Author(s)
Bowden, Brian E.
Title
First principles used in orbital prediction and an atmospheric model comparison
Publisher
Monterey, California. Naval Postgraduate School
Issue Date
1994-06
URL
http://hdl.handle.net/10945/28219
This document was downloaded on May 04, 2015 at 22:20:41
Approved
for public release; distribution is unlimited.
First Principles
Used
in Orbital Prediction
and an Atmospheric Model Comparison
by
Brian £. Bowden
Lieutenant United States Navy
B.S., The Military College of South Carolina, 1986
,
Submitted
in partial fulfillment
of the requirements for the degree of
MASTER OF SCIENCE IN ASTRONAUTICAL ENGINEERING
from the
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1994
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5.
USED IN ORBITAL PREDICTION AND AN
ATMOSPHERIC MODEL COMPARISON
FIRST PRINCIPLES
6.
AUTHORS
Bowden, Brian
7.
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Naval Postgraduate School
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WORDS)
This thesis develops an orbital prediction model based on fundamental principles of orbital dynamics and drag.
FORTRAN based orbital prediction scheme was designed to provide accurate ephemerides for a particular DoD sateU
program. The
satellite
program under study has
satellites at
650 and 800 kilometers with high inclinations. In order
models had to be conducted in order to determine wh
obtain the highest accuracy possible, a comparison of atmospheric
model was more accurate. Mathematical formulation for three widely used earth atmospheric models are presented;
60, JACCHIA 71, and MSIS 86 atmospheric models. The MSIS 86 atmospheric model was not evaluated due
computer problems. Comparison of the two JACCHIA models proved that the JACCHIA 71 model provided much m<
accurate ephemerides. It is believed that this is due not only to the incorporation of variations in density caused by so
flux, but also geomagnetic activity and a better modeling of the polar regions. Further work on this project would incli
incorporation of the MSIS 86 model for evaluation, incorporation of the full WGS-84 geopotential model, and using m<
accurate observed vectors in order to obtain a better comparison. Incorporating a subroutine which will vary the B-factor
a function of latitude will greatly increase accuracy. This is a major deviation from current operational practice, in that
JACCHIA
i
B-factor
14.
is
often used as an error catch-all and does not truly represent
its
dynamical purpose.
SUBJECT TERMS
15.
NUMBER OF PAGES
83
Atmospheric Models, Orbital Prediction, atmosphere, thermospheric
processes, B-factor, solar flux, geomagnetic activity, density
16.
OF REPORT
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of this page
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1
ABSTRACT
This thesis develops an orbital prediction model based on fundamental principles of
orbital
dynamics and drag.
A FORTRAN based
DoD
provide accurate ephemerides for a particular
program under study has
satellites at
orbital prediction
satellite
scheme was designed
program. The
650 and 800 kilometers with high
to
satellite
inclinations.
In
order to obtain the highest accuracy possible, a comparison of atmospheric models had to
be conducted
order to determine which model
in
was more
accurate.
Mathematical
formulation for three widely used earth atmospheric models are presented; the
60,
JACCHIA
71,
and
was not evaluated due
proved that the
to
work on
71 model provided
geomagnetic
this project
which
will
in
86 atmospheric model
much more
activity
JACCHIA models
accurate ephemerides.
in density
full
It is
caused by
and a better modeling of the polar regions.
would include incorporation of the MSIS 86 model
evaluation, incorporation of the
observed vectors
MSIS
due not only to the incorporation of variations
is
solar flux, but also
atmospheric models. The
computer problems. Comparison of the two
JACCHIA
believed that this
Further
MSIS 86
JACCHIA
WGS-84
geopotential model, and using
for
more accurate
order to obtain a better comparison. Incorporating a subroutine
vary the B-factor as a function of latitude will greatly increase accuracy. This
a major deviation
from current operational practice,
error catch-all and does not truly represent
its
in
in that the
B-factor
dynamical purpose.
is
is
often used as an
/MOO
0,1
TABLE OF CONTENTS
I.
II.
INTRODUCTION
\
BACKGROUND
A.
B.
C.
ATMOSPHERE
3
1.
Troposphere
4
2.
Stratosphere
4
3.
Mesosphere
4
4.
Thermosphere
5
5.
Homosphere
5
6.
Heterosphere
5
7.
Time Dependent Variations
7
a.
Diurnal Variation
b.
27-day Variations
7
c.
Semi-Annual Variations
8
d.
Long Term
8
Variations
THERMOSPHERIC PROCESSES
7
9
1.
Solar
EUV Radiation Effects
9
2.
Solar
Wind
\\
3.
Geomagnetic Storms
12
4.
Gravity Waves, Planetary Waves, and Tides
13
PROGRAM DEVELOPMENT
14
Earth's Geopotential
14
2.
Drag
16
3.
Third
1.
D.
3
Body
Attractions
ATMOSPHERIC MODEL DEVELOPMENT
1.
Lockheed Densel
a.
2.
3.
20
21
Mathematical Basis
Jacchia-Roberts 71
a.
19
23
26
Mathematical Basis
MSIS 86
36
39
IV
0UDLE1
IU.
IV.
ATMOSPHERIC MODEL EVALUATION
47
A.
JACCHIA
60
MODEL EVALUATION
50
B.
JACCHIA
71
MODEL EVALUATION
51
SUMMARY
52
APPENDIX A FIGURES
54
APPENDIX B TABLES
71
LIST OF
REFERENCES
INITIAL DISTRIBUTION LIST
73
75
ACKNOWLEDGMENT
The author would
project
like to
thank Brian Axis without whose help and ideas,
would have never gotten offthe ground. Special thanks goes
Stanford University
who
required of the project.
provided the
A special
to
this
Cary Oler from
much needed F10.7 and Ap environmental
heartfelt thanks to
continuous enthusiasm and encouragement kept
VI
my
me on
family and friends
the path.
whose
data
INTRODUCTION
I.
Orbital prediction has
programs. Exact
satellite
become an
satellite
essential science
more accurate
more accurate
the satellite ephemerides can be
the mission analysis becomes.
objective being to pinpoint the satellites current or future position.
Sputnik
in
the late
fifties,
DoD
ephemerides provide for a more accurate means of
mission analysis. Put more directly, the
calculated or predicted, the
needed for several of the
The main
Since the launch of
a great deal of effort has been placed in trying to model the
space environment. In particular, the upper atmosphere, or neutral atmosphere, has been
of great
interest in the study
attempts have been
made
in
of artificial
satellite orbit theory.
modeling the thermosphere
in
Since 1957, several
order to aid
in satellite
mission
Several atmospheric models, or satellite drag models, are currently used for
analysis.
practical applications such as lifetime estimates, reentry prediction, orbit determination
tracking, attitude dynamics,
Atmospheric drag affects
beyond geosynchronous
is
and most recently, mass analysis of a particular
all satellites, in all
altitudes.
altitude regimes,
For many
satellites,
empirical models.
MSIS
different
orbits to well
on the
satellite.
This thesis will compare three of the more widely used
The models used
for the comparison are the Jacchia 60, the Jacchia 71,
86 earth atmospheric models. Each of these models
manner and are unique for the
being to find out which model
satellite
from low earth
drag models can be divided into two categories, the empirical models and
the general circulation models.
and the
satellite.
the modeling of atmospheric drag
the largest error source in describing the forces acting
Satellite
and
is
altitude
more accurate
is
formulated
in
a
bands that were tested. The ultimate goal
in
terms of orbit prediction for a particular
program. Since the modeling of atmospheric drag
is
the largest error in orbital
prediction, finding the
more accurate model
for the satellite
program
in
question will
greatly improve the predicted satellite ephemirides and hence mission analysis.
The atmospheric models being used
input that each
model
requires.
in this
comparison
all
differ
with respect to the
Currently the Air Force Satellite Tracking Center (STC)
uses the Jacchia 60 atmospheric model which uses not only date and time as inputs but
also the
measurement of the solar
flux,
F10.7.
The J71 model, considered
to be an
improvement of the Jacchia 60 model, adds the measurement of geomagnetic
Ap, to
its
inputs.
The MSIS 86 atmospheric model
is
formulated
than the Jacchia atmospheric models, and uses both F10.7, and
The atmospheric models
in
Ap
activity,
or
a different manner
as
its
inputs.
calculate the density and constituency of the
atmosphere
based on the current and predicted or average environmental conditions. The accuracy of
these models has been calculated to be 80 to 85 percent accurate. This percentage drops
off substantially as the altitude increases. Accuracy also seems to decrease during periods
of high solar and geomagnetic
1 1
year solar cycle. This
is
activity.
At the moment, the sun
on the downside of the
an advantage for the comparison of the atmospheric models,
because there were fewer and weaker solar
upper atmosphere.
is
flares
and geomagnetic storms to perturb the
II.
A.
BACKGROUND
ATMOSPHERE
The
earth's
atmosphere
is
classically divided into four different regions
based on
temperature and pressure gradients. These four regions are the troposphere, stratosphere,
mesosphere and the thermosphere. The corresponding boundary
or upper limits of
layers,
each of the regions are the tropopause, stratopause, mesopause, and thermopause
Figure
respectively.
1
(Akasofu, 1972,
p.
109), illustrates the
atmospheric regions. Beyond the thermopause
The exosphere
is
a region of extremely
is
breakdown of the
earth's
the region delineated as the exosphere.
low density and temperature, and
is
the transition
region into space.
Another way
in
two regions known
between the regions
heterosphere
is
which the
as the
is
earth's
atmosphere
divided,
labeled the turbopause.
labeled the exosphere.
Once again
at this time, that the
by the classification of
two
altitude
band
boundary
p.
