnott et al. (1983), see Tables 7.7 and
7.4, while the other orbital parameters are well within the error bars. The non-
precessing ellipse is such a good fit that it suggests that the ephemeris of Pandora
CHAPTER 7. PANDORA 150
Table 7.9: Orbital elements for Pandora from a fit to the SP kernel vg2 sat.bsp:Epoch 2444839.6682 JED. 160 positions at 3600.0 sec. intervals starting at2444836.6682 JED.
para- Precessing Ellipse Model Non-Precessing Ellipse Model
metera β = γ = 0.0
a 141699 ± 4 141700.0001± 0.0003
acalc 141712.8 ± 0.1 141712.547715± 0.000009
λ 82.203 ± 0.002 82.2023529± 0.0000001
n 572.7873 ± 0.0008 572.78910045± 0.00000006
e (4.19 ± 0.02) × 10−3 (4.199996 ± 0.000001) × 10−3
i (1.3 ± 0.2) × 10−2 (1.27906 ± 0.00002) × 10−2
65.74 ± 0.09 66.522367± 0.000006
Ω 357 ± 9 356.3948± 0.0007
rms 6.43 0.00046
aDistances are in km (rms is a distance), longitudes in degrees and rates in degrees/day. Errors
for elements from this work are the formal errors from the fitting process. All longitudes measured
from the ascending node of Saturn’s equator at epoch on the Earth mean equator at J2000, i
measured from Saturn’s equatorial plane at epoch.
in vg2 sat.bsp may well have been generated using a non-precessing elliptical model
with the parameters of Synnott et al. (1983).
The precessing ellipse fit to sat081.4.bsp in Table 7.10 has identical mean motion,
n, eccentricity, e, and semi-major axis, a, to the combined Voyager 1 and Voyager
2 fit of Jacobson (private communication), see Table 7.7. The precessing ellipse is
such a good fit that it suggests that the ephemeris of Pandora in sat081.4.bsp may
well have been generated using a precessing elliptical model with the parameters of
Jacobson (private communication). Also worth noting in the precessing ellipse fit
to sat081.4.bsp is that the semi-major axis from the fitting process, a, and the semi-
major axis derived from the mean motion, acalc, are within ∼ 200 metres of each
other. This close agreement between a and acalc is also apparent in the precessing
ellipse fit for Pandora from sat081.4.bsp (see Table 6.13) where it is ∼ 500 metres.
In all other cases of fitting ellipses, both to real observations and to ephemerides,
CHAPTER 7. PANDORA 151
Table 7.10: Orbital elements for Pandora from a fit to the SP kernel sat081.4.bsp:Epoch 2453371.00 JED. 160 positions at 3600.0 sec. intervals starting at 2453368.00JED.
para- Precessing Ellipse Model Non-Precessing Ellipse Model
metera β = γ = 0.0
a 141712.60± 0.09 141712 ± 4
acalc 141712.579± 0.003 141712.9 ± 0.1
λ 94.39098± 0.00004 94.391 ± 0.002
n 572.78891 ± 0.00002 572.7870± 0.0008
e (4.3710 ± 0.0004) × 10−3 (4.35 ± 0.02) × 10−3
i (1.144 ± 0.005) × 10−2 (1.1 ± 0.2) × 10−2
284.381 ± 0.001 285.22 ± 0.06
Ω 1.8 ± 0.2 1 ± 10
rms 0.15 6.68
aDistances are in km, longitudes in degrees and rates in degrees/day. Errors for elements
from this work are the formal errors from the fitting process. All longitudes measured from the
ascending node of Saturn’s equator at epoch on the Earth mean equator at J2000, i measured
from Saturn’s equatorial plane at epoch.
the difference in on the order of 10’s of kilometres.
7.8 The 3:2 near-resonance with Mimas
Up to this point no consideration had been given to perturbations on the orbit of
Pandora due to other satellites. The precessing ellipse model of Taylor (1998) used
to model the orbits of Atlas, Prometheus and Pandora is only valid if there are no
significant perturbations due to other satellites. Synnott et al. ( 1983) and Jacobson
(private communication) made no allowance for the proximity of Pandora to the
3:2 co-rotation eccentricity resonance (CER) with Mimas. Pandora’s semi-major
axis is ∼ 60 km from the location of the exact resonance. For details of resonances
in general and the Pandora-Mimas 3:2 CER in particular see Murray and Dermott
(1999).
CHAPTER 7. PANDORA 152
Allowance can be made for this resonance by modifying the mean longitude
term, Eqn. 2.59 to
λ = λ0 + nt + A sin ((2π/T )t+ θ) (7.1)
where A sin ((2π/T )t+ θ) is a periodic term introducing the effect of the resonance
with A as the amplitude, T the period and θ the phase of the periodic effect of the
resonance.
French et al. (2000) fitted data from HST observations in the period 1994-2000
and found that Pandora was lagging behind its position expected from Synnott
et al. (1983) by 1.27/year. In addition there was a periodic term with amplitude
0.78 and period ∼ 600 days. French et al. (2000) stated that the period, amplitude
and phase of this periodic term were entirely consistent with the effects of the
Pandora-Mimas 3:2 CER when calculated from theory. The relevant theories are
the planetary disturbing function and Lagrange’s planetary equations.
