Modeling Planetary Motions: Why We Care and How We Do It

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1. INTRODUCTION
1.1. Why do we need precise planetary positions?
There are several good reasons why we need precise planetary positions.
I will mention just a few that impact the U.S. Naval Observatory and its
various scientific (and military) programs and objectives. There are two
broad requirement categories: Department of Defense (DoD) requirements
and astronomical requirements, which overlap significantly. First, for ord-
nance guiding and targeting purposes, the DoD requires stellar positions to
MODELING PLANETARY MOTIONS:
WHY WE CARE AND HOW WE DO IT
M
ARC
A. M
URISON
Astronomical Applications Dept., U.S. Naval Observatory, 3450 Massachusetts Ave,
NW
,
Washington DC 20392 USA
murison@aa.usno.navy.mil
March 5, 1999
ABSTRACT
Presented is a review of high-precision modeling of the mo-
tions of solar system objects, especially the planets. Motiva-
tions are discussed, as are various astronomical reference
frames and their interdependence. We then discuss, in broad
terms, how one generates a high-precision ephemeris and the
kinds of observations that such an ephemeris depends upon.
Following that, we review Newcomb, a new high-precision
ephemeris program currently under development at the U.S.
Naval Observatory. This new program takes full advantage of
modern programing design and techniques, including object
orientation and graphical user interfaces. We close with a
brief description of the three main modules of Newcomb and
how they work.
Key words:
solar system — ephemerides — celestial mechan-
ics

better than 20 milliarcseconds (mas)
1
. Placing accuracy requirements
upon a stellar reference frame has implications for how accurately we must
then know, among other things, the positions of the planets and other solar
system objects. The sequence of connections that joins the two seemingly
disparate accuracies (that of the stars and that of the planets) requires a dis-
cussion of dynamical and astronomical reference frames, which I will get
to in a moment.
The second broad requirement category is the astronomical need for ac-
curate planetary positions. Besides the intrinsic interest of astronomers in
planetary, asteroidal, and cometary positions, knowledge of these positions
over time — called an
ephemeris
— fundamentally affects many areas of
solar system and even stellar astronomy:
1. To the general public, perhaps the most apparent astronomical need
for precise planetary positions is in spacecraft navigation.
2. Solar system celestial mechanics depends greatly on accurate posi-
tions. Theories of planetary and satellite motions live or die according
to how well their predictions agree with observational knowledge of
positions. These theories are the means by which we develop our
most fundamental understanding of the many complicated dynamical
processes and interactions in the solar system.
3. Another area where accurate knowledge of planetary positions is cru-
cial is stellar occulations. If a planet is passing in front of a star, and
we can predict where on the Earth’s surface this event is visible, then
we may learn several things, including density and composition of the
planetary atmosphere, certain facts about the atmosphere of the oc-
culted star, and, of course, better knowledge of either the star’s posi-
tion, the planet’s position, or both. Further, if the occulting body is an
asteroid, we can even determine the projected shape of the asteroid.
MURISON: MODELING PLANETARY MOTIONS 2 of 20
1
One arc second is equal to one 3600th of a degree. One milliarcsecond equals one one-
thousandth of an arc second. To provide some context, consider that 1 arcsecond corre-
sponds to 1/16 of an inch around the perimeter of the 1000-foot radius Observatory Cir-
cle here at USNO. How much is 50
micro
arcseconds [the nominal accuracy of the pro-
posed space astrometry mission FAME (http://aa.usno.navy.mil/FAME/)]? That is the
angle subtended by the width of a typical strand of human hair as seen from a distance
of
65 miles
.

4. Finally, General Relativity has been tested by observing starlight that
grazes a body — the Sun, Jupiter, and Earth have all been used thus.
Again, accurate knowledge of planetary positions is essential.
This is not an exhaustive list of the astronomical benefits of accurate
planetary positions, but it gives a flavor of the value of such positional
knowledge and prediction.
1.2. Dependencies
Theory, observation, and application are interdependent, as illustrated in
Figure 1. Observations can be interpreted only in the context of our under-
standing — represented by theoretical models — of the solar system and
its dynamics. The observations can then be used to correct and update our
theoretical models. The fusion of the two results in planetary (and
satellite) ephemerides as well as a better solar system reference frame.
The combination of planetary system model — as determined by theory
and refined by observations — plus stellar position catalogs, gives rise to a
multitude of practical applications in many areas, including astronomy,
geophysics, and military.
1.3. Reference Frames
Although we here on
the Earth’s surface might
often prefer to work with
a nearly inertial frame of
reference tied to the dis-
tant stars, we must in-
stead put up with the
various noninertial refer-
ence frames to which we
find ourselves affixed.
Defining and/or connect-
ing them is both observa-
tionally and theoretically
a complex undertaking.
We must tie together a lo-
cal reference frame, at-
MURISON: MODELING PLANETARY MOTIONS 3 of 20
Figure 1 — Interplay between theory, observation, and
application.
Planetary
Ephemerides
Solar System
Reference
Frame
Star Positions
Geodynamics
Asteroid
Motions
Planetary
Motions
Gravity Model
General
Relativity
Time
MilitaryAstronomy
Planetary and
Satellite
Observations
Application
Theory
Observation