1 1
1) illustrates
the
defining systems. This figure provides
altitude regimes.
atmospheric region of interest during
band between 600 to 800 km. This
transition
the region outside the
Figure 2 (Akasofu, 1972,
breakdown of altitude versus temperature for the various
noted
is
homosphere and the heterosphere. The
various atmospheric regions as derived by the
a
is
is
this
It
should be
study was an altitude
encompassed within
thermosphere or exosphere, depending on which classification scheme
is
either the
being used. For
the sake of continuity, the altitude band of interest will be considered to be within the
thermosphere.
Troposphere
1.
The troposphere
to roughly 10 to 15
surface and
is
km
is
that portion
above the
of the atmosphere that extends from the surface
surface.
equilibrium with the sun-warmed
It is in
characterized by intense convection and cloud formations. In this region,
both temperature and density decrease with increasing altitude, with an occasional
inversion layer. (U.S. Air Force, 1960,
p.
1-3)
Stratosphere
2.
The stratosphere
region of the atmosphere
is
is
located above the tropopause and extends up to 50 km. This
extremely important
in that
it
responsible for the absorption of the extreme ultraviolet
sun.
Due
to the absorption
of this
EUV radiation,
temperature gradient. The density, however,
consequence of the absorption of the
radiation cannot be
be addressed
3.
still
contains the ozone that
(EUV)
is
radiation produced by the
the stratosphere has a positive
decreases with altitude.
One
other
EUV radiation by the ozone layer is that the EUV
measured from the
earth's surface.
This presents a problem which will
later in this paper.
Mesosphere
The Mesosphere
is
that region
of the atmosphere located above the stratopause
and extends up to 80km. Once again, both the temperature and density are decreasing
with altitude. The mesosphere
is in
radiative equilibrium
between the
ultraviolet
ozone
heating by the upper fringe of the stratosphere, and the infrared ozone and carbon dioxide
cooling by radiation to space. (U.S. Air Force, 1960,
4.
p.
1-3)
Thermosphere
The thermosphere extends from the mesopause
altitude limit.
The thermosphere
is
to higher altitudes with
no
characterized by a very rapid increase in temperature
4.
Thermosphere
The thermosphere extends from the mesopause to higher
altitude limit.
The thermosphere
is
between -600 to 1200
also be heated
known
EUV radiation.
The temperature
increase
as the exospheric temperature, the average values being
K over a solar cycle.
by geomagnetic
no
characterized by a very rapid increase in temperature
with altitude due to the absorption of the sun's
reaches a limiting value
altitudes with
activity,
(Larson, 1992, p.208)
The thermosphere may
which transfers energy from the magnetosphere
and ionosphere to the thermosphere. The heating of the thermosphere causes an increase
atmospheric density due to the expansion of the atmosphere. Figure 3 (Hess, 1965,
in the
p.
679), illustrates the variation of temperature versus altitude for the various atmospheric
regimes.
5.
Homosphere
The homosphere extends from the surface
characterized by
its
trace
down
into the following:
amounts of other gases. The uniformity of the region
turbulent mixing of the gas constituents. (Adler, 1993, p. 10)
uniformity, of the
homosphere changes
at
~100km
of the oxygen molecules. Because the density
monatomic oxygen
100km.
It is
uniform composition and relatively constant molecular weight. The
composition of this region can be broken
At and
to approximately
is
altitude
78%
is
N,
21%
02,
1%
created due to the
The composition, hence
the
due mainly to the dissociation
at this altitude is
low, recombination of the
very infrequent; even more so as altitude increases.
The
dissociation
of oxygen causes the molecular weight to decrease substantially.
6.
Heterosphere
The heterosphere
region
is
exists
from - 100km outward, with no
altitude limit.
The
characterized by diffusive equilibrium and significantly varying composition.
The
molecular weight of the atmosphere decreases rapidly, from ~ 29
at
90km
500km. Above the region of oxygen dissociation, nitrogen begins to
to
~ 16
dissociate.
at
Diffusive
equilibrium begins to take place, and the lighter molecules and atoms rise to the top of the
atmosphere. The distribution functions, or scale heights, of each of the constituents of the
heterosphere are found by equating the pressure gradient of the atmosphere with the
gravitational force, as described
by the
ideal gas law,
P = {-\t
where p = gas
pressure, k
= Boltzman
(1)
constant,
m=
molecular mass,
T=
temperature,
and p = density. For a small cross sectional area of thickness dh
dP = -pgdh
(2)
therefore
P
o
T
dp
mg {P
By assuming
+
dT
(4)
T
isothermal variation and defining the scale height to be
differential equation
is
H=
kT
,
a simple
mg
obtained described by the following;
^ = ->
(5)
with the solution being;
-1*
p = p oe »
Each molecule
will
(6)
have a different scale height depending upon
to diffusive equilibrium, in that the density
its
mass. This gives rise
of the varying constituents
will
decrease at
certain levels.
The
diffusion process takes place
amongst Ar, O2, N2, O, He, and
respectively as altitude increases. (Adler, 1992, p.
illustrates the densities
7.
1
Figure 4
1)
(
Akasofu, 1972,
of the various atmospheric constituents versus
H
p.
110)
altitude.
Time Dependent Variations
Several time dependent density variations are present in the earth's atmosphere.
The
density of the earth's atmosphere varies according to the time of day, day of the year,
and which year
a.
it
happens to be
year solar cycle.
1 1
Diurnal Variation
The atmospheric
called the diurnal variation.
at
in the
density variation
p.
dependent upon the time of day
minimum around 0200
becomes more pronounced with an increase
(Hess, 1968,
is
is
The maximum density of the upper atmosphere can be found
approximately 1400 local time, with the
variation
which
99) The diurnal variation
is
local time.
in altitude as
The density
can be seen
in
Figure
5.
caused by the alternate heating during the day
and cooling during the night of the upper atmosphere. Often called the diurnal bulge, the
density variation occurs mainly over the equator, with an elongation in the north-south
direction
due to the
tilt
of the
ecliptic.
of the sub-solar point. (Jacchia, 1963,
atmosphere during
The peak of the phenomena occurs
p.
983) The temperature change
this diurnal correction parallels that
in
at the latitude
the upper
of the density, except that the
temperature change lags behind the density change by approximately two hours. This lag
seems opposite of what would be expected, and
b.
is
not completely understood.
27-day Variations
It
has been established that there
is
27-day solar cycle. This was shown by Jacchia,
actual satellite drag
a density variation that
who
is
caused by the
determined the correlation between
measurements and the solar decimetric
flux,
or 10.7
cm
flux.
It
was
found that the density of the atmosphere not onJy had a daily variation, but also a monthly,
or 27 day cycle, that corresponded to the 27 day rotation cycle of the sun. (Hess, 1968,
P 101)
j
c.
Semi-Annual Variations
One of the
least
understood upper atmosphere density variations
annual variation. This density variation
is
characterized by a pronounced
the June-July time frame, with a secondary
maximums occur during
maximum
minimum occurring
the semi-
minimum during
in January.
The
the September-October time frame, with a lesser secondary
occurring during March-April. Several theories exist as to what causes the
semi-annual variation. The most controversial,
wind. Another theory
summer pole
that the variation
is
to the winter pole.
is
is
that the variation
is
a\
Long Term
It
was found
an effect of the solar
caused by the convective flows from the
The flows would descend
at the
winter pole, transporting
heat to the cooler mesosphere from the higher altitudes. (Hess, 1968,
1 1
is
p.
103)
Variations
that there
was
a long-term density variation associated with the
year solar cycle. Measurements of the solar flux, or F10.7, taken over several years
provide evidence of this effect.
has an
1 1
year cycle with
measurement, explained
As can be seen
maximum and minimum
in detail later, is
a
periods of high solar activity, the solar flux
density due to
Figure 6 (Larson, 1992,
measurement of the
is
p.
209), the sun
values of F10.7. The F10.7
EUV radiation.
During
high, thus causing an increase in atmospheric
EUV heating and the resultant expansion.
figure, the density
maximum.
in
As can be assumed from
the
of the upper atmosphere has a corresponding eleven year minimum and
THERMOSPHERIC PROCESSES
B.
The upper atmosphere or thermosphere undergoes a change
in
composition and
The majority of the change
is
caused
density due to several external inputs.
the activity of the sun.
The
in
response to
radiation from the sun, both thermal and ultraviolet, cause
varying rates of change in the composition and hence the density of the atmosphere.
The
sun also causes changes in density due to the solar wind. The impingement of the solar
material
the
upon the
earth's
magnetosphere and upper atmosphere causes several changes
make-up of the thermosphere and subsequently the
overall density at altitude.
other major contributor to density variation of the earth's atmosphere
activity.
Geomagnetic storms, although
is
short-lived, cause a significant
in
The
geomagnetic
change
in
the
atmospheric composition and density.
1.
Solar
The
EUV Radiation
first
Effects
source of density change that will be addressed
Ultraviolet Radiation
produced by the sun. The sun's
density variation in the earth's thermosphere.
at
low
latitudes in the
The
structure of the thermosphere at the
caused by the absorption of the
EUV radiation is the main cause of
solar
summer hemisphere. (Marcos,
low and middle
EUV radiation is deposited mainly
1993,
p.
1991, p.3)
circulation
and
by the heating
UV and visible radiation reach the lower
atmosphere
is
absorbed
enters the atmosphere through photoionization.
is
The
latitudes are controlled
atmosphere and hence heat the earth's surface. (Burns, 1991,
electrons and ions
2)
EUV radiation by the ozone layer in the lower
thermosphere. The longer wavelength
radiation reaching the earth's
the Extreme
is
at the
p.