7.8.1 The planetary disturbing function
While the general three (and indeed n > 2) body problem is analytically insoluble,
it is possible to express the equations of motion of planets that are mutually inter-
acting in term of their orbital elements, thus obtaining a set of variational equations.
This technique was developed independently by Laplace (1772) and Lagrange (1776)
and the variational equations developed are called Lagrange’s planetary equations.
Consider a mass m orbiting a central mass M in an elliptical orbit. If these were
the only two bodies in the system, and treating the masses as point-masses, the
analytical solution is an ellipse with constant orbital elements a, λ, n, e, , i and Ω
(section 2.2). Adding a third mass m′ to the system, orbiting M in an elliptical orbit
exterior to that of m, introduces additional gravitational accelerations in addition
to the two-body accelerations due to M (see Fig 7.4).
This is the three-body problem and is not analytically soluble (section 2.2).
Now if both M ≫ m and M ≫ m′ then these additional accelerations due to the
introduction of m′ can be treated as perturbations of the two-body orbit. The ad-
ditional accelerations of m and m′ relative to M are derived from the gradient of
the perturbing potential, usually called the disturbing function, R. The disturb-
CHAPTER 7. PANDORA 153
r
r
r − rM
m
m/
/
/
Figure 7.4: The position vectors r and r′ of masses m and m′ with respect to thecentral mass M .
ing function can be derived from Universal Gravitation and Newton’s three laws
of motion, although we shall only give a brief summary complete derivations are
available in Murray and Dermott (1999) and Ellis and Murray (2000) for example.
The disturbing function for the inner mass can be written as
R =µ′
|r′ − r| − µ′ r · r′r′3
(7.2)
where µ′ = Gm′. Note that primed (′) quantities always denote the outer mass and
r′ > r at all times. The disturbing function for the inner mass is
R′ =µ
|r− r′| − µr · r′r3
(7.3)
where µ = Gm. While Eqns. 7.2 and 7.3 are for two orbiting bodies a similar anal-
ysis can be performed for any number of bodies. The accelerations associated with
the disturbing function can arise from sources other the point-mass gravitational
forces e.g. effects due to the oblateness of M .
The disturbing function can be analysed in several ways, but we shall follow
Murray and Dermott (1999) and consider the series expansion of the disturbing
function in Legendre polynomials in terms of the orbital elements. Murray and
Harper (1993) give the complete expansion of the disturbing function to eighth
order in e and i.
The orbital elements used are a, λ, e, , i and Ω for the inner body, mass m,
and similar primed (′) quantities for the outer body, mass m′. R is the disturbing
CHAPTER 7. PANDORA 154
function for perturbations on m due to m′ and R′ is the disturbing function for
perturbations onm′ due to m. Murray and Dermott (1999) show that the expansion
of R has the form
R = µ′∑
S (a, a′, e, e′, i, i′) cosφ (7.4)
where φ is a linear combination with the general form
φ = j1λ′ + j2λ+ j3
′ + j4 + j5Ω′ + j6Ω (7.5)
where the ji’s are integers and6∑
i=1
ji = 0 (7.6)
(Eqn. 7.6 only applies if the angles are referred to a fixed reference direction). The
S term in Eqn. 7.4 has the form
S ≈ f (α)
a′e|j4|e′|j3|i|j6|i′|j5| (7.7)
where α = a/a′ and so α < 1.0. Eqn. 7.7 is the D’Alembert relation. The function
f(α) can be expressed as a function of Laplace coefficients and their derivatives.
The Laplace coefficient b(j)s (α) is defined by
1
2b(j)s (α) =
1
2π
∫ 2π
0
cos jψdψ
(1 − 2α cosψ + α2)2 (7.8)
The terms in Eqn. 7.4 for which j1 and j2 are non-zero, i.e. those which involve
the mean longitudes, produce periodic perturbations while those where j1 = j2 = 0
are called ‘secular’ perturbations. Over timescales of 100-1000s of years these secular
perturbations can appear to be monotonically rising or falling. The expansions of
Eqn. 7.4 and also expansions for R′ can be found in Murray and Harper (1993) to
eighth order and Murray and Dermott (1999) to fourth order in the appropriate
orbital elements.
7.8.2 Lagrange’s planetary equations
Expanding the disturbing function provides the disturbing potential given the or-
bital elements. The orbital variations of the perturbed body can be analysed by
using the Lagrange planetary equations. These equations require an additional an-
gle, ǫ, where
λ = nt + ǫ (7.9)
CHAPTER 7. PANDORA 155
Lagrange’s equations for the variations of the orbital elements are (Murray and
Dermott 1999) to lowest order in e and i
dn
dt= − 3
a2
∂R∂λ
(7.10)
de
dt= − 1
na2e
∂R∂
(7.11)
di
dt= − 1
na2 sin i
∂R∂Ω
(7.12)
d
dt=
1
na2e
∂R∂e
+sin 1
2i
na2
∂R∂i
(7.13)
dΩ
dt=
1
na2 sin i
∂R∂i
(7.14)
dǫ
dt=
e
2na2e
∂R∂e
(7.15)
For a full derivation of Lagrange’s planetary equations see Brouwer and Clemence
(1961) and Roy (1988). In Eqn. 7.10, a has been substituted for by n = −32a/a
(Kepler’s third law) and partial derivatives involving ǫ by those involving λ (Murray
and Dermott 1999).