tached to a specific location
(and a specific time) on the
Earth’s surface, to a refer-
ence frame that takes into
account the spinning and
wobbling motions of the
Earth. This spinning and
wobbling frame can be
connected via lunar laser
ranging to a frame that en-
compasses the dynamic so-
lar system, with all its com-
plicated planetary and sat-
ellite motions, each body
affecting to various degrees
the motions of every other
body in accordance with
Newton’s and Einstein’s
theories of motion in gravi-
tational fields. We then try
to join the solar system dy-
namical frame to a Galactic
frame, which takes the
form of standard catalog
frames such as FK5
2
or
HIPPARCOS
3
, which are
stellar catalogs, or the new
ICRF
4
, based on extragalac-
tic objects. The Naval Ob-
servatory has been and con-
tinues to be among the
world’s foremost contribu-
tors to and creators of these
MURISON: MODELING PLANETARY MOTIONS 4 of 20
4
For an informative introduction to the International Celestial Reference System (ICRS),
see http://aa.usno.navy.mil/AA/faq/docs/ICRS_doc.html. The ICRF catalog and related
information is available at http://hpiers.obspm.fr/webiers/results/icrf/README.html.
3
http://astro.estec.esa.nl/Hipparcos/hipparcos.html
2
http://adc.gsfc.nasa.gov/adc-cgi/cat.pl?/catalogs/1/1149A/
Figure 2 — A reference frame hierarchy, showing the
context in which modeling planetary motions (shaded
boxes) resides.
Galactic
Reference Frame
Solar System
Reference Frame
Earth Reference
Frame
Extragalactic
Reference Frame
Quasars
Local Reference
Frame
Planetary
Ephemerides
Natural Satellite
Positions
Dynamical
Models
Earth Rotation
VLBI
GPS
GPS
HIPPARCOS
FAME
Where am I? What time is it?
NPOI
- Spacecraft navigation
- Tests of GR
- Celestial mechanics
- Asteroid masses
- Stellar occultations
- Geophysics
- Geodesy
VLBI, optical
Star Catalogs
Planetary
Positions
Figure 3 — Connections between reference frames.
data types that
connect the frames
observations that
determine the
frame
topographic
frame
geocentric
frame
solar system
dynamical
frame
galactic frame
extragalactic
frame
y
planets
y
nat. satellites
y
stars
y
radio stars
y
quasars
y
VLBI
y
space astrometry
y
s/c ranging
y
transit circle
y
radar
y
LLR
y
VLBI
y
transit circle

kinds of fundamental position catalogs. Figure 2 is an illustration showing
various reference frames, from the largest scale to the smallest, and the
kinds of processes or observational methods that they rely on. The first
column is a hierarchy of frames, from the size scale of the universe down
to the local and very practical question of “Where am I right now?”. We
must attach the notion of time to that of position, since in any dynamical
frame the two are inextricably linked. The second column contains the
major input category or dynamics type that corresponds to the associated
reference frame. The third column lists the most important observation
types that determine the reference frame. The activities associated with
planetary ephemeris generation correspond to the shaded boxes, and these
are the areas we will concentrate on here. Figure 3 shows these same ref-
erence frames, but organized to show how they are related to each other
observationally.
1.4. Generating Precise Predictions of Planetary Positions
We have established the
need for accurate planetary
positions. Therefore, we
need to be able to
generate
accurate
predictions
of
planetary positions. These
tabulated predictions we
call
ephemerides
.
5
How
does one generate an
ephemeris? This is a three-
stage process, as illustrated
in Figure 4.
First and foremost, we
must obtain accurate obser-
vations. Historically, ob-
servations consisted mainly
of ground-based optical po-
sitions of planets and their
satellites. Satellite posi-
MURISON: MODELING PLANETARY MOTIONS 5 of 20
5
From the Greek
ephemeros
, meaning
daily
. That is, a table of coordinates of a celestial
body at specific times.
Figure 4 — Generating a planetary ephemeris. Nu-
merical integrations of a solar system model are com-
pared with observational data, resulting in O-C residu-
als. Based on these residuals, model parameters are
refined and the entire process repeated until conver-
gence.
Observations
Numerical
Integration
Parameter
Estimation
Done
Start
Solar System
Model
O-C

tions are the more valuable since satellites generally have no atmospheres
(and hence no limb-darkening) to contend with. Since they orbit their par-
ent planets in a manner predictable by Newton’s law of gravity, their posi-
tions can form the basis for determining the parent planet positions. This
is complicated considerably, however, by the difficulty in constructing
highly accurate theories of satellite motions, caused by the complex inter-
actions of the satellites with each other, with the planets, and with the non-
spherical parent planet whose mass distribution we don’t always know as
well as we’d like. The modern era has seen the advent of other types of
observations, including ground-based radar, lunar laser ranging, spacecraft
telemetry, and space-based astrometry. The European HIPPARCOS
mission
6
is a successful example of a space astrometry mission. We hope
other missions, such as
FAME
(
USNO)
7
, and
SIM
(
JPL)
8
or
GAIA
(
ESA)
9
,
will follow.
The second step in generating an ephemeris is to develop a comprehen-
sive solar system model that we then integrate numerically. We must in-
clude complications, such as planetary (especially Earth) rotation dynam-
ics and lunar motion, as well as more subtle effects, including general rela-
tivity, tidal interactions between Earth and Moon, and planetary topogra-
phy models (for better resolution of radar data). The state of the art has
advanced to the point that it is becoming necessary to include the masses
of individual asteroids
10
as well as a mass model for the asteroid belt.
Both of these kinds of masses are in general very poorly known, yet aster-
oidal mass uncertainties are now the largest source of error in high-
precision ephemerides of the inner planets. Currently, the
JPL
ephemeri-
des (specifically, DE405) include mass estimates for 300 asteroids. These
masses are based on
IRAS
magnitudes, albedo estimates, and mean density
estimates.
The solar system model contains many adjustable parameters, such as
masses, orbital elements, initial positions and velocities, gravity model pa-
rameters, and so on. The third step in generating an ephemeris is to simul-
taneously fit all of these model parameters to the available observations.
MURISON: MODELING PLANETARY MOTIONS 6 of 20
10
James Hilton of USNO is the world’s foremost expert in determination of asteroid
masses. See http://aa.usno.navy.mil/hilton/asteroid_masses.htm
9
http://astro.estec.esa.nl/SA-general/Projects/GAIA/gaia.html
8
http://sim.jpl.nasa.gov/
7
http://aa.usno.navy.mil/FAME/
6
http://astro.estec.esa.nl/Hipparcos/hipparcos.html