300
The energy
3)
km
Most of the
level
that has
EUV
and the energy
been absorbed by the
passed to the neutral atmosphere by collisional processes. (Burns,
The
solar
EUV radiation also
momentum
imparts a
to the neutral gas by the
creation of pressure gradient forces that drive the neutral winds from the day to night
regions and from the
summer
to winter hemisphere. (Burns, 1991, p. 3)
Variations
EUV radiation interacting with the thermosphere lead to
strength of the
changes
in the
in the
composition and density of the neutral atmosphere. During the period of solar minimum,
the
EUV radiation produced by the sun is much less than that being produced
maximum. Therefore,
during a solar
1992,
p.
209)
the thermospheric temperatures and neutral densities will be lower
minimum than during
It
during solar
can be seen from
solar max.
This fact
is
Figure
illustrated in
this figure that the variation in density
7.
(Larson,
becomes much
greater at higher altitudes during periods of high solar activity versus low solar activity.
Due
makes
it
to the fact that the solar
difficult to
obtain a measurement of the
problem was solved by Jacchia
found
at
EUV radiation is absorbed by the thermosphere,
EUV
in the early sixties.
impinging on the atmosphere. This
Jacchia discovered that the intensity
10.7cm closely corresponded to the amount of solar
Currently the accepted measurement of solar flux
Figures 6 and
as high as
is
290 during the
proportion would
mean
solar
maximum
activity being witnessed.
the F10.7 index.
the typical value for F10.7 during solar
7,
it
minimum
period of 1958.
is
A change
As can be seen
75,
in
and has reached
in solar activity
of this
a change in density by a couple orders of magnitude at the altitude
regime of interest.
One of the drawbacks of using Fl 0.7
that
it
lies at
the other end of the spectrum
as a
from the
gauge of EUV
radiation,
EUV radiation,
and
is
is
the fact
not a direct
measure of the amount of EUV radiation reaching the thermosphere. Figure 8 (Hess,
1968,
p.
668)
illustrates the solar
frequency on the
F10.7,
is
on the
left
right.
spectrum.
It
can be seen that the
EUV radiation is at a
hand side of the peak spectrum, whereas the solar flux measurement,
Because the F10.7 index
10
is
not a direct measurement of the amount
EUV radiation entering the atmosphere,
of solar
solar activity have
remedy
this
available,
2.
some
accurate density calculations based on
Several programs have been initiated in order to
inherent error.
problem, but currently the F10.7 index
and consequently
Solar
is
is
the best indicator of solar flux
most widely used
the index
in
atmospheric modeling.
Wind
Another of the driving forces causing composition and density variation
thermosphere
is
the solar wind.
outward from the
sun's corona.
The
solar
wind consists of protons and electrons flowing
Higher density plasma streams are also ejected from the
sun during periods of flare and sunspot activity. (Fleagle, 1963,
blows the interplanetary magnetic
the sun. (Burns, 1991,
p.
field lines
Field-aligned currents flow
down
at
The up welling and
field
a direction
away from
encounters the earth's magnetic
high latitudes. The ions
them
4) These collisions produce heat,
the auroral zone.
in
to the ionosphere, closing the circuit, and
collide with the neutral particles, driving
p.
236) The solar wind
4) This in turn causes a potential drop across the earth's
producing an ion-convection pattern
1991,
p.
across the polar cap
magnetic polar cap as the interplanetary magnetic
field.
in the
in a similar
which
in turn
in this
convection pattern
convection pattern. (Burns,
produces a
rising
motion around
the convection-driven neutral winds produce a
composition and density change which spreads from the high latitudes to the lower
latitudes.
The convection driven
neutral
winds also produce another
significant heat
source. Joule heating results from the frictional heating of the plasma as
it is
through the neutral upper atmosphere by the auroral electric
(Marcos, 1993,
p.
3) This Joule heating
is
a substantial heat source, but
field forces.
dragged
becomes even more prevalent
during periods of solar flare activity. Flares on the sun cause the solar wind to accelerate,
driving the interplanetary magnetic field faster across the earth's magnetic field.
11
This
increases the cross-cap potential and the rate of particle precipitation, and can also
produce a magnetic substorm. (Burns, 1991,
p. 5)
Geomagnetic Storms
3.
Another of the major contributors of composition and density variation are
geomagnetic storms. Geomagnetic storms are produced by the interaction of the
When
interplanetary magnetic field with the earth's magnetic field.
occurring, shock
magnetic
waves
in
monitored by means of the
solar flare activity.
Ap
minutes
later,
is
magnetic
geomagnetic
The geomagnetic storm presents
The
onset, or
field.
This effect
itself as
of
a world-wide transient
sudden commencement, of a
Ap
Approximately 20
index.
the temperature and density of the auroral zones begins to increase.
manifestations of geomagnetic storms
is
the generation of waves
propagate from the auroral regions to the lower latitudes. These waves take
approximately eight hours to reach the lower latitudes. (Alder, 1992,
magnetic storm, illustrated
The
or longer.
quite
p.
is
activity index, during periods
characterized by a rapid increase in the
One of the
that
in the earth's
index, or
variation in the earth's magnetic field.
magnetic storm
is
the interplanetary magnetic field are driven into the earth's
causing rapid transients
field
solar flare activity
effects
pronounced
35)
It
in
Figure 9 (Akasofu, 1972,
p.
can be seen from Figure
10, that
This
is
bombardment of EUV
A typical
last
even longer and are
illustrated in Figure 10. (Ratcliffe, 1972,
during periods of geomagnetic storms, the
density at the altitude regime of interest increases several fold.
intense
18)
557), lasts from 24 to 48 hours
of the geomagnetic storm on the density,
at the higher altitudes.
p.
radiation, increases the density
magnitude.
12
This fact coupled with
by several orders of
4.
Gravity Waves, Planetary Waves, and Tides
The
source of composition and density variation to be discussed are
final
propagating tides and gravity waves. Atmospheric tides are caused primarily by the
absorption of ultra-violet radiation by the ozone in the stratosphere, while gravity waves
are caused by a variety of mechanisms
which occur
in the
troposphere.
A couple of these
mechanisms are the shears associated with cold fronts and winds blowing over mountains.
(Burns, 1991,
p.
density variation.
3)
EUV
altitudes, the semi-diurnal tide is the
major contributor to
This effect becomes less apparent at higher altitudes due to the fact that
the semi-diurnal tide
diurnal tide
At lower
is
dissipated by viscosity and ion drag.
becomes the
driver of density variation and
radiation in the thermosphere.
Overall, these
is
At higher
altitudes, the
caused by the absorption of the
waves and
tides propagate
up from the
lower altitudes affecting the composition and density of the upper altitudes.
As
a consequence of the
that the temperature
above mentioned thermospheric processes,
is
and hence the density of the upper atmosphere vary with the
following parameters:
solar flux (solar cycle and daily
geomagnetic
activity
local time
day of the year
latitude
longitude
wave
it
structures
13
component)
known
C.
PROGRAM DEVELOPMENT
The propagation program created
FORTRAN based
model comparison
for the atmospheric
program which uses a Runge-Kutta
integrator.
is
a
The propagator uses
a
form of Cowell's Method of orbital prediction. The program uses the position and
velocity vectors in Cartesian coordinates as
its
inputs,
and integrates over a designated
time frame to produce the position and velocity vectors, also
Method of orbital
Cowell's
of round off error.
in
Cartesian coordinates.
prediction has been found to be inaccurate due to the build-up
A more accurate method would
have been to convert the Cartesian
coordinates into normalized spherical coordinates in order to minimize the integration
error build-up.
Further research into this conversion
is
being conducted
in
order to obtain
greater accuracy. For the purpose of the comparison, however, the accuracy of the result
was not
in question,
but the accuracy of the atmospheric model, and
relative ease
its
of
use.
The propagation program
perturbing forces acting on the
applied to the
result
satellite's
of the perturbing
consists of a series of subroutines which calculate the
These
satellite.
motion resulting
forces.
Figure
forces, in the
in a position
1 1
form of accelerations, are
and velocity vector reflecting the
and Table l(Milani, 1987,
p.
14-15) illustrate
the various perturbing accelerations and their relative magnitude as a function of altitude.
of orbital motion
In order to obtain an accurate reflection
it
was decided
attractions
1.
at
the altitude band in question,
that in addition to the drag force, the earth's geopotential
would
and
third
body
also be included as perturbing forces.
Earth's Geopotential
As can be seen from Figure
earth orbiting satellites
is
1 1
,
the main perturbing force encountered by
the earth's gravity
field.
14
If the earth
was
low
a perfectly round and
.
smooth
planet, then the forces
gravity law
shown
in
equation
of gravity could easily be modeled by the inverse square
7.
g=
a
~r
(7)
r
However,
since the earth
is
not perfectly spherical, and not smooth, the gravity field must
be derived by obtaining the solution to Laplace's equation:
v
2
V=
(8)
where
„ _,
V=G
dm
am
—
t
J
(9)
volume
5 being the distance from the
is
satellite to
an incremental mass, dm, inside the earth, and
the factor of proportionality in Newton's
basic equation for the geopotential
Law
G
of Gravitation. Laplace then derived the
shown below:
G
V = -\lPn (cose)[P)dm
By
converting to spherical coordinates and applying Rodriguez' formula, the earth's
geopotential takes the form of equation
V=
GM
1+XX
m=0 r
rr=2
In this equation,
constant, r
the
(10)
model
is
^ im (sin0')(C„ m cosmA + sin/;;A)
potential function,
GM
is
the radius vector from the earth's center of mass, a
n and
geocentric latitude,
and
X
Pnm
is
(ID
\
V is the gravitational
ellipsoid,
coefficients,
1 1
the earth's gravitational
is
the semi-major axis of
m designate the degree and order of the coefficients,
the geocentric longitude,
are the associated
Cnm
in
use
15
in
is
the
and Snm are the harmonic
Legendre functions. (Ross, 1993,
Earth gravitational models are currently
(j)
both the
civilian
and
p. 5-3)
DoD
Several
communities.