7.8.3 The effects of the Pandora-Mimas 3:2 CER from theory
If the resonance is sufficiently ‘shallow’ the simplified approximation developed later
in Eqn. 7.20 may be used. The resonance is shallow if the ratio
3nMimas − 2nPandora
nMimas
is much larger than MassMimas/MassSaturn. From Harper and Taylor (1993) the
mean motion ratio is ∼ 3 × 10−4 and the mass ratio ∼ 1 × 10−7. This three orders
of magnitude difference indicates that the resonance is shallow and the derived
Eqn. 7.20 is valid.
Using Eqn. 7.10, the time variability of the mean motion, n is given by
n = − 3
a2
∂R∂λ
(7.16)
If a term of the averaged expansion of the disturbing function is of the form
〈R〉 = µ′S cosφ (7.17)
then∂〈R〉∂λ
= −j2µ′S sinφ, (7.18)
CHAPTER 7. PANDORA 156
using the definition of φ from Eqn. 7.5. Substituting Eqn. 7.18 into Eqn. 7.10 gives
n =3j2a2µ′S sinφ (7.19)
We now assume, that at least to lowest order, the orbital elements on the right-
hand side of Eqn. 7.19, with the exception of λ and λ′ remain constant (i.e. are time
independent). We further assume that Eqn. 2.12 is valid i.e. λ ≈ n∆t + constant1
and λ′ ≈ n′∆t+ constant2. The solution for n from Eqn. 7.19 then becomes
n = n0 −1
j1n′ + j2n
3j2a2µ′S cosφ (7.20)
Both de/dt and di/dt will have similar solutions since they also involve ∂〈R〉/∂λ.
Taking starting values for the orbital elements at epoch of Pandora and Mimas
we numerically integrated Eqn. 7.20 using a fourth order Runge-Kutta technique
to determined the value of φ, λ and n for Pandora at any time. We used a stepsize
of 0.002 days ∼ 0.01 of Pandora’s orbital period. The resonant argument for the
Pandora-Mimas 3:2 CER is
φ = 2λ′ − 3λ−′ (7.21)
The value of φ was calculated at each iteration. The elements for Mimas were
calculated using the theory of Mimas of Harper and Taylor (1993), which takes into
account the effects of the 4:2 Mimas-Tethys resonance. The elements at epoch for
Pandora were taken from the combined Voyager 1 and Voyager 2 fit in Table 7.7.
The mean longitude for Pandora at each iteration was obtained using a modified
form of Eqn. 2.59
λi = λi−1 + ni−1∆t (7.22)
where i is the number of the iteration step and ∆t is the stepsize. The value of ∆λ
was also calculated at each step where
∆λi = λi − (λ0 + n0t) (7.23)
Therefore ∆λ is the difference between the longitude with the perturbation due to
the resonance included and the unperturbed mean longitude. The results obtained
for φ, n and ∆λ are shown in Figs. 7.5, 7.6 and 7.7 respectively
CHAPTER 7. PANDORA 157
Figure 7.5: Time variation of the resonant argument of Pandora. Epoch:2444839.6682 JED
Figure 7.6: Time variation of the mean motion of Pandora. Epoch: 2444839.6682JED
CHAPTER 7. PANDORA 158
Figure 7.7: Time variation of ∆Λ for Pandora. Epoch: 2444839.6682 JED
The amplitude, A, period, T and phase, θ, of ∆λ are
A = 0.85
T = 633 days
θ = 42.7
when Eqn. 7.1 is used with these values in fitting a precessing ellipse to the data
instead of Eqn. 2.59, we obtained the fit to the full set of Voyager 1 and Voyager 2
observations shown in Table 7.11.
When the values of A and T derived from theory was used but the phase, θ was
fitted as well the best fit orbit to the data is shown in Table 7.12 with the ‘best’ θ
being θ = 28.7.
7.8.4 Discussion
When the effects of the 3:2 CER with Mimas are included in the orbit fit the mean
motion drops from 572.78859 ± 0.00009/day (Table 7.7) to 572.78439 ± 0.00009
(Table 7.11) or 572.78549± 0.00009 (Table 7.12). There is a corresponding change
in the mean longitude at epoch from 82.12±0.01 to 114.63±0.01 or 114.82±0.01.
The effects of the 3:2 resonance clearly have a significant effect on the motion of
CHAPTER 7. PANDORA 159
Table 7.11: The orbital elements of Pandora: epoch2444839.6682 JED, including the calculated effects ofthe 3:2 CER with Mimas.
parametersa Orbit Fit
a 141731 ± 13
acalc 141713.32± 0.01
λ 82.13 ± 0.01
n 572.78439 ± 0.00009
e (4.4 ± 0.1) × 10−3
i 0.054 ± 0.007
56 ± 1
Ω 243 ± 4
rms error 0.2976
aDistances are in km, longitudes in degrees, rates in de-
grees/day and the rms error in arcseconds. Errors for elements
from this work are the formal errors from the fitting process. All
longitudes measured from the ascending node of Saturn’s equator
at epoch on the Earth mean equator at J2000, i measured from
Saturn’s equatorial plane at epoch.