This requires performing a
nonlinear least squares
analysis of a comparison
between a numerical inte-
gration of the solar system
model and the observa-
tional data. This analysis
results in (hopefully minor)
adjustments to the model
parameters. We then inte-
grate the model again, us-
ing the adjusted parameter
values, then compare again
to the observations. We it-
erate this process until the
parameters stop changing
appreciably. At that point,
we have the best fit of the
solar system model to the available observations.
1.5. Observation Types
For observing planetary positions, the various observational data types
fall naturally into the two broad categories: timing (in a sense, the radial
coordinate from the observer) and positions on the sky (i.e., transverse to
the radial direction). The hierarchy of types is illustrated in Figure 5.
1.6. Example: Space-Based Asteroid and Natural Satellite Observations
Figure 6 shows space-based astrometric observations by HIPPARCOS of
the 48 asteroids and 3 natural satellites it was able to reach. For the bright-
est asteroids, the single-measurement accuracy is less than 10 milliarcsec-
onds. The accuracy of the satellites is degraded by the fact that at the reso-
lution of the HIPPARCOS telescope these objects are not point sources
but extended bodies, introducing centroiding difficulties. This figure also
shows the projected single-measurement accuracy of the
FAME
satellite
11
.
(The
USNO
is hoping to launch FAME in 2003 or 2004 as a
NASA
MIDEX
MURISON: MODELING PLANETARY MOTIONS 7 of 20
11
See the FAME homepage at http://aa.usno.navy.mil/FAME/
Figure 5 — Observation types.
Observation
Types
Transverse
Radial
Transit Circle
Differential Satellite-Satellite
Satellite-Planet
Doppler
Time Delay
One-Way
Two-Way
Doppler Radar
Pulsar
Two-Way S/C
S/C Ranging
One-Way S/C
Differential Radar
LLR
Radar RangingRadar
Spacecraft
Global
Occultation
Spacecraft-Planet
Star-Planet
Satellite-Planet

mission.)
FAME
will be
able to do an order of mag-
nitude better than HIP-
PARCOS in positional
measurements of solar sys-
tem objects.
FAME
will
also go substantially
fainter, allowing observa-
tions of many more aster-
oids and natural satellites
than HIPPARCOS. Natu-
ral satellite observations
— especially in the outer
solar system — are impor-
tant because, combined
with integrations of their
motion, they can be used
to obtain the positions of
the parent planets much more accurately than observations of the planets
themselves.
FAME
will potentially be able to reach over 20 natural satel-
lites and upwards of 2100 asteroids.
2. NEWCOMB: A SOLAR SYSTEM EPHEMERIS PROGRAM
Newcomb
12
is a new Solar System Ephemeris program currently under
development at the U.S. Naval Observatory. In terms of use at the
USNO
,
Newcomb will be the successor of
PEP
, the Planetary Ephemeris Program
maintained at the Smithsonian Astrophysical Observatory
13
, and of the DE
MURISON: MODELING PLANETARY MOTIONS 8 of 20
13
See http://cfa-www.harvard.edu/~reasen/ssd.html for information about PEP.
12
See the official Newcomb program web site at http://aa.usno.navy.mil/Newcomb/. As
reviewed elsewhere in these Proceedings, Simon Newcomb (1835-1909) was a remark-
able force in 19th century American mathematics and astronomy. He was Superinten-
dent of the Nautical Almanac Office from 1877 to 1897, and he devoted much of his
prolific career to (in his words)
...a systematic determination of the constants of astronomy from the best existing
data, a reinvestigation of the theories of the celestial motions, and the prepara-
tion of tables, formulae, and precepts for the construction of ephemerides, and
for other applications of the same results.
See also the biography page located at
http://www-history.mcs.st-and.ac.uk/~history/Mathematicians/Newcomb.html
Figure 6 — HIPPARCOS single-observation accuracy
of solar system objects, with a comparison to projected
FAME single-observation accuracy.
56789101112131415
Mean Magnitude of Object
1.0
10.0
0.6
0.7
0.8
2.0
3.0
4.0
5.0
6.0
7.0
8.0
20.0
30.0
Mean Uncertainty in Position (milliarcseconds)
Hipparcos Single-Observation Mean Position Uncertainties
asteroids
FAME
Europa
Titan
Iapetus
Solar System Objects