The
DoD
currently uses the
VVGS-84 Earth
tesserai
DoD
gravitational
harmonic
satellite
WGS-84
geopotential model as
(EGM)
model
programs including the
GPS
The
gravitational model.
contains a 180 by 180 matrix of zonal and
The WGS-84
coefficients.
its
EGM
currently being used for several
is
satellite constellation,
and
is
considered to be
an extremely accurate representation of the earth's geopotential. For most cases, the
matrix
is
not needed, and the matrix
is
truncated
down
to a 41 by 41 matrix.
full
The
truncated matrix provides an ample representation.
Currently, the propagation
coefficients.
4
1
matrix.
Future iterations of the propagation program
At the moment, attempts
84 coordinate system have
from the
first
program contains the
The
failed.
eight zonal terms
at
first
8 sets of zonal and tesserai
will contain the
complete 41 by
configuring the propagation program
first
eight sets
in the
of coefficients vary only
WGS-
slightly
of the more basic geopotential models and can be used
without incurring significant errors during propagation.
2.
Drag
For
low earth
satellites in
must be modeled. Drag
affects
orbit,
drag
is
all satellites in all
the other major perturbing force that
altitude regimes, but the affect
is
considered to be insignificant at altitudes greater than 1000 km. The following equation
represents the atmospheric drag acceleration which
aD
In this equation,
CD
is
aD
2
placed on an orbiting body.
2
(12)
m
represents the drag acceleration imparted
the coefficient of drag,
the direction of motion,
atmospheric density.
=^CD -pV
is
A
is
geocentric velocity of the
Fis the
satellite velocity
velocity used in this equation
satellite
orbiting satellite,
the cross sectional area of the satellite perpendicular to
m is the satellite mass,
The
upon an
is
and p
is
the local
computed by combining the
with the contribution due to earth rotation and the wind
16
velocity at altitude.
It is
important to note that the wind component of the velocity cannot
be ignored, and can be quite significant
is
at altitudes in
excess of 600 km. The neutral wind
a consequence of the activity caused by a geomagnetic disturbance.
can be
5%
activity, the neutral
25%
200m/s of wind
for every
change
velocity.
winds can reach velocities
in density.
(Marcos, 1993,
p.
The change
in
drag
During a period of intense geomagnetic
in
excess of
6) In order to
1
km/s which
model
equivalent to a
is
this behavior, several
atmospheric wind models have been developed. Models of the neutral winds begin with
the efforts of Sissenwine in the late sixties, to the Horizontal
Wind Model 90
(HWM 90).
Currently these wind models are not incorporated into the empirical atmospheric drag
models, but are incorporated
The
coefficient
in
the general circulation models.
of drag,
determine the coefficient of drag,
CD
it
,
is
a difficult quantity to obtain. In order to
must be determined whether the
continuum flow, or a free molecular flow. This
number. The Knudsen number
the average mean-free-path
Knudsen number
flow.
is
The
is
much
satellite is
greater than one.
is
a
done by determining the Knudsen
is
the ratio between a typical dimension of the satellite and
of the molecules found
less
satellite is in
than one, the
considered to be
in
in the local
satellite is
is
When
the
considered to be in a continuum
a free molecular flow
Since the atmospheric density
atmosphere.
when
the
Knudsen number
so low at orbital altitudes, satellites
in
the upper atmosphere are characterized by free-molecular-flow aerodynamics. (Ross,
1994,
p.
16)
The
collision
of the atmospheric
particles with the spacecraft
produce the
atmospheric drag force. The collisions can be classified under three categories: (1) elastic
or specular reflection, (2) diffuse reflection and (3) absorption and subsequent emission.
In elastic or specular reflection, the molecule collides with the satellite
away without
transferring energy to the satellite.
17
and
is
reflected
In diffuse reflection, the molecule
collides with the satellite
to the satellite
satellite is
from a
when molecules
on impact and
mechanical calculation. (Ross, 1994,
statistical
p.
are absorbed
is
also transferred
later emitted.
Since the
considered to be in a free-molecular-flow, the coefficient of drag
makeup of the
1992,
and transfers a portion of its energy. Energy
satellite,
the coefficient of drag can vary from 2.0 to 6.0.
illustrates the variation
207)
this thesis, the coefficient
of drag
will
remain constant
all
make up what
is
Inverting equation 13 will result in
is
to guess
is
used. For the purpose of
a value of 2.2.
at
called the B-factor.
B=
Current practice
Table 2 (Larson,
of drag, the cross sectional area, and the mass of
In equation 12, the coefficient
the satellite
what
is
^-
(13)
m
more widely known
what the correct B-factor
B-factor until the correct position and velocity
orbital prediction, but rather orbital correction.
is
is
as the Ballistic coefficient.
for that given
day and adjust the
achieved. This does not
If the size
seem
and shape of the
of drag. However, most of the current
catch-all for any other errors in the
actual value.
It
must be noted
In this comparison,
schemes use the B-factor as a
it
was decided
to
keep the B-factor
is
at
more accurate comparison between atmospheric
at this point that
keeping the B-factor
major change, vice using the B-factor as an error
The
coefficients
modeling program. Hence the value of the B-factor
a constant value in order to obtain a
models.
orbital prediction
to be
satellite are
known, as well as the mass, then the B-factor should only vary with varying
nowhere near the
and
size
of the drag coefficient for various orbiting platforms.
of drag cannot be determined, a value of 2.2
If the coefficient
obtained
Depending on the
16)
p.
is
at its actual
is
a
catch-all.
density for the drag acceleration calculations
is
of course derived from the
atmospheric models. In the calculation of the drag acceleration, the density
18
value
is
the most
difficult to obtain,
and frequently the one parameter with the greatest error. As a
most current atmospheric models have a
increasing as altitude increases.
Due
1
5 to
20%
inaccuracy
rate,
rule, the
with this inaccuracy
to this inaccuracy, the propagation
scheme
is
only as
accurate as the atmospheric model being used.
3.
Third Body Attractions
The
that
last
of third body
of the perturbing forces currently included
In the case
attractions.
perturbing bodies are the sun and moon.
moon
contribute
some
is
if
As can be seen
usually ignored for the
the altitude band in question (600
ignored
satellite
-
the propagation
program
Figure
in
small portion of disturbance force to a
practice, this disturbance force
in
of the
in
in question,
1
1,
medium
program
is
the
both the sun and the
altitude satellite.
In
low earth orbiting platforms, but
800km), these disturbance forces must not be
a truly accurate representation
of orbital motion
moon
used to find the perturbing acceleration due to the
is
is
to be modeled.
The equation
described by equation 14
below.
v = --^r-/i,
r
In this equation, r
moon
is
(r
\
r
3
(14)
3
'
\'ms
'
m® )
the radius from the earth to the satellite, rnts
to the satellite, rm9)
is
the radius from the earth to the
gravitational parameter of the
moon. (Bate, 1971,
p.
is
the radius from the
moon, and
\i m is
the
389)
Currently the propagation scheme does not contain several perturbations that are
important for accuracy purposes. At the moment, the sun's radiation pressure
as well as the earth's albedo effect.
is
ignored
Also, relativistic effects are ignored, which must
eventually be incorporated in order to improve accuracy.
19
D.
ATMOSPHERIC MODEL DEVELOPMENT
As mentioned
previously, atmospheric
models can be divided
the empirical models, and the general circulation models.
The
into
two
categories,
general circulation models
are dynamic representations of the earth's atmosphere, and require extensive
computational time
in
order to model the atmosphere.
The
general circulation models that
have been recently created require Cray computers to run simulations. Efforts are
currently under
way
may be used on
smaller computers.
and take
the
little
models
The
G. Jacchia
is
to convert these
models to a more user friendly format, so
The empirical models, however,
One drawback
computational time per simulation.
is
that they
are readily available,
that the accuracy
of
quite a bit less than that of the general circulation models.
history
of the empirical earth atmospheric models dates back to the
in the late fifties.
Once
efforts
of L.
the early satellites such as Sputnik and Pioneer had
been launched, immediate drag analysis was performed, and atmospheric models
developed based on
this analysis.
The
early
models were crude, and only represented the
regions where the satellites were orbiting. Little
was understood of the
environment and the density fluctuations being encountered. As more
variability
satellites,
greater understanding of the sun's interaction with the earth's atmosphere
the accuracy of the atmospheric models increased.
Figure 12
was
illustrates the
of the
and a
obtained,
developmental
history of the various earth atmospheric drag models. (Marcos, 1993, p. 20)
The
mathematical development of the Lockheed-Densel or Jacchia 60 model, the Jacchia 71,
and the
MSIS 86
earth atmospheric models are described below.
20
Lockheed Dense!
1.