Pandora and must be included in any orbit fitting process. The fits for Pandora of
Synnott et al. (1983) and Jacobson (private communication), which do not include
the effects of this resonance, are unsatisfactory.
The derived orbit of Pandora of French et al. (2000), which does include the
effects of the resonance is shown in Table 7.13.
The mean motion and eccentricity of the fit using the best fit phase (Table 7.12)
are consistent (within ∼ 1σ) with those of French et al. (2000) (Table 7.13).
CHAPTER 7. PANDORA 160
Table 7.12: The orbital elements of Pandora: epoch2444839.6682 JED, including the calculated effects ofthe 3:2 CER with Mimas. The best fit value of thephase is used.
parametersa Orbit Fit
a 141731 ± 13
acalc 141713.14± 0.01
λ 82.15 ± 0.01
n 572.78549 ± 0.00009
e (4.4 ± 0.1) × 10−3
i 0.053 ± 0.007
56 ± 1
Ω 243 ± 4
rms error 0.2971
aDistances are in km, longitudes in degrees, rates in de-
grees/day and the rms error in arcseconds. Errors for elements
from this work are the formal errors from the fitting process. All
longitudes measured from the ascending node of Saturn’s equator
at epoch on the Earth mean equator at J2000, i measured from
Saturn’s equatorial plane at epoch.
CHAPTER 7. PANDORA 161
Table 7.13: The orbital elements of Pandora of Frenchet al. (2000): epoch 2449940.0 JED, including the cal-culated effects of the 3:2 CER with Mimas.
parametersa Orbit Fit
acalc 141713.1119± 0.0084
λ 96.03 ± 0.05
n 572.785574± 0.000051
e (4.53 ± 0.36) × 10−3
395.8 ± 6.3
rms error ∼ 0.5
aDistances are in km, longitudes in degrees, rates in de-
grees/day and the rms error in pixels. The quoted errors are three
times the formal errors from the fitting process (i.e. 3σ). All lon-
gitudes measured from the ascending node of Saturn’s equator at
epoch on the Earth mean equator at J2000.
Chapter 8
Planning Future Observations
8.1 Introduction
A great deal of time and effort goes into the planning of the observations to be
made by a spacecraft on a mission to a Solar System object. A wide variety of
instrument packages are usually carried. The Cassini spacecraft has the following
instruments
• Composite Infrared spectrometer (CIRS).
• Imaging Science Subsystem (ISS).
• Ultraviolet Imaging Spectrograph (UVIS).
• Visual and Infrared Mapping Spectrometer (VIMS).
• Cassini Radar (RADAR).
• Radio Science Subsystem (RSS).
• Cassini Plasma Spectrometer (CAPS).
• Cosmic Dust Analyser (CDA).
• Ion and Neutral Mass Spectrometer (INMS).
• Dual Technique Magnetometer (MAG).
• Magnetospheric Imaging Instrument (MIMI).
162
CHAPTER 8. PLANNING FUTURE OBSERVATIONS 163
• Radio and Plasma Wave Science (RPWS).
Each of these science packages has its own team of investigators and instrumen-
tation on the spacecraft (RADAR and RSS use some elements of the spacecraft’s
communications links as well).
Data gathered by the science packages is stored on the spacecraft Solid State
Recorders (SSRs) before being regularly down-linked to one of the receivers of
the Deep Space Network (DSN). The instruments are capable of generating vast
amounts of data which could very quickly exceed the memory capacity of the SSRs,
this places constraints on the amount of data that can be gathered. Hence all the
instruments cannot be gathering data all of the time. Another restriction is the
orientation of the spacecraft. All the instruments are fixed to the spacecraft bus
and are pointed at the desired target by actually changing the spacecraft orienta-
tion. Some instruments are mounted orthogonally to others, e.g. ISS and RADAR,
so it is impossible to take simultaneous ISS images and RADAR observations of
the same target. So it is possible that different teams could want the spacecraft
pointing in entirely different directions at the same time.
Space on the SSRs between successive downlinks is allocated to individual in-
struments. Even if no other team wants to take data simultaneously and the space-
craft can be pointed in the direction a particular team wants the space restrictions
on the SSRs may mean that data cannot be collected. Also, down-linking data
costs money, for use of the DSN etc., so financial constraints require the number
of downlinks to be minimised. The amount of data collected should be kept to the
minimum needed to achieve the required scientific goals.
Each team puts in bids, stating what observations they want made, when they
want them, how much SSR space is required and how critical the observations
are. Each bid has to be fully supported by a scientific argument. The bids of all
the teams are compared and time and SSR space allocated, bearing in mind the
aforementioned financial constraints on the number of downlinks. Time and SSR
space is allocated years before the actual time of the observations. At the time of
writing (Oct. 2000) this process for Saturn for Cassini is nearing completion even
though Saturn Orbit Insertion (SOI) is not until July 1st 2004.
CHAPTER 8. PLANNING FUTURE OBSERVATIONS 164
To ensure that a bid stands the best chance of being accepted it must require
the minimum number of observations to achieve the stated scientific goals. An
advantage is also to schedule observations when no other team is likely to want to
take data. If such scheduling is achieved the bid is then accepted or rejected solely
on the strength of the scientific case and the amount of data required.