series of programs from the Jet Propulsion Laboratory. DE and
PEP
are
the only currently existing high-precision solar system ephemeris pro-
grams.
2.1. Motivation: Why a New Program?
The developmental origins of both
PEP
and the DE programs dates from
the early to mid 1960s. Both computer program design and language ca-
pabilities, as well as the precision of both observational data and the prac-
tical needs for that data, have advanced far beyond the anticipations of
three and a half decades ago when
PEP
and the
JPL
DE programs were
originally developed. Program technology that is several generations out
of date, combined with the practical inability to add further significant ca-
pabilities or modifications to
PEP
, has been deemed sufficient cause for
development of a new ephemeris program. Additional motivations are
that it is to the
USNO
’s great advantage to have a comprehensive ephem-
eris capability in-house (especially since the
NAO
publishes the
Astro-
nomical Almanac
), and that Newcomb will provide a check against
PEP
and the JPL
DE
programs.
Finally, the creation of a modern ephemeris program provides an oppor-
tunity to simultaneously develop a flexible research tool for investigation
of solar system dynamics. The integration module of Newcomb is in large
part already completed and exists as a standalone program called
Newton.
14
This program is currently being used in the Astronomical Ap-
plications Department
15
for investigations of the dynamics of inner solar
system asteroids, the asteroidal “noise” in the motions of Earth and Mars,
and the dynamics of trans-Neptunian objects.
To highlight the difference between “old” and “new” programming, con-
sider the task of setting program input parameters. Appendix A contains a
typical input file used by
PEP
. Figure 7 shows how it is done using a GUI.
The intuitiveness of the
GUI
approach leads to substantial time savings in
coming up to speed in program usage, as well as actual use of the program
day to day. Even more valuable is that it allows a much more sophisti-
cated interface and a much more sophisticated set of program capabilities.
MURISON: MODELING PLANETARY MOTIONS 9 of 20
15
http://aa.usno.navy.mil/AA/
14
http://aa.usno.navy.mil/Newton/

2.2. Advantages
of Modern Pro-
gramming and
Design
Chief among
the advantages
of writing a
new program is
the opportunity
to make use of
both modern
programming
and modern de-
sign technolo-
gies, namely
object-oriented
programming (
OOP
) and object-oriented design (
OOD
), as well as graphi-
cal user interfaces (GUIs). Recently, the highly productive “components”
programming associated with rapid application development (
RAD
) envi-
ronments has greatly enhanced the efficiency, sophistication, and depend-
ability of
GUI
programming. Additionally, modern integrated develop-
ment environments (IDEs) have matured into a powerful and reliable
means of rapidly developing, testing, and debugging complex and sophisti-
cated programs. None of these powerful technologies was available until
the 1990s. Hence, design and construction of modern programs is faster,
safer, and more intuitive. Also very important is the fact that all of the nu-
merical algorithms used in a high-precision ephemeris program — e.g.,
numerical integrators, nonlinear estimation, etc. — are now mature tech-
nologies, which was certainly not the case thirty-five years ago.
Consequently, the Newcomb computational back end is written entirely
in
ANSI
C++, and development and testing are done entirely within the
best C++
RAD
environment currently available.
16
Throughout the
program, we take full advantage of standard
OOP
/
OOD
concepts and tech-
niques, including full data encapsulation, template and nested template
classes, polymorphism, and, where necessary, multiple inheritance.
MURISON: MODELING PLANETARY MOTIONS 10 of 20
16
http://www.borland.com/bcppbuilder/
Figure 7 — Examples of a graphical user interface (from Newton).

The benefits of a completely object-oriented approach are many, includ-
ing faster prototyping and development, fewer and more easily locatable
coding errors, vastly simpler and more intuitive design, more sophisticated
functionality, easily extensible architecture, and (most importantly) drasti-
cally reduced long-term maintenance costs. Another major benefit is that
the program can be brought up and running with minimal functionality, al-
lowing further capability to be easily and relatively painlessly incorporated
as need arises.
Ease of extensibility is largely a result of object-oriented design, but it is
also directly related to how good that design is. Hence, considerable effort
has gone and is still going into the design of Newcomb. Experience in the
software industry over the last one to two decades abundantly shows that
the payoff later on in terms of maintenance and extensibility is far out of
proportion to the effort expended early on — in the design stages — of the
program life cycle.
The benefits of a
RAD
environment for development and testing are also
very attractive. Chief among the attractions is the ease by which it is pos-
sible to create highly sophisticated graphical user interfaces. During de-
sign, graphical interface components — such as buttons, edit fields, tool-
bars and so on — are “dropped” onto a window form or dialog box. Use-
ful properties of the components are settable at design time, in addition to
being available during runtime. It is easy to create custom components as
well. For example, for Newcomb we designed a custom component that is
in fact a fully functional and self-contained power spectral density (
PSD
)
analysis package, including plots and file output. All that is needed to add
a
PSD
module to a program is to drop the
PSD
component onto a form or
dialog. Hence, building, changing, and extending the graphical user inter-
face of a program is astoundingly easy once a good overall design has been
created. This of course spills over and makes changing or extending major
program structural elements correspondingly painless.
2.3. Newcomb Project Outline
In these beginning stages of the Newcomb project, tasks naturally fall
into three main categories: program design, documentation, and science
applications. A rough outline of the most obvious subjects that must be
addressed is:
MURISON: MODELING PLANETARY MOTIONS 11 of 20