The Lockheed Densel, or Jacchia 60 earth atmospheric model was
model to be implemented
was one of the
earliest
atmospheric models contrived, and hence
models; the
ARDC
is
its
actually a combination
accuracy
nautical miles, the density
is
> 76
nautical miles.
obtained from the
ARDC
is in
12,
it
great
of two atmospheric
1959 density model for low altitudes (h < 76 nautical
Jacchia 60 density formulation for h
first
scheme. As can be seen from Figure
into the prediction
The Lockheed Densel model
question.
the
miles),
and the
For the density below 76
1959 model which contains a table
of density values as a function of altitude. The discussion on obtaining the density below
76
nm
not be discussed in this paper.
will
60 formulation
desired, the Jacchia
is
When
The
used.
the density at an altitude above 76
first
requirement
is
nm
is
to define the unit
vector to the diurnal bulge using the solar position and the bulge lag angle. In order to
accomplish
this the solar longitude
must be determined by the following equation,
^ = (2 '%5.25)- 1 4,+00335sin (2,,%65.2s)
-
where
d represents
sun
obtained from the following series of equations.
is
the
£=
where e
is
number of days
cosA-
m,
December
since
=
sin A.
31, 1957.
cose
the obliquity of the ecliptic. (Lockheed, 1992,
The
unit vector to the diurnal bulge
is
by the following matrix
21
//,
p.
=
The
sin
<
15 >
unit vector to the
A sine
(16)
B-100)
then calculated
in true
of date coordinates
£,cos6-m,sm 6
uB
= c m.cosO+Cs'm 6
(17)
n,
where C = J2000.0
9 = 0.55 radians which
to true to date transformation matrix, and
Two
equals the bulge lag angle.
value of F10.7. The user
may
when using
options are available to the user
either specify a value, or
one
the solar flux
calculated using the
is
following formula,
F,o,
where once gain d
is
the
=
number of days
%
1.5+.8cos( 2;i
December
since
(18)
020 )
31, 1957.
an approximation of the F10.7 value on any given day over the
(Lockheed, 1992,
p.
1
1
This equation allows
-year solar cycle period.
B-103) The Jacchia 60 model divides the atmosphere into a
three bands for density calculation.
The
equation used to calculate the density
pW
= (p)«
is
in this altitude
108-// '
(76
ft
band
first
32
I
from 76
band
+ 0.85F.O7
is
ft
1.0
+
108 nautical miles. The
given by
fft-76Y3
32
\
j
-
-76
(l.O
+ coscp)
3
1224
/?
=
7.18
=
(p):6
ft
=
Fw
density from
altitude in
=
7
COS(p
ARDC
1959 model
at
76
measurement
R = SV true of date
=
nm
nm
solar flux
R
Ub = diurnal bulge vector (equation
17).
22
series
position vector
(19)
of
!
The second
calculation
is
given by equation 20;
p(^)
= p.(/;)0.85F
l
o.7[l.O
+ 0.02375(e
00102 '
a - 0.00368,
-1.9)(l.O + cos(p)
3
(20)
]
b
final altitude
1000 nautical
p(/,)
-
15.738, and k
band that
miles.
is
calculated in the Jacchia 60
is
1000 nautical
miles.
it
is
model
lies
used to calculate the density
assumed
3
(A
<p)
that the density
3
is
between 378 and
in this region.
-6.0xl0 6 ) + 6.0 xlO 6
zero
third
when
<
]/.-
the altitude
is
21
)
above
Mathematical Basis
background
In order to provide a mathematical
the partial derivative equations are illustrated below.
calculated density with respect to position
foM
=
d
J
dR
is
The
the conversion factor from slugs to kg/km^-
= (0.00504 F.oV)[o. 125(1.0 + cos
60 model
a.
is
Equation 21
In the Jacchia
where k\
,
)0(mh
)\ogAo]
p {h) = kexp[{-b-ab+6.363e^
where
and
band ranges from 108-378 nautical miles, and the density
altitude
the conversion factor
The
for the
above density equations,
partial derivative
of the
described by equation 22;
is
M
dh
d
dft
.
,
*
dR
M
dip
dip
t
*
(
between nautical miles and kilometers. The
derivative of the altitude with respect to
R
k-A-
is
22)
dR
partial
best estimated by the following equation,
R 4T- 7
-
(23)
,
•y/l-e'cos* 0'
R = position vector magnitude
R,
23
= earth
equatorial radius
e
The
=
0' =geocentric latitude
earth eccentricity
differentiation yields equation 24;
\\-e
R'
dh
-K,
2
e cos0'
dcos<p'
(24)
3
R
6R
2
2
(l-e cos :
<p')
dR
2
where
y/X
A =
COS0
,
2
+Y
2
dcos<p'
Z
R
a/2
fl
X, Y, and
1
and
R
is
1
'R*U.yr
:
dR
and
1
<p
+Y
2
)]
and
The
partial
of
(p
described by the following relationship,
d(p
The
2
cos<p'
represent the geocentric coordinates of the orbiting object.
with respect to
respect to
-[XZ\ YZ 2 -Z(
4
partial derivative
sin<p|_
R
3
Ub^'
K
of p(h) with respect to h and the
are different for each of the altitude bands.
08 nautical miles, the
(25)
R
partial derivative
partial
of p(h) with
In the altitude band
of p(h) with respect to h and
(p
between 76
are given by
the following equations:
dp(h)
=
dh
p pii.)
AC
h
32
(h-16
1-1.133^10.7
\
32
AR
+ cos<pl
+^Hl
1224
J
L
where p(h)
is
the density found from equation 19,
follow (Lockheed, 192,
A=(p),
76
~h
p.
p=
7. 18,
ip
+ 0.85Fio7
V
32
24
(26)
and the variables A, B, and
B-105):
B=
x
32
;
C
+
c= ioJ£i2*\ l+cos<py
df(h)
r
1224
{
-3^[^^](l + cos(p)
2
sin(p
(27)
dip
For the
altitude
band between 108 and 378 nautical
miles, these
two equations
take on this form;
= -f{h)[a + 0. 0305424 exp(-0. 0048/?)]&;l
dh
+0.0002059125p.(/7)Fio.7[exp(0.0102/;)](l
+ cos(p)
-^-^ = -0.0605625a(/7)Fio7[exp(0.0102/7)-1.9l(l + cos(p)
L
dip
dp(h)
-
p.
—^—
8p(/?)
378 and 1000
0.00189Fio
7
6
0.00189Fio
(l
dip
+ cos<p)
ri
[l
+ cos<p]l3 k
numerous
on
satellites
their empirical
(29)
2
,
3
sin(p(/;
(30)
-6.0xl0 6 )U
(31)
h*
prediction program, and in fact are currently used for that purpose.
model was one of the
sin(p
nautical miles, the equation
These equations can be used to implement an atmosphere
density
2
B-106)
dh
dp(h)
(28)
J
In the highest altitude band between
take on the form (Lockheed, 1992,
3
first
empirical models
in
an orbital
The Jacchia 60
on the market. With the launching of
throughout the recent years, Jacchia
et. al.
have been able to expand
model. As can be seen from figure 12, several Jacchia models have been
developed, each model building on
its
predecessor.
25
2.
Jacchia-Roberts 71
The next model
to be discussed
is
the Jacchia-Roberts 71 atmospheric model.
The Jacchia-Roberts 71 atmospheric model takes
geomagnetic
activity during the time in question.
profiles to represent
One
profile
into account both the solar
L. G. Jacchia defined
empirical
temperature as a function of altitude and exospheric temperature.
was defined between the region of 90
to 125 kilometers,
125 kilometers. Jacchia then used these temperature functions
thermodynamic
two
and the
differential
in
and the other above
the appropriate
equations to obtain density as a function of altitude and
exospheric temperature. (Lockheed, 1992,
p.
B-81) The Jacchia model as
it
stood was
very cumbersome and required a great amount of data storage to hold the data required,
so C. E. Roberts provided a method for evaluating the Jacchia model analytically. This
lead to a faster and
more manageable model. The following
determining the atmospheric density as derived
in the
are the equations used in
Jacchia-Roberts 71 atmospheric
model.
The
first
step in the
model
is
to calculate the nighttime
minimum
global
exospheric temperature for zero geomagnetic activity,
rc
= 379 +3°.24Fio7 +
l°.3[/r io7-F.o7]
(32)
where Fio 7=81 day running average of the F10.7 centered on the day
=
solar flux
measurement as obtained from the solar observatory
at
in question.
F\o
Ottawa, Canada.
(Jacchia, 1970, p. 16)
The value of the nighttime minimum exospheric temperature
calculating the uncorrected exospheric temperature as follows,
26
is
then used
in
i
r,= rJl + 0.3 sin"0+(cos
::
r}-sin
::
e)cos
30
^
(33)
I
where
r=//-37°.0 + 6°.0sin(// + 43 o .0)
6 = -\<p+&\
jj-I|0_a|
,
-k< r< k
=
5,
the sun's declination
0=the
geodetic latitude of the satellite in true of date coordinates
(
includes
earth flattening)
//
~ "2^1 /
Wl-^2
= 180°.W
juS
x
X
2
—S
2
cos
X
J,
-i
A +o,A,
(34)
(tf+s
]
\
2
/2
2 )
The X variables
date
sun
(TOD)
in
TOD
It
are the
(A? +
2\X2
;tf)
components of the unit position vector of the
coordinates, and the
S variables
satellite in true
are the components of the unit vector to the
coordinates. (Lockheed, 1992, p.
B-83)
has been found through analyzing the effect of geomagnetic activity on the
atmosphere, that there
is
a lag of approximately 6 to 7 hours from a detection of a density
change from the actual geomagnetic disturbance. In order to account for
value of Kp
It
of
is
this lag, the
obtained for a period of 6.7 hours prior to the integration time
must be noted
that the
Kp
A7/~
is
calculated using the following formulas,
= 2S°.0Kp + 0°.03e Kp
=
question.
value only exists at a three hour resolution. At this point the
correction for the exospheric temperature
A7/»
in
14°. OKp
+
o
.02e
(Z> 200 kilometers)
A>
The corrected exospheric temperature
27
(Z< 200 kilometers)
is
then
(35)
r-
=
7\
+ Ar~
and the inflection point temperature
is
given by the following formula (Jacchia, 1970,
p.