In this chapter we investigate the minimum number of observations of a satellite
that are required to achieve a specific accuracy in the determination of its orbital
elements. Any derived relationships between number of observations and accuracy
in orbit determination can then be used to justify observations requested during
the bidding process.
8.2 The error in e from geometry
Consider an observer, O, making observations of a satellite, S, orbiting a planet,
P. The observer is a distance, d, from the planet while the satellite is a distance, r,
from the planet. The geometry of the situation is shown in Fig. 8.1.
O
P
S
d
r
θ
Figure 8.1: Geometry of observation 1
CHAPTER 8. PLANNING FUTURE OBSERVATIONS 165
The angle at O between the lines OS and OP is θ. Now for θ ≤ 0.08 radians
we have
r ≃ dθ (8.1)
r and θ are then increased by small amounts, δr and δθ, as shown in Fig. 8.2
O
P
S
d
r
θ
δr
δθ
Figure 8.2: Geometry of observations 2
r + δr ≃ d(θ + δθ) (8.2)
Which with a little rearranging becomes
δr ≃ dθ + dδθ − r (8.3)
putting Eqn. 8.2 into Eqn. 8.3 gives
δr ≃ dθ + dδθ − dθ (8.4)
leading to
δr ≃ dδθ (8.5)
For an ellipse
rapocentre = a(1 + e) (8.6)
CHAPTER 8. PLANNING FUTURE OBSERVATIONS 166
making a constant and differentiating Eqn. 8.6 gives
δrapocentre = aδe (8.7)
or
δe =δrapocentre
a(8.8)
for e≪ 1 we can say rapocentre ≃ r, hence
δrapocentre ≃ δr (8.9)
thus
δe =δr
a(8.10)
Putting Eqn. 8.2 into Eqn. 8.10 gives
δe ≃ dδθ
a(8.11)
Now what if we have N observations where δθ is the observational error. Further-
more, assume that δθ has a Gaussian distribution. Eqn. 8.11 becomes
δe ≃ dδθ
a√N
(8.12)
Eqn. 8.12 gives the relationship between the absolute error in e for the orbit de-
termination of a satellite using N images taken at a distance d from a planet with
the mean observational error being δθ. So from geometrical arguments we have an
expression for the absolute error that can be expected in the determination of the
eccentricity of a satellite.
∆e ≃ d∆θ
a√N
(8.13)
It is important to note that ∆e does not depend on e at all.
Due to the assumptions that have been made, Eqn. 8.12 is only valid if e ≪ 1
and θ ≤ 0.08 radians. Using this last restriction in Eqn. 8.2 means that d ≥ 12.5r.
So far no consideration has been given to angle between the observer-planet line
and the orbital plane of the satellite. For a satellite orbiting in, or near to, the
equator plane of the planet this is the sub-spacecraft latitude (SSL). When the SSL
is taken into account Eqn. 8.13 is expected to be of the form
∆e ≃ d∆θ
a√NF(SSL) (8.14)
where F(SSL) is a function of the sub-spacecraft latitude.
CHAPTER 8. PLANNING FUTURE OBSERVATIONS 167
8.2.1 Determining F(SSL)
Using the precessing elliptical model of Taylor (1998), sets of observations of objects
orbiting Saturn were generated. It was assumed that the observations were evenly
spaced in time along one orbital period of the body. The observation point was
stationary with respect to the center of Saturn with
x = d cosSSl
y = 0
z = d sinSSL
The parameters used to generate the sets of observations were a, e, d, SSL, N and
∆θ. At all times it was ensured that d ≥ 12.5r. For the purpose of this analysis the
longitudes giving the position of the body in its orbit at the start of the observation
period were unimportant.
The sets of observations were then processed though the orbit determination
software outlined in Chapter 5 to derive a fitted orbit for the object corresponding
to each set of test observations. A record was made of the formal errors in e and i
in each case.
A least squares polynomial fit was performed on the data and this is shown in
Fig. 8.3 . The least squares fit to the data has the form
F(SSL) = 1.3481678−1.852335 |SSl|+1.098612 SSL2−0.1946722 |SSL|3 (8.15)
where SSL is measured in radians. Therefore Eqns. 8.14 and 8.15 together can be
used to estimate the number of images needed to achieved a required accuracy in
e.
8.2.2 Other orbital parameters
A similar analysis can be performed on the other orbital elements to determine the
relationship between number of images and accuracy achieved.
CHAPTER 8. PLANNING FUTURE OBSERVATIONS 168
Figure 8.3: Relationship between F(SSL) and SSL
8.2.3 Limitations
The analysis was performed using the assumption that the input parameters d,∆θ, a
and SSL remained constant throughout. This of course is not the case when the
images are being taken from a moving spacecraft. During an observation run the
spacecraft-planet distance, d, can change quite dramatically as can the SSL. When
this is the case Eqn. 8.14 , with F(SSL) given by Eqn. 8.15, is invalid. For actual
observations from a spacecraft orbiting a planet a different approach must be taken.
8.3 Expected accuracy of realistic observations
It has been noted that when observations are made from a spacecraft orbiting
a planet, parameters effecting the accuracy of orbit determinations don’t remain
constant but can be subject to quite large variations. The expected accuracy of de-
termined orbital parameters cannot be obtained via simple analysis as in Eqn. 8.14.