I. Design Issues
A. numerical integration scheme
1. object-oriented design
2. Integrable objects have
knowledge of dynamical en-
vironment as well as the
ability to dynamically
evolve in that environment.
B. exception handling
1. all exceptions fully recover-
able
2. procedure stack traceback
C. robust parameter estimation
1. Singular Value Decomposi-
tion (
SVD
)
2. use a mature package from
elsewhere
D. graphical user interface
E. reduction of observations
F. individual class design and test-
ing
II. Science Issues and Projects to Con-
sider
A. asteroids
1. masses from orbital interac-
tions
2. provide ephemerides (serv-
ices to the community)
3. cumulative effects on plane-
tary motions
a. Asteroids are the larg-
est source of “noise” in
the orbits of Mars and
the Earth-Moon
system.
B. lunar motion
1. chaotic dynamics
a. predictions from nu-
merical models
b. comparisons with
LLR
data
2. radiation pressure
3. resonant interaction be-
tween tidal and
GR
terms
4. lunar librations
C. Nordtvedt
h
parameter (anoma-
lous gravitational field energy
effects — i.e., a difference be-
tween gravitational and inertial
mass proportional to the gravita-
tional binding energy of a body)
D.
GR
precession
1. lunar orbit
2. Earth’s spin
E. bounds on time variation of the
gravitational constant
F. millisecond pulsars
1. derive Earth orbit
G. bounds on dark matter in the so-
lar system?
H. planetary satellites?
1. centroiding vs. satellite-
derived center of mass
I. other science?
III. Documentation
A. code
1. source documentation
model
2. interface (user manual)
B. algorithms
C. physics
1.
GR
and partial derivatives
2. Earth-Moon tidal interac-
tions
D. parameter estimation and error
and correlation analysis
E. numerical integration design
F. reduction of observations
MURISON: MODELING PLANETARY MOTIONS 12 of 20

2.4. An Overview of the Newcomb Program Structure
The top level process structure of Newcomb is shown in Figure 8. Basic
operation is as follows.
The observations module is responsible for reading input astrometric ob-
servations and reducing (“massaging”) them as necessary. The observa-
tions will be of various types (Figure 5), taken at various observing loca-
tions (Figure 11), including spacecraft. The reduction process corrects for
various instrumental and other effects (e.g. from the atmosphere) that are
specific to a particular set of observations.
The integration module is responsible for numerically integrating a so-
phisticated dynamical model of the solar system — including general rela-
tivistic terms, a detailed Earth-Moon system, planetary spin vectors includ-
ing precession and nutation, and an unlimited number of asteroids — to
produce an ephemeris.
The model ephemeris is then compared with the observations in the O-C
section of the parameter adjustment module to produce a set of residuals.
The parameter estimator uses the partial derivatives of the model equations
with respect to the model pa-
rameters (including initial
conditions) to solve the asso-
ciated nonlinear least squares
problem for the most prob-
able set of model parameter
values that minimizes the
O-C residuals.
The adjusted model pa-
rameters are then fed back
into both the ephemeris gen-
erator and the observation
transformation methods. The
data are rereduced as neces-
sary, and a new ephemeris is
generated by the integration
module, using the updated pa-
rameter values. These are
again combined to produce a
MURISON: MODELING PLANETARY MOTIONS 13 of 20
Figure 8 — Major program processes.
iteration
loop start
massage
observations
integrate eqs.
of motion
form O-C
residuals
parameter
estimator
O-C
eval
iteration
loop end
observations
reduced
observations
ephemeris
parameters
correlation
matrix
observations module
parameter
adjustment
module
integration module
physical
model
observation
models
calculate
observables
data
data flow
process
parameter feedback
process flow

new set of residuals. This process is iterated until the parameters satisfy
predetermined success criteria.
At the end of the iterative process, we will have produced an ephemeris
that best fits the observations, given the model used, as well as the best-fit
model parameters, formal error estimates of those parameters, and the pa-
rameter cross correlations. The parameter error estimates and parameter
correlations are derived from the partial derivatives and the correlation
matrix from the least squares analysis. Experience with
PEP
has shown
that, normally, at most only a couple or a few iterations are needed
2.5. The Integration Module
The integration module of Newcomb is relatively straightforward, as
shown in Figure 9. After choosing which bodies to integrate, one sets all
the initial conditions for all the integrated bodies, as well as both the
physical model parameters (G,
masses, etc.) and the integrator
parameters (accuracy limits,
step size limits, etc.). The inte-
grator then integrates the equa-
tions of motion, providing inter-
mediate output along the way.
The intermediate output varies
in complexity, from simple di-
agnostics to runtime graphics of
orbital elements, close ap-
proaches, mean-motion reso-
nance angles, and so on. As
previously mentioned, the inte-
gration module is such an in-
trinsically useful tool that it has
been broken out as a standalone
solar system dynamics applica-
tion, called Newton.
MURISON: MODELING PLANETARY MOTIONS 14 of 20
Figure 9 — The Integration Module.
Integration
initialize the
physical model
integrate the
equations of
motion
parameters
initial
conditions
ephemeris
- body masses
- gravity model
- planets
- asteroids
- natural satellites
- planetary spins
initialize the
integrator
parameters
- integrator type
- accuracy
- start/stop times
- output interval
- planets
- asteroids
- natural satellites

2.6. The Parameter Adjust-
ment Module
The parameter adjustment
module is relatively straight-
forward. The processed obser-
vations from the Observations
Module and the calculated
ephemeris data from the Inte-
gration Module are compared,
thus forming the O-C
residuals. First, coordinate
frame compatibility between
the observations and the syn-
thetic ephemeris is reconciled.
The calculated ephemeris
must be transformed to appar-
ent positions in order to match
the observations. The residu-
als are characterized, with sta-
tistical and descriptive output
going to disk as well as to an
output window on-screen. At
this point, outlying data points
can be automatically — or
manually — detected and
removed.
The core of the module fol-
lows with the determination of
parameters via a nonlinear
maximum likelihood estimator
(e.g., Levenberg-Marquardt). The normal equations are formed and
solved, and the parameters and associated formal error estimates are saved.
Finally, the residuals are evaluated, and the module exits with a solution
“acceptability” code. Figure 10 illustrates the process.
Matrix inversion is accomplished via singular value decomposition
(
SVD
), which is very robust and offers useful diagnostics for ill-
MURISON: MODELING PLANETARY MOTIONS 15 of 20
Figure 10 — The Parameter Adjustment Module.
Parameter
Adjustment
retrieve
massaged
observations
retrieve
calculated
ephemeris
calculate O-C
quantities
statistical
characterization
of residuals
statistics
form the
normal
equations
solve the
normal
equations
output
parameters
correlations
determine
parameters
evaluate
residuals
exit TRUEexit FALSE
removal of
outliers
calculate
apparent
positions