21)
7V
= 371 o .6673 + 0.5188067/~-294°.3505^ 0021622r
These values for the temperatures and the
satellite altitude are
~
(36)
used
in the calculation
presented by Roberts for the atmospheric density. However, a number of corrections
must be applied due to several atmospheric processes presented by Jacchia. These
corrections will be presented before continuing with the Roberts calculations.
One of the
density
corrections deals with the direct effect of geomagnetic activity
below 200 kilometers. This correction
(Alogp) G =
The next
is
on the
calculated by the following relation,
O.OUKp + 1.2 x 10-V'
correction deals with the semi-annual density variation.
(37)
This correction
|
is
calculated from the following relations,
where Z
is
the altitude in kilometers.
(A\ogp)sA=f(z)g(t)
(38)
where
7
2331
+ 0.06328)^ OO2868Z
/(Z) = (5.876xl0- Z
g(/)
= 0.02835 + [0.3817 + 0.17829sin(2^i^ + 4.137)]sin(4^is^ + 4.259)
-1
Xsa-
+ O.O9544<
1
(40)
65
- + -sin(2/r</> + 6.035)
2
<p
(39)
(41)
2
=
(42)
365.2422
where JDm*
is
the
number of Julian days since January
84)
28
1,
1958. (Lockheed, 1992,
p.
B-
The next correction
deals with the seasonal latitudinal variations in the lower
thermosphere. Equation 43 represents the general density variation, whereas equation 44
represents the correction for
Helium
specifically.
(Alogp)ir = 0.014(Z-90)e
(
(Alogp)//*
where e
is
=
0.65
-OOOI3(Z-90)
sin(27T0+1.72)sin(/>|sin(/>|
k
<p&
4
2&
sin
2
\
-0.35355
the obliquity of the ecliptic. (Lockheed, 1992, p.
Below 125
kilometers, Roberts uses the
=
r(z)
v
(44)
B-84)
same temperature
profile as Jacchia,
4
dx
7V+-^-yCnZ'
4
35
(43)
^
(45)
n=Q
where
d\
=
Tx
-
To,
To
=
183°.
where
Tx
is
and the coefficients follow
2
Ci = -526S7.5knr
Co = -89284375.0
C.
OK
C* = -0.8^;^
3
Cs = 340.5A77;-
= 3542400.0^;-*
(46)
the inflection point temperature at 125 kilometers calculated by equation 36.
Roberts then substituted the temperature profile obtained from equation 45 into the
barometric differential equation and integrated by partial fractions to obtain the following
expression. Equation
47 represents the density found
100 kilometers. (Lockheed, 1992,
p.
Ps(Z) =
in
the altitude band between 90 and
B-85)
p.To
M(Z)
k
F,
Mo
where the "o" indicates the conditions
at
T{Z)
exp(kF2
)
(47)
90 kilometers. The mean molecular weight
calculated by the following equation,
29
is
M(Z) = ^AnZ"
(48)
where
i
Ai = \\.0433S7545knr
Ao = -435093.363387
^1
A* = -0.08958790995*7//^
= 28275.5646391^-'
Ai = -765. 334661 OSkni
~2
A6 =
As = 0.00038737586*7//
-0. 000000697444 km*
These constants give a value of the mean molecular weight
which
is
close to the sea level
in
equation 47
at
90 kilometers of 28.82678
mean molecular weight of 28.960. (Lockheed,
86) The density of the lower limit
The constant k
"5
is
is
assumed to be constant
at
p = 3.46 x
1992,
-9
10
p.
B-
gm/cm*
evaluated by the following formula
k =
-
(49)
Rd\C.
where g
is
assumed
to be
F
F
The functions
and
x
2
9.80665m/sec^
in
=
90 +
R
F =(Z-90)
2
The
variables
90- rJ
A6
r,
the acceleration due to gravity at sea level.
equation 47 are determined from equations 50 and 51.
ft
^
,
+
YY
Z -2XZ + X + Y
[90-rJ [&100-ISQX + X +Y
Z-r,
2
2
A
2
2
(50)
2
)
Y{Z- 90)
+ ^tan-'
Y
{Z + Rj90 + Rj
Y 2 +(Z-X){90-X)
and rz are the two
real roots
and
X and Y are the
real
and
imaginary parts (Y>0), respectively, of the complex conjugate roots of the following
quadratic,
P{Z) =
±CU Z"
n=0
with the following coefficients
30
(52)
(51)
X
35
4T
c
rr ^c;
4
c
C4
+ ^- and
C4
Q/,
l<n<4
C;=-^-
The parameters p
for values of C„ used in equation 45.
evaluated by the following relations (Lockheed, 1992,
Pz
=
S(r
x
)
U{r
x
S(r
2)
=
^
Pi
p4 ={B -r r2 Rl[BA + B {2X + r^r2
5
x
-{r/2 B
Ra (x +Y
2
5
p6
=B
P]
=B -2p -p,-p
4
+ B {2X + r +r
i
4
5
2
]
2
)
-R
a
p.
equations 50 and 5
)]^W{r )p2 }lX'
x
]
-R )-p -2(X + R
5
a
)p4 -{r2
-Y
2
)p 5 }/ X'
+Rjp
3
-(r
2
where
r=-2r,r Ra (Ra +2XR a + X + V
2
2
2
2
V = {Ra +r ){Ra +r
2
]
U(r,)
= (r,
)(R
2
2
^)=r,rAK+/;)
)
+ 2XR a + X 2 +Y 2 )
+Ra f(r -2Xr,+
2
+7% -r
2 )
\+x±?
J
v
and
B = cc n +pn
n
Tx
1
S(Z) =
-T
1
(//
=
0, 1,.... 5)
o
±B„Z"
n=0
with the coefficients
a n =3144902516.672729
a,
ft
=-123774885.4832917
-52864482.17910969
-16632.50847336828
31
are
B-86)
+ W(r2 )p,+r r2 (Ra2 - X 2
a
1
S(-K)
V
=
Ps
~
U(r2 )
)
in
t
]
+Rjp
2
ft
= -1.308252378125
a = -1 1403.31079489267
ft
=
a 4 =24.36498612105595
ft =0.0
a = 0. 008957502869707995
ft
a =
2
1816141.096520398
3
5
Equation 47
to be predominant.
it
is
is
a valid equation
0.0
=0.0
below 100 kilometers where mixing
However, above 100 kilometers,
diffusive equilibrium
is
is
assumed
assumed, and
necessary to substitute the profile given by equation 45 into the diffusion differential
equation (one for each constituent of the atmosphere), and integrated by partial fractions.
This was done by Roberts to yield the density for the altitude band between 100 and 125
kilometers, given by equation 52.
Ps{z) = JlP,(Z)
(52)
/=i
It is
computationally expensive to calculate the density
different exospheric temperatures
km were
at
100 kilometers
at
by using equation 47. Thus values for the density
at
100
pre computed at several values of the exospheric temperature, and these values
could then be extracted instead of using valuable computer time and memory. Next the
atmospheric constituent mass densities are calculated by the following,
1+or,
p,(Z)
The index
/
= p(l00)— L/i,
M
s
r(ioo)
FM
'k
3
T(Z)
corresponds to the values
1
«p(A/,tfJ
through 6 of the various atmospheric
constituents of N2, Ar, He, 02, O, and H, respectively.
The constants found
53 are the characteristics of these species and are tabulated
Hydrogen
is
in
in
equation
Table 3 of Lockheed 1992.
not a significant constituent below 125 kilometers, so
32
(53)
it
is
not included in the
(Lockheed, 1992,
calculations.
p.
B-89) The temperature
at
100 kilometers
is
calculated
from the following,
4
li\00) = Tx +Qd,
Q = 35~
i
where
^C {lOO)" =-0.94585589
n
(54)
n=0
and
F
}
F
and
4
^3
are calculated by equations 55 and 56
=
Z-r,
a
100-rJ ^100-r2
H,+iw
v
^4
<?:
z+R
=
</ 5
(z-ioo)
+
(Z + /?J(^ + 100)
where the parameters
<7,
—Y
Z -2AZ + X +f
YY
v
2
ioo
2
2
-2oox+x +r
2
V
2
4
(55)
2
y
r(z-ioo)
tan"
(56)
2
r +(z-A-)(ioo-^)
are calculated by
1
<7 2
=
0&i)
?3
</4
All
-1
=
U(r2 )
=
{l
+ rxh [R\
-X -Y
2
2
)q5
Ve
= "?5 ~2(X + Ra )q4 -{r2
<7i
=-2<7 4
of the variables
-?
3
+ W{r
x
+R
a
)q 2
h -(^+R
a
is
)q 2
-tf2
in these
equations have been defined
For the region above 125 kilometers,
equilibrium
+ W{r2 )q3 } I X'
it
is still
earlier.
a valid assumption that diffusive
dominant, but the temperature given by equation 45
is
no longer
valid.
Jacchia defined the temperature profile of the upper altitude regions by the following
empirical asymptotic function:
33
T{Z)
= T .+-(Z-Tx )t<m
K
5
^^j^l+4.5xltf(Z-l2? ]|
]
x
\o.95ii
In order to integrate the diffusion differential equations in closed form,
equation 57 with the following (Lockheed, 1992,
T(Z) =
The parameter
T_-(Tm
I will
T
-Tx )exp
be discussed
-T
p.