In such more realistic cases a different approach was used.
CHAPTER 8. PLANNING FUTURE OBSERVATIONS 169
The technique employed requires an accurate ephemeris for the observing space-
craft and approximate elements for the orbit of the target object. The number of
observations to be made and the time of each must be decided before any calcula-
tions can be made.
The position of the target object at any time is calculated using Eqns. 2.66, 2.67
and 2.68. The position of the spacecraft at the observation time is obtained from
the spacecraft ephemeris. The apparent position of the target object as seen from
the spacecraft is obtained from the positions of each object with allowances being
made for planetary aberration and light travel time, given by Eqn. 2.69.
In this way a set of theoretical observations is calculated. Random errors are
introduced into this observation set with a mean error of ∆θ. Observations are then
processed through the orbit determination process outlined in Chapter 5 . The orbit
determination includes formal errors for all the (assumed) orbital parameters of the
target object. The number of observations and the observation times are changed
until the required accuracy in the determination of a particular orbital parameter
is reached.
Using this method it is possible to plan a series of observations for a forthcoming
space mission, provided the accuracy requirements are known
8.4 Planning Cassini observations of the small
saturnian satellites.
A technique similar to that just outlined was employed to plan Cassini observa-
tions of the small saturnian satellites. Using an accurate ephemeris for the Cassini
spacecraft and a generated ephemeris for the satellite of interest, a series of syn-
thetic observations of the satellite were generated. The set of synthetic observations
also included synthetic ‘observational errors’ with mean value σ arcseconds. The
ephemeris for the satellite was generated using the precessing ellipse model of Taylor
(1998) (section 2.5.2).
Having an accurate ephemeris for Cassini and the elements of the satellite’s
orbit enabled the generation of a series of synthetic observations of the satellite,
CHAPTER 8. PLANNING FUTURE OBSERVATIONS 170
P(ti), at times ti. The number of observations made, i, the times at which they
are made, t(i) and the mean observational error to be introduced, σ, are the only
input variables. The varying observational geometry and positions of Cassini and
the satellite depend on t(i).
8.4.1 Introducing synthetic ‘observational errors’
For inclusion of generated ‘observational errors’ the unit pointing vectors, P(ti)
were used instead of the pointing vectors P(ti). The unit pointing vectors were
converted into sets of Right Ascension and Declination coordinates, RA(i) and
DEC(i). From the mean error, σ, a set of random right ascension and declination
errors were calculated
δRA(i) = σ cos(xi) (8.16)
δDEC(i) = σ sin(xi) (8.17)
where xi is a random number with values between 0 and 2π. Since the mean error,
σ, is small we assumed that
RAobserved(i) = RA(i) + δRA(i) (8.18)
DECobserved(i) = DEC(i) + δDEC(i) (8.19)
and then converted the sets of RAobserved(i) and DECobserved(i) coordinates back
into unit pointing vectors, Pobserved(i) which now included synthetic observational
errors.
8.4.2 Fitting an orbit
The set of observations Pobserved(i), along with their associated observation times
t(i), were then processed through the orbit determination software as described
in section 5.6. The formal errors in each of the orbital elements as determined
by the fitting process were used as the accuracy of the orbit determination. For
observations of a particular satellite the dependency of the accuracy of the fit on i,
t(i) and σ could be determined.
CHAPTER 8. PLANNING FUTURE OBSERVATIONS 171
8.4.3 Examples of calculated orbit determination accuracy for Cassini
The following examples are all sets of observations of small saturnian satellites
which were proposed to the Cassini ISS team by the author. The assumed and
determined orbital elements are fully listed.
Table 8.1: 90 observations of Prometheus starting at 2005 APR 29 10:00:00.0 JED,each observation at intervals of 12 mins. The epoch of the orbits is 2005 APR 2910:00:00.0 JED
parametera Assumed orbit Determined Orbit
a 139377.41 139376± 1
acalc 139377.37 139377.1± 0.4
λ 65.78 65.780 ± 0.001
n 587.28916 587.2906± 0.0022
e 0.00229 (2.267 ± 0.010) × 10−3
304.71 304.7 ± 0.2
i 0.011 0.0117 ± 0.0006
Ω 1.98 6 ± 3
σ 2.0
aDistances are in km, longitudes in degrees and rates in degrees/day. Errors for elements
from this work are the formal errors from the fitting process. All longitudes measured from the
ascending node of Saturn’s equator at epoch on the Earth mean equator at J2000, i measured
from Saturn’s equatorial plane at epoch. σ is in arcseconds.
CHAPTER 8. PLANNING FUTURE OBSERVATIONS 172
Table 8.2: 90 observations of Pandora starting at 2005 APR 30 10:00:00.0 JED,each observation at intervals of 12 mins. The epoch of the orbits is 2005 APR 3010:00:00.0 JED
parametera Assumed orbit Determined Orbit
a 141712.578 141713± 1
acalc 141712.578 141713.1± 0.3
λ 102.9610 102.9630± 0.0008
n 572.78891 572.7858± 0.0016
e 0.004370 (4.368 ± 0.007) × 10−3
228.320 228.02 ± 0.06
i 0.01143 0.0124 ± 0.0005
Ω 17.4 19 ± 2
σ 2.0
aDistances are in km, longitudes in degrees and rates in degrees/day. Errors for elements
from this work are the formal errors from the fitting process. All longitudes measured from the
ascending node of Saturn’s equator at epoch on the Earth mean equator at J2000, i measured
from Saturn’s equatorial plane at epoch. σ is in arcseconds.