conditioned matrices. Singularities are automatically detected and cor-
rected, and the problem parameters are identified. In essence, if the algo-
rithm encounters an ill-conditioned matrix, it safely steps around the prob-
lem point(s) and proceeds in such a way as to mine the matrix for the
maximum amount of information. When a singularity (rare in practice) or
degenerate column (not rare!) is encountered, the combination of parame-
ters that led to the fault is easily extracted. Thus, not only are singularities
safely handled, but — more importantly — parameter combinations to
which the data are insensitive are automatically identified.
It is unusual to encounter a computational method that is this reliable
and blowup-proof. I have already developed and tested matrix inversion
using
SVD
and incorporated it into the Matrix utility class. With regard to
Newcomb,
SVD
is a “plug’n’play” capability.
2.7. The Observations Module
Perhaps the most difficult section of the program is the module that
processes input observations and reduces them to a form suitable for pas-
sage to the O-C section of the parameter adjustment module (see Figure
10). In essence, the observations are sent to the O-C section in the form of
apparent positions, corrected for various biases, including (but not limited
to):

catalog corrections

delay/doppler bias corrections

coordinate frame fiducialization

aberration corrections

nutation and precession
Integral to this section are the specific types of observational datasets
and the specific types of observational platforms. The data and platform
types vary widely.
2
.7.1. Observing Platforms
One must consider the various observing platforms presently available in
the solar system. They are
MURISON: MODELING PLANETARY MOTIONS 16 of 20

I. Planet
A. Earth
1. Earth-based observa-
tories
2. Earth orbiters
B. Planetary landers
C. Planetary orbiters
II. Deep space probes (i.e., gravi-
tationally unbound from all
planets and satellites)
Figure 11 shows the object
hierarchy of observing plat-
forms.
17
The C++ code
classes reflect this hierarchy.
Each input data stream will
contain relevant observing
platform information. An
appropriate observing plat-
form object will encapsulate this information. Each type of platform ob-
ject also encapsulates the necessary functionality (referred to as
methods
)
to provide information needed to manipulate or transform data of the cor-
responding type (see Figures 5 and 11).
For example, planetary observing platform objects know how to precess
and nutate coordinates to a specified epoch. Each base class contains pa-
rameters and functionality common to all subclasses derived from it. The
derived classes contain only the additional or specialized parameters and
functionality required to handle platforms of a specific kind. For example,
since all planetary platforms have a basic precession and nutation capabil-
ity, these methods reside in the base class
PlanetPlatform
. An
EarthPlatform
object automatically inherits all the functionality and data of
PlanetPlatform
.
The
EarthPlatform
object therefore contains only additional abilities, data, or
refinements, for example precession parameters specific to the Earth.
Proper use of inheritance eliminates code duplication for common tasks in
a natural and intuitive way. The inheritance mechanism is built into the
C++ language and therefore requires no enforcement by or special disci-
pline from the programmer.
MURISON: MODELING PLANETARY MOTIONS 17 of 20
17
Arrows in Figures 11 and 12 point
from
derived classes
to
parent (also called
base
)
classes. This is the standard notation.
Figure 11 — Observing platform class hierarchy.
Platform
PlanetPlatform
SpacecraftPlatform
EarthPlatform OrbiterPlatform ProbePlatform
EarthOrbiterPlatform
epoch
central object
local position
local coord origin
precess
nutate
class method
encapsulated data
multiple inheritance

I. Transverse (position)
A. Optical observations
1. Global positions
a. Transit circle
2. Differential positions
II. Radial (timing)
A. Doppler observations
1. Oneway
a. Pulsars
b. Spacecraft
2. Twoway
a. Radar
b. Spacecraft
B. Time delay observations
1. LLR
2. Radar
a. Differential radar
3. Spacecraft
a. Single
Figure 11 intentionally shows only the major class types, in accord with
the introductory nature appropriate to this Chapter. It is a simple matter to
derive further specialized classes from the base classes shown. For exam-
ple, one would derive a
VikingOrbiter
from
OrbiterPlatform
.
2.7.2. Observation Types
As previously mentioned, the various observation data types fall natu-
rally into the two broad categories: timing and position. For reasons hav-
ing mainly to do with datasets that are currently insufficiently large or in-
sufficiently accurate to have a substantial effect on ephemeris accuracy,
early versions of Newcomb will not include some of the observation
types.
Newcomb will include the following subset types
:
MURISON: MODELING PLANETARY MOTIONS 18 of 20
Figure 12 — Observational class hierarchy.
ObservationGroup
(virtual)
TransverseObs RadialObs
multiple inheritance
OffsetObs TransitCircleObs
SatSatObs SatPlanetObs
DopplerObs
SpacecraftObsRadarObs
TimeDelayObs
OneWayDoppler TwoWayDoppler
PulsarObs
RangeRadarObs LLRObs
DiffRadarObsDopplerRadarObs
OneWayDopplerSCObs TwoWayDopplerSCObs RangeSCObs