(57)
Roberts replaced
B-90)
Z-125
^1
e
(58)
L-Tx
v**+Z
35
later.
Integration of equation 58 yields the following equation,
which
is
valid for
all
of
the constituents except hydrogen.
l+a + r/
(
p,(Z) = p,(l25)(i
T-T
(59)
T-T,xj
where
M
>2
lgRj
it
At
Rcr
(
-Tx
(
Tm
V
Tx~1q
this point several corrections are
\^
35
(60)
JV
648 1.766,
made
for the particular constituent densities
due to seasonal changes. The value of the helium density
that
is
calculated by using
equation 59, must be corrected due to the seasonal latitudinal variation as given by
equation 44.
The
specific
form
is
presented below
Wz)L--p.Wio^where
/
corresponds to the index of helium presented above.
concentration becomes significant, therefore
it
p6 (Z) = p 6 (500)
where the hydrogen density
at
Above 500
kilometers, the
must be accounted for by the following
-il+tt 6 +r,s
r(5oo)
(61)
r
L-T{Z)
(62)
L-T(500)
~T(Z)
500 kilometers can be calculated from
34
=^-10^ ,H394 - 551ogr-
p6 ( 5 oo)
The temperature
constituents are
summed and
following (Lockheed, 1992,
(63)
A
500 kilometers
at
llogr'~ 1
is
calculated by using equation 58.
the standard density above 125 kilometers
p.
is
The
given by the
B-94):
P.(Z) =
5>,(Z)
(64)
i=i
So
far,
the standard atmospheric density at any given altitude has been calculated,
but the densities calculated using equations 47, 52, or 64 must
geomagnetic
activity, the
now
be corrected for
semi-annual variation and the seasonal latitudinal variation by
equations 37, 38, and 43 respectively.
The
of these variations can be summed
effects
logarithmically to obtain the following relation
(Alogp) corr =(Alogp) G +(Alogp)^ +(Alogp) Lr
The
final
corrected density at any altitude can then be determined by
f(Z) = p t \0
Due
(65)
to the introduction
of equation 58
ii,
°»
>
-
in the place
(66)
of Jacchia's equation (57) for
temperature, the results of the Roberts portion of the model did not exactly concur with
those that were observed by Jacchia. This effect
was only
in
the upper portion of the
model, whereas the lower altitude bands agreed exactly(as should be the case). This
where the
C parameter
comes
into play in equation 58.
calculated in order to obtain the best least squares
2500km)
values
is
to the Jacchia tabulated data.
less that
in the
Lockheed
p.
Values of the t parameter were
of density versus altitude (125
The maximum
6.7%. (Lockheed, 1992,
parameter C can be found
fit
is
-
deviation from the Jacchia density
B-94) The derivation and values of the
reference, and will not be discussed in this
paper.
35
a.
Mathematical Basis
The following
section contains the partial derivatives of the Jacchia-
Roberts model. Starting from equation 66, the partial derivative of density with respect to
local position can be found.
to better
model the
This can then be used in orbital prediction schemes in order
satellite's orbital trajectory.
The following equation
is
a simple
restatement of equation 66.
p(z) = p 5 (z)Ap_
The
desired partial derivative then
*g
dR
The
I
(67)
becomes
=Ps
r'
*^A +APc
?i±
rc
dR
(68)
dR
variation of the correction factor with respect to the satellite position
is
derived from
equations 65 and 37 through 43.
^
Apc
-.
0.4342944819
dR
L(/)/'(z)^
+ 0.14sin(2^
6
J
Y + 1.72K°°
R
V
l3(
Z-90
'
I
2
dZ
(l-0.0026{(Z-90)} )sin<^|sin(^|-= + 2(Z-90)|sin(|)|cos^-i
dR
oR
(69)
where
/
The
,
(Z)
= -0.002868/(Z) + 2.33l(5.876xl0- 7 )z ,33,
variation of altitude with respect to position,
calculated in equation 24.
The
variation
of geodetic
dZ/dR
,
^
is
0O2868Z
the
(70)
same as
latitude with respect to position
derived by differentiating the following equation
<p
=
X,
tan"
(l-/)
1/2
:
(xt + xl)
to obtain
36
that
(71)
is
-lT
-X,
2
X] +
+x
x2
2
_
dty
sin
x
20
2
W~2~
(72)
xl+xl
J_
X,
The
variation of standard density with respect to position
from the barometric
differential equation for altitudes
3R
Mt!
]_dT_
TdR
RT dR
n=l
calculated directly
below 100 kilometers
dZ_
Ps
is
(73)
and from the diffusion differential equation above 100 kilometers
dZ
dR
'
^dT
f
(74)
R(Z+Rj
T
where
p'
The
45 for
(75)
altitudes less than 125 kilometers
T-Z ( dTx
dT
ZaM
of the temperature with respect to position are derived by
partial derivatives
differentiating equation
=
A
dT„
dT dR
dR
-1
+
(76)
J -J
dR
~—\
or equation 58 for altitudes above 125 kilometers
dT
dR
dT
dR
(
r-r/
T-ZX J
{(
JdTT
Z-Zx
*
X
V
az
Ra +Z
37
,35,
(
Tx
dL
T„-Tx
j*
Tm
and
finally,
-T
3T.x
_f
dTx
dT
( 11-
-Tx
the derivatives of
-T )
Tx
i
TX
-T 1
-l
dR
K+zx
\
(Tx
]
(±\dZ
(77)
35JdR
and 7^with respect to position are derived by
36 and 33 respectively,
differentiating equations
dTx _
= 0.0518806 + (294.3505)(0.0021622)e -0 0021 622
dT
^
dR
7"_
(78)
= O.3rc J2.2sin 12 0cos0fl-cos 3o -l^£
* *
-2.2cos
12
2JdR
{
\
30
•
rjsinrjcos
T dfl
3
t dt
,2 /^
2 V
rj-sin" 0]cos~-sin--=r
/
.
--=--(cos
2dR 2
.
2
.
}
(79)
2dR
where
dr\
_
1
(p-8s
d(j)
dR~2\<f>-8s \dR
de _
i
<j)+8 s
d<$>
dR ~2\<j>+5s \dR
=
dX.
dH
dX
{
180|
n(H + 43.0)
dH_
80
dX.
COS
lH
30
(/=1,2)
lsoffs.x,-^*,)
K \\S X2 — S2 x^
}
t
i
-£
x<
2
[(X,
+
2
2
AT2 )(5,
+5
2
2
)-(^,+^J
2 )
38
2
2
]l/2
*,
+*2
('=1,2)
d
*=o
dX,
dH
=o
dX
3
3.
MSIS
86 (Mass Spectrometer and Incoherent Scatter 86)
The Lockheed Densel model and the Jacchia 71 model, both developed by
Luigi Jacchia, use drag analysis data from earth orbiting satellites to derive the densities of
the various altitude bands associated within the models. A. E. Hedin chose a different
approach to the problem of modeling the atmosphere. Hedin's approach was to use actual
density data retrieved from orbiting instruments aboard several satellites and
combine
this
data with measurements taken by ground based incoherent scatter radar stations. Hedin,
et. al.,
began the
original
models, however,
it
MSIS model
was found
using the total densities inferred from Jacchia's
that the
measurements of the temperature found by the
incoherent scatter method differed significantly. (Hedin, 1972,
were noted
in the
time of daily
variations of the density.
maximum, and
p.
2139) These differences
the amplitude of the daily and annual
Hedin used the data that was obtained from the mass
spectrometers on board several orbiting satellites to confirm
this.
The measurements from
the satellites as well as the ground based incoherent scatter measurements provide
information at different altitudes, latitudes, longitudes, solar activities
The MSIS 86 model
can be seen
in figure 1.
is
,
and seasons.
the culmination of years of research and development as
The formulation of the MSIS models
is
based on a spherical
harmonic expansion of exospheric temperature and effective lower boundary densities
39
which uses
local solar time
and geographic
a special case of a
expansion
is
where the
coefficients
latitude as the independent variables.
more general expansion based on longitude and
of the expansion are represented by a Fourier
This
latitude,
series in universal
time. (Hedin, 1979, p. 2)
This leads to the following expansion
AG = £ X X pm
[
a cos( w n't + »&) +
"n
K
sin(
w <o't + //A)]
(80)
m=0 /=0 n=-/
where
P =
ln
/
=
co'
Legendres associated functions
the
= 2^/864005"'
X = geographic longitude
in
This particular expansion
was used
more
more incoherent
MSIS
geographic latitude
universal time in seconds
launching of several
several
in
radians
for the
initial
MSIS
models. With the
equipped with mass spectrometers, and the addition of
satellites
scatter radars, the
model evolved
into the
MSIS83, and
eventually
86.
The main emphasis now
is
the formulation of the
MSIS
86 model. Hedin
explains that the models uses a Bates temperature profile as a function of geopotential
height for the upper thermosphere, and an inverse polynomial in geopotential height for
the lower thermosphere. (Hedin, 1987,
p.
4649) The exospheric temperature and other
key quantities are expressed as functions of geographical and solar/magnetic parameters.
The temperature
profiles allow for the exact integration
of the hydrostatic equation for a
constant mass to determine the density profile based on a density specified at 120
kilometers as a function of geographical latitude and solar/magnetic parameters.
MSIS
The
83 model used the expansion formula (Equation 80) to model the variations due to
40
local time, latitude, longitude, universal time, F10.7,
enhances the
MSIS
MSIS 86 model
and Ap. The
83 model by adding terms to express hemispherical and seasonal
differences in the polar regions and local time variations in the magnetic activity effect.