CHAPTER 8. PLANNING FUTURE OBSERVATIONS 173
Table 8.3: 90 observations of Epimetheus starting at 2005 MAY 17 15:00:00.0 JED,each observation at intervals of 12 mins. The epoch of the orbits is 2005 APR 1715:00:00.0 JED
parametera Assumed orbit Determined Orbit
a 151412.76 151412± 1
acalc 151412.76 151412.4± 0.7
λ 236.3553 236.355 ± 0.001
n 518.49171 518.493 ± 0.003
e 0.009868 (9.845 ± 0.008) × 10−3
344.024 344.10 ± 0.05
i 0.3349 0.3353 ± 0.0006
Ω 143.94 143.93 ± 0.08
σ 2.0
aDistances are in km, longitudes in degrees and rates in degrees/day. Errors for elements
from this work are the formal errors from the fitting process. All longitudes measured from the
ascending node of Saturn’s equator at epoch on the Earth mean equator at J2000, i measured
from Saturn’s equatorial plane at epoch. σ is in arcseconds.
CHAPTER 8. PLANNING FUTURE OBSERVATIONS 174
8.4.4 The effects of varying the number of observations and assumed observational
error
The following tables show the effects of changing the number of observations and
the assumed observational error on the accuracy of the orbit determination. The
assumed orbits are the same as in section 8.4.3. The orbital elements of both the
assumed orbit and the determined orbit are not listed in the tables. The formal
error from the fitting process for each of the determined elements is listed as a
fraction of that element e.g. ∆e = σe/edetermined
Table 8.4: Observations of Prometheus starting at 2005 APR 29 10:00:00.0 JED,ending at 2005 APR 30 04:00:00.0 JED. Constant number of images. The epoch ofthe orbits is 2005 APR 29 10:00:00.0 JED
parametera
N 160 160 160 160
σ 1 2 3 4
∆a 5.2 × 10−6 1.0 × 10−5 1.6 × 10−5 2.1 × 10−5
∆acalc 1.3 × 10−6 2.5 × 10−6 3.8 × 10−6 5.1 × 10−6
∆λ 7.9 × 10−6 1.6 × 10−5 2.4 × 10−5 3.2 × 10−5
∆n 1.9 × 10−6 3.8 × 10−6 5.7 × 10−6 7.7 × 10−6
∆e 2.2 × 10−3 4.3 × 10−3 6.5 × 10−3 8.7 × 10−3
∆ 2.5 × 10−4 4.9 × 10−4 7.3 × 10−4 9.9 × 10−4
∆i 3.0 × 10−2 5.7 × 10−2 8.9 × 10−2 0.11
∆Ω 0.5 0.8 1.5 × 10−2 0.65
rms 0.103 0.204 0.309 0.414
aDistances are in km, longitudes in degrees and rates in degrees/day. N is the number of
observations. All longitudes measured from the ascending node of Saturn’s equator at epoch on
the Earth mean equator at J2000, i measured from Saturn’s equatorial plane at epoch. σ and rms
are in arcseconds.
Table 8.4 clearly illustrates that the formal errors in a, acalc, λ, n, e, and i are
linearly dependent on the assumed mean error, σ, in the observations. This is in
agreement with Eqn. 8.14 for ∆e derived from elementary geometrical arguments
(where ∆θ = σ). It appears that the geometrical argument for the dependency on
CHAPTER 8. PLANNING FUTURE OBSERVATIONS 175
Table 8.5: Observations of Prometheus starting at 2005 APR 29 10:00:00.0 JED,ending at 2005 APR 30 04:00:00.0 JED. Constant σ. The epoch of the orbits is2005 APR 29 10:00:00.0 JED
parametera
N 20 40 80 160
σ 2.5 2.5 2.5 2.5
∆a 2.4 × 10−5 1.8 × 10−5 1.3 × 10−5 8.8 × 10−6
∆acalc 4.1 × 10−6 2.9 × 10−6 2.0 × 10−6 1.4 × 10−6
∆λ 4.2 × 10−5 3.1 × 10−5 2.2 × 10−5 1.5 × 10−5
∆n 6.2 × 10−6 4.4 × 10−6 3.0 × 10−6 2.1 × 10−6
∆e 9.4 × 10−3 7.2 × 10−3 5.1 × 10−3 3.5 × 10−3
∆ 1.1 × 10−3 8.2 × 10−4 5.7 × 10−4 4.0 × 10−4
∆i 0.19 9.7 × 10−2 6.6 × 10−2 5.1 × 10−2
∆Ω 0.69 0.51 0.54 8.8 × 10−3
rms 0.492 0.400 0.272 0.191
aDistances are in km, longitudes in degrees and rates in degrees/day. N is the number of
observations. All longitudes measured from the ascending node of Saturn’s equator at epoch on
the Earth mean equator at J2000, i measured from Saturn’s equatorial plane at epoch. σ and rms
are in arcseconds.