Because extensibility is built into the design of Newcomb, adding further
capabilities as they become necessary will involve minimal effort — there
is no need, from a maintenance standpoint, to include capabilities that are
anticipated to go unused for a long time. That is, with a good object-
oriented design we do not have to worry so much about “making room”
for anticipated future capabilities. Figure 5 shows the observation types
hierarchy. Figure 12 shows the proposed corresponding object class hier-
archy used in Newcomb.
Each type of input data stream will contain embedded type information,
and instantiations of the appropriate data objects will handle the data. The
specific objects shown in Figure 12 encapsulate not only the correspond-
ing observational data but also the functionality required to reduce that
data type. For example, notice that all datatype objects have, via inheri-
tance from the base class
Observation
, platform information and the ability
to handle (say) aberration.
As with Figure 11, Figure 12 is intentionally not complete, especially re-
garding encapsulated data and method details. However, all the important
base classes, and their inheritance dependencies, are shown.
3. SUMMARY
We have given a brief description of the field of high-precision modeling
of solar system planetary and natural satellite motions. Motivations for
high-precision ephemerides stem from — perhaps surprisingly to many —
military as well as astronomical requirements. The latter category includes
such areas as spacecraft navigation, celestial mechanics, occultation pre-
dictions, tests of General Relativity, etc. Several kinds of observations go
into determining high-precision ephemerides — essentially, we use any-
thing we can get our hands on. We have also discussed in broad terms the
method of generating high-precision ephemerides, making use of both ob-
servational data sets and comprehensive models to solve for the “best”
model parameter values.
Given that the extant first generation of high-precision ephemeris pro-
grams is antiquated, the U.S. Naval Observatory has begun development
of a new, highly flexible ephemeris program called Newton. This modern
program takes full advantage of design and programming techniques de-
veloped in the 1980s and early 1990s and now available as a mature set of
technologies. An overview of the program design has been presented,
MURISON: MODELING PLANETARY MOTIONS 19 of 20