∆θ holds for all the orbital elements. No conclusions can be reached regarding ∆Ω’s
dependency on σ since Ω ≈ 0 (Table 8.1). This makes accurate estimation of ∆Ω
problematic since a small change in Ω has a disproportionally large effect on ∆Ω.
If the error in the determination of an element is indeed proportional to 1/√
(N)
i.e.
∆ ∝ 1√N
(8.20)
then a graph of ∆ plotted against 1/√N should be a straight line. In Fig. 8.4
the values from Table 8.5 are plotted. Fig. 8.4 shows that for all the elements,
∆ ∝ 1/√N , is an accurate description. The value of i deviates slightly from this
relationship but it is accurate enough to make an estimate.
This indicates that Eqn. 8.14 is valid for predicting the rms errors in the deter-
CHAPTER 8. PLANNING FUTURE OBSERVATIONS 176
Figure 8.4: Sets of plots of the ∆ values from Table 8.5 on the y-axis against thecorresponding 1/
√N values on the x-axis. In the semi-major axis plot the bottom
line is acalc.
mined values of all the orbital elements, not just e. Hence,
∆ ≃ d∆θ
a√NF(SSL) (8.21)
where ∆ is the error in any orbital element. Care should be taken since Eqn. 8.21
is probably only valid for fixed F(SSL). The value of the sub-spacecraft latitude
(SSl) remained fairly constant for the synthetic observations generated in Table
8.5.
Eqn. 8.21 can be used to support requests for specified numbers of images of
satellites. It indicates the requirements for achieving the accuracy desired in the
determination of an orbit.
Chapter 9
Summary and Discussion
9.1 Summary
This work was an attempt to derive orbits for the saturnian satellites Atlas, Prome-
theus and Pandora at the Voyager 1 and Voyager 2 epochs. The Voyager Saturn
data set was searched with the aim of identifying images of these satellites not used
by Smith et al. (1981) and Synnott et al. (1983). A number of explanations for the
∼ 19 mean longitude lag of Prometheus observed in 1995 were examined in light
of the the derived Voyager 1 and Voyager 2 orbits.
The orbits for Prometheus and Pandora derived by Synnott et al. (1983) and
Jacobson (private communication) were approximately the same as derived in this
work using the same image set. The Prometheus orbital elements from this work
were determined using three times as many images as those of Synnott et al. (1983)
and Jacobson (private communication). The rms error of the fit to the combined
Voyager data set was half that achieved by Synnott et al. (1983).
There was a difference in the determined mean motion of Prometheus at the two
Voyager epochs of ∼ 3σ, while the eccentricities differed by ∼ 6σ. We concluded
that not only did Prometheus’ orbit change between the 1980’s and 1995 but also
underwent a change between the two Voyager encounters. None of the derived
orbits were compatible with the observed mean longitude in 1995. There was no
theory that adequately explained the changes in Prometheus’ orbit and produced
the observed mean longitudes at various epochs.
177
CHAPTER 9. SUMMARY AND DISCUSSION 178
For Pandora the number of images used in this work was not significantly greater
than used by Synnott et al. (1983). There were no indications of any change in
Pandora’s orbit between the two Voyager encounters. When the effects of the 3:2
CER with Mimas were included the mean motion and mean longitude at epoch
for the derived orbits changed significantly. The simple precessing ellipse model
was not representative of the actual dynamics of Pandora’s orbit. An additional
periodic term had to be included in the calculation of the mean longitude.
For Atlas we used three times as many images as in the determination of Smith et
al. (1981). Our derived orbit differed significantly from that of Smith et al. (1981)
but was comparable with that in the current JPL ephemerides. There was no
indication that Atlas’ orbit changed between the two Voyager encounters.
The orbits of Pandora and Atlas did not show significant change between the
two Voyager epochs while that of Prometheus did. This indicated that the change
in Prometheus’ orbit was real and not a flaw in the whole image navigation/orbit
determination process.
9.2 Accuracy of determinations
The simulations showed that the formal error from the fitting process for each of
the orbital elements is modeled by the equation
∆ ≃ d∆θ
a√NF(SSL)
This can be used to support request for imaging sequences of satellites during the
Four year Cassini tour of the Saturn system.
9.3 Future work
Prometheus’ anomalous motion indicates that there is undetected mass in the F
ring region, either small satellites or within the F ring itself. In the course of deter-
mining the orbits of Atlas, Prometheus and Pandora in excess of 150 navigated and
geometrically corrected images of the F ring region were produced. These images
are ideal for use in a search for previously undetected small satellites. Additional
CHAPTER 9. SUMMARY AND DISCUSSION 179
images of the Pioneer Gap region could be easily located, geometrically corrected
and navigated.
Combining the precessing ellipse model of Taylor (1998) with numerical inte-
gration of the full equations of motion along the lines used by Harper (1987) would
prove useful. The trajectory of the satellite would be fit to the observations using
numerical integration of the equations of motion with all bodies in the system being
included. From the derived state at epoch the numerical integration can be used
to produce an ephemeris. The precessing ellipse model can then be fitted to the
ephemeris. All the subtle effects due to the perturbations of other objects would
be included in the ephemeris while the precessing ellipse model produces easily
understandable orbital elements.
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