Ephemerides improvement for the first 4 asteroids.
&NMLST1
EXTPRC= 0, $ Use hardware extended precision
ICT(1)= 10,
ICT(3)= 1,
ICT(4)= 1, $ Compute partial derivatives
ICT(5)= -1, $ Do not used saved normal equations
ICT(9)= 0,
ICT(10)= -2, ICT(11)= -2,
ICT(12)= 2, $ prediction or harmonic analysis
ICT(34)= 3,
ICT(39)= 1,
ICT(50)= 1, $ USE BROWN MEAN MOON IF NO IPERT
ICT(80)= 0,
JCT(13)= 1, $ Use J2000.0 coordinates.
JCT(33)= -1, $ Use USNO UT1 and wobble
JCT(27)= 1, $ Use * commands
JCT(28)= 7,
MASS(1)= 6023600.D0, $ Use DE200 masses for the planets
MASS(2)= 408523.5D0,
MASS(3)= 328900.550000000047D0,
MASS(4)= 3098710.D0,
MASS(5)= 1047.35001090551827D0,
MASS(6)= 3497.99999984177066D0,
MASS(7)= 22960.0000007059389D0,
MASS(8)= 19314.0002382557432D0,
MASS(9)= 130000000.238686755D0,
MASS(10)= 0.012150581D0,
MASS(11)= 2.239D9,
MASS(12)= 9.247D9,
MASS(13)= 8.7D10,
MASS(14)= 7.253D9,
MASS(17)= 1.849D11,
AULTSC= 499.0047837D0, $ AU in light seconds
ECINC= 23.439281083D0, $ Use DE118 Obliquity
PRMTER(47)= 0.0D0, $ RA OF ASC. NODE OF BELT
PRMTER(48)= 23.4433D0, $ INCLINATION OF BELT
PRMTER(49)= 2.9D0, $ DISTANCE OF BELT FROM SUN
PRMTER(50)= 8.773725302941010D-10, $ MASS OF BELT
PRMTER(81)= 0.0D0,
MDSTSC= 0., $ MOON TAPE DISTANCE UNIT IN AU
NBODY= 0, IPERT= 10,
NUMOBT= 1,
IOBS = 30,
IOBS1= 14, IOBS2= 15,
$ EPS(3)= 100,
$ EPS(4)= 100,
LPRM(1)= 11, LPRM(2)= 12, LPRM(3)= 14,
*OBJECT EARTH-ROTATION
CON(22)= 5029.0966,
CON(23)= 84381.4119,
*OBJECT 11
NAME= ' CERES ',
INCND= 0, ITAPE= 31, NCENTR= 0, JTYPE=6,
A=2.767121817D0, E=0.07749262D0, INC=27.116375D0,
ASC= 23.471566D0, PER= 133.40890D0, ANOM=2.08129D0,
JD1=2378801, JD2=2450001, JD0=2444801,
K(31)=1, K(32)=1, K(33)=1, K(34)=1, K(35)=1, K(36)=1, K(37)=1,
K(38)= 1, K(39)= 1, K(40)= 1, K(41)= 1, K(42)= 1,
K(43)= 1, K(44)= 1,
K(61)= 1 $ Include GR
K(87)= 2, INT= 2, $ INTERVALS
K(88)= 2, K(89)= 6, $ ADAMS-MOULTON, 7 TERMS
K(91)= -3, K(92)= -6, EPS(3)=1E-9 $ STARTING INTERVALS
K(98)= -500, K(99)= 0, K(100)= -1, $ PRINT + TAPE; ORDI-
NARY EQNS OF MOTION
KI= 1, 1, 1, 1, 1, 1, 1, 12, 13, 14,
L= 1, 1, 1, 1, 1, 1,
*OBJECT 12
NAME= ' PALLAS ',
INCND= 0, ITAPE= 32, NCENTR= 0, JTYPE=6,
A= 2.771672932D0, E= 0.23398027D0, INC= 11.809637D0,
ASC= 161.02570D0, PER= 322.78775D0, ANOM= 298.543057D0,
JD1= 2379251, JD2= 2450001, JD0= 2449601,
K(31)= 1, K(32)= 1, K(33)= 1, K(34)= 1, K(35)= 1,
K(36)= 1, K(37)= 1,
K(38)= 1, K(39)= 1, K(40)= 1, K(41)= 1, K(42)= 1,
K(43)= 1, K(44)= 1,
K(61)= 1, $ Include GR
K(87)= 2, INT= 2, $ INTERVALS
K(88)= 2, K(89)= 6, $ ADAMS-MOULTON, 7 TERMS
K(91)= -3, K(92)= -6, EPS(3)= 1E-9 $ STARTING INTERVALS
K(98)= -500, K(99)= 0, K(100)= -1, $ PRINT + TAPE; ORDI-
NARY EQNS OF MOTION
KI= 1, 1, 1, 1, 1, 1, 1, 11,
L= 1, 1, 1, 1, 1, 1,
*OBJECT 13
NAME= ' JUNO ',
INCND= 0, ITAPE= 33, NCENTR= 0, JTYPE= 6,
A= 2.670660949D0, E= .25626106D0, ANOM= 156.782239D0,
INC= 10.814499D0, PER= 46.75209D0, ASC= 11.27760D0,
JD1= 2380151, JD2= 2450001, JD0= 2444801,
K(31)= 1, K(32)= 1, K(33)= 1, K(34)= 1, K(35)= 1, K(36)= 1,
K(37)= 1,
K(38)= 1, K(39)= 1, K(40)= 1, K(41)= 1, K(42)= 1, K(43)= 1,
K(44)= 1,
K(61)= -1,
K(87)= 4, INT= 4, $ Intervals
K(88)= 2, K(89)= 6, $ Adams-Moulton integra-
tor, 7 terms
K(91)= -3, K(92)= -6, EPS(3)= 1E-9, $ Starting intervals
K(98)= -500, K(99)= 0, K(100)= -1, $ Print & tape, ordi-
nary eqs. of motion
KI= 1, 1, 1, 1, 1, 1, 1,
L= 1, 1, 1, 1, 1, 1,
*OBJECT 14
NAME= ' VESTA ',
INCND= 0, ITAPE= 34, NCENTR= 0, JTYPE= 6,
A= 2.362114063D0, E= 0.08961581D0, INC= 22.717580D0,
ASC= 18.187358D0, PER= 237.48420D0, ANOM= 308.51911D0,
JD1= 2381051, JD2= 2450001, JD0= 2444801,
K(31)= 1, K(32)= 1, K(33)= 1, K(34)= 1, K(35)= 1,
K(36)= 1, K(37)= 1,
K(38)= 1, K(39)= 1, K(40)= 1, K(41)= 1, K(42)= 1,
K(43)= 1, K(44)= 1,
K(61)= 1, $ Include General Relativity
K(87)= 2, INT= 2, $ INTERVALS
K(88)= 2, K(89)= 6, $ ADAMS-MOULTON METHOD, 7
TERMS
K(91)= -3, K(92)= -6, EPS(3)= 1E-9 $ STARTING INTERVALS
K(98)= -500, K(99)= 0, K(100)= -1, $ PRINT + TAPE; ORDI-
NARY EQNS OF MOTION
KI= 1, 1, 1, 1, 1, 1, 1, 11,
L= 1, 1, 1, 1, 1, 1,
*OBJECT 27
NAME= ' Arete ',
INCND= 0, ITAPE= 37, NCENTR= 0, JTYPE= 6,
A= 2.73942088D0, E= 0.1630220D0, INC= 26.08786D0,
ASC= 20.12111D0, PER= 310.03548D0, ANOM= 168.77030D0,
JD1= 2407351, JD2= 2450002, JD0= 2450001,
K(31)=1, K(32)=1, K(33)=1, K(34)=1, K(35)=1, K(36)=1,
K(37)=1,
K(38)= 1, K(39)= 1, K(40)= 1, K(41)= 1, K(42)= 1,
K(43)= 1, K(44)= 1,
K(61)= 1, $ Include GR
K(87)= 2, INT= 2, $ INTERVALS
K(88)= 2, K(89)= 6, $ ADAMS-MOULTON, 7 TERMS
K(91)= -3, K(92)= -6, EPS(3)= 1E-9 $ STARTING INTERVALS
K(98)= -500, K(99)= 0, K(100)= -1, $ PRINT + TAPE; ORDI-
NARY EQNS OF MOTION
KI= 1, 1, 1, 1, 1, 1, 1, 14,
L= 1, 1, 1, 1, 1, 1,
*SITES
APPENDIX: TYPICAL
PEP
INPUT
As an example of the old-style program interface, following is an excerpt
taken from a typical
PEP
input file. Compare to Figure 7. This particular
file (courtesy James Hilton) was used in the generation of ephemerides for
the four largest asteroids for use in the Astronomical Almanac for the year
2000.
which serves also to provide further insight into how a modern, high-
precision solar system ephemeris can be generated, as well an indication as
to some of the complexity of such an undertaking. It is relatively simple
and straightforward to write a program that makes low-precision predic-
tions. However, generation of a high-precision ephemeris is another mat-
ter altogether.
MURISON: MODELING PLANETARY MOTIONS 20 of 20