The following equations
atmospheric model. The
temperature
first
are the complete formulation
step in the formulation
is
of the
MSIS 86
to obtain an expression for the
profile.
T[Z) = Tm -(Tm - 7;)exp
T{z) = l/{\/T +
"^ (zz,)]
where
[
Z>Z
TB x + Tc x 4 + TD x 6 ) where
2
the matching temperature and temperature gradient at
Ta = T{Za ) = Tm -(21 - ^exp^
1
Za
(81)
a
Z<Z
a
(82)
equals
^
(83)
2
I = r(Za ) = (71 - Ta )<{(R p + Z,)/(Rp + Za )]
(84)
2
rD = o.66666^(z ,zJr;/ra -3.iiiii(i/ra -i/r )+7.iiin(i/7; -i/r )
::
Tc =
(85)
2
!;(Z
,ZX'/(2Ta )-(VTa -\/T )-2TD
TB ={VTa -\/T )-Tc
(86)
-TD
(87)
x = -[^(Za ,Z)-^(Z ,Zj]/^(Z ,Zj
(88)
Tn=T + TR {Ta -T
(89)
)
&Z,Z = (Z-Z ){RP + Z
I
)
I
I
)/(R P + Z)
(90)
#Z,Zj = (Z-Zj(*, + zJ/(*,+z)
(91)
=
cr=7;7(7:-7;)
7;
Z =t{[i+g(l)]
t = t [\+g(l)]
L
Z =Z
= Z[\ + G(L)]
7;[i
+g(z,)]
[\
+ G{L)]
TR = TR [l + G(L)]
where
(all
temperatures
in
Kelvin)
T= ambient temperature
T -
41
average mesopause temperature
Ta = temperature
Ta
T,
=
(Z +
at
Za
at
Z
;
in
Za = altitude
Z=
) /
at Z,
average mesopause shape factor
altitude in
km
= 120 km
Z,
at Z,
of temperature profile junction;
1
average mesopause height
Hedin now explains
that the density profile
mixing profiles multiplied by one or more factors
dynamics flow
average temperature gradient
TR =
K/km
average exospheric temperature
= average temperature
Z =
=
T,
=temperature gradient
T]2 = temperature
T„
Za
at
effects. (Hedin, 1987, p.
is
17.2
km
Rp =
6356.77
km
a combination of diffusive and
Cv ..Cn
to account for chemistry or the
4638)
,/
//(Z,M) = [/;,(Z,M)%/;m (Z,M)'] "[C,(z)...C„(z)]
A=
M /(M
h
-M)
(92)
(93)
where
n = ambient
nd =
M
density
M
diffusive profile
nm = mixing
diffusive profile
is
= 28.95
M = molecular weight of gas species
profile
The Following equations
The
= 28
h
represent the diffusive profile and the mixing profile.
given by the following
a
nd (Z,M)
=n,D{ZM)[T(Z
DB (Z,M)
D{Z,M) =
D{ZM) = z)5 (Z
fl
,M)exp
^
U - ,)/ro+r
{ri
l
3
l
)/T{z)]*
Z>Z
(95)
a
- ,)/3+rcU5 - ,)/5+rBU7 - ,)/7
l
DB {ZM) = [T(Z,)/T(z)]
42
(94)
-a
ri
exp
}
[
^
(Z <
z
'
)]
Za )
(96)
(97)
2
=Mgl /{oR T„)
y2
ga=gJ(l + ZJR p )
g
7x=MgJ&»Zm )l*t
g,
=/7,exp
//
;
=g./(l+Z,/R,Y
[G(L)]
(98)
where
=
n,
the average density at Z,
2
g =9.80665xl0" km / s
3
s
Rg = 8. 3 14 x
and
a
is
0" 3
1
g
km 2 /mols
s
the thermal diffusion coefficient
2 deg
of -0.4 for He and zero for the
rest
of the
constituents.
The mixing
profile
is
given by the following
a
//JZ,M) = /; [Z)(Z,,M)/Z)(z,,A7 )][r(zJ/r(zJ] D(z,A7 )[7tZ )/r(z)]
Hedin (1987), goes
into a
procedure to simulate the chemistry and dynamic flow
of the turbopause which provides a specified mixing
effects
C,(Z) = exp{^/[l
R
l
where
Q.
and H]
is
is
+ exp
correction
is
is
(z)
Z\
is
H2
is
Z2
is
is
is
Cj
is
R/2,
a peak in the mixing ratio in the
modeled by the following
the altitude
the scale height of the correction.
43
(101)
the altitude at which logjo
= exp{/?2 /[l + exp(Z-Z2 )///2 ]}
the density correction parameter,
R2/2, and
(100)
]}
the scale height for this correction. There
2
with respect to nitrogen.
=\og[Cln m (Zl1 2*)/n m (Z,,M)]
the mixing ratio relative to N2,
C
ratio
(z - Zl)//fl
lower thermosphere due to 0, N, and H. This peak
where R2
(99)
/
/
(102)
where logjo density
As mentioned above,
the
MSIS 86
density.
atmospheric model attempts to model
This
is
all
the
known causes of the
done by the expansion function (Equation
expansion function for the
MSIS
86 model
80).
variations of
The following
is
the
quantities:
Time independent
G = a PW + a
\0
/>
20
+ #40^40
20
Solar activity
+ F£AF + F£(aF)
2
2
+f£AF + f£(AF) +f£P AF
20
Symmetrical annual
+cl
cosQ J (td -t£)
Symmetrical semiannual
+(4+c
2
2
/?20 )cos2n
2
(/,-4 )
c/
Asymmetrical annual (seasonal)
P +c30 P,
l
+(clQ
lQ
)F cos2ad (td
l
-4)
Asymmetrical semiannual
+c
2
^
cos2ft ,(/,-/
(
c2
)
1
Diurnal
+[a u Pu
+a P +a„P + c P cosQ,^ -t*j\F2 coscoz
+[*„/>,
+b P +b P +d2\P21 cosQ,(^
31
i]
5l
31
3]
5]
2l
2l
5]
C
-/, o)]/s sin cur
Semidiurnal
+[a 22 P22
+aA2 P42
+[^22 +
+(c\ 2 P32 +cl 2 P52 )cosQ d {t d
M« +(<4^32 +4^2
44
)
~C)]f
2
™sQ d {< d - t£ )]F
2
cos2cot
Sin
2(OT
)
)
Terdiumal
+[a 3i PJ3
+(da P43 +ca Pa )cosSl d (t d -C)]F
+[^33^33
+(4^3 +da Pa )cosCl d (t d -t;l)]R sm3m
x
2
cos3cot
l
Magnetic
activity
+ [^00 + ^20^20 + ^40^40
"+"
V
+^p +^p
+(*„/>,
51
31
^10 ^10
+ ^30^30 + *50M0 J C0S "<A^
^10 /
)coso)(t-^)]a^
Longitudinal
+[a°2l P2] +a°4] P4l
+ a 6°,P6! + a °
x(l
+[**/>,
J>
1 1
1
+<P
31
+a°/>51 +{a«Pu + a*P3l )cosQ d (t d -C)]
+ F * AF)cosA
2
+b^ + b°„p
+ k pm + b°6l p6l +b? pu
5]
l
x
x(\ +
+ (b;;pu +%ip»)cosa d {td -/-)]
F° AF)smX
UT
4^o+*iA+aio^J(l +
^
J
xcosa/(/ -t^) + (a\ 2 P}] +al 2 P52 )co^w'{t - / 3 2 ) + 2A
UT/longitude/magnetic activity
+(*2?/> + **° />, +
,
+*„'/>,
61
)(
1
+ r2\°P10 )M cos( A - A*°
cosQ^ -^A/lcc^A-A*';)
+[*io flo
/•
C^
+ *i' ^30 + *£ P» ]M cosq)'(/ - /£
=1 + ^AF + /^AF + /^(AF)
45
( 1
03)
2
(104)
F
2
= + F°AF + f£AF + f£(AF)
2
(105)
\
where
AF = F]07 -\50
AF = F, 07 -Fl07
the solar flux
measurement
for the previous day,
measurement centered on the day
Pnm
are the
F]Q7
is
F10.7
is
the 81 day average of the solar flux
in question.
non normalized Legendre-associated functions equal to the following
{\-x 2
2
Y
tt
/2 n\\(d
n+m
/dx
n+m
)(x
2
-\y
with x equal to the cosine of the geographic latitude,
^=2^/365
universal time in seconds, tj
Q)
is
= 2k/24
o)'
= 2^/86400,
the day count in year, and
X
is
lis local time
in hours,
t is
the geographic longitude
with eastward being positive. Hedin explains that there are two choices for magnetic
activity,
AA={Ap-4) + (k^-\){Ap-4 + Q xp{-k^{Ap-4))-\]/k^}
[
where Ap
is
the daily magnetic index, or there
is
(106)
another method illustrated that
establishes the magnetic activity that uses the average values over an extended period
time. (Hedin, 1987, p.
4661)
46
of
ATMOSPHERIC MODEL EVALUATION
III.
The evaluation of atmospheric drag models
to the fact that there
no
is
real
a relatively difficult procedure
is
comparison tool to be used. Until enough
satellites
owing
can be
placed into various orbits and each of these satellites carries density measuring equipment,
As mentioned
the density of a particular orbit cannot truly be obtained.
modeling of the
encountered
earth's
in orbital
atmosphere
motion
is
previously, the
the most inaccurate of all the perturbations
The
analysis.
of this project was to obtain known
intent
vectors of a particular satellite program, and run a comparison of an orbital prediction
scheme using each of three atmospheric models
make
for
two
the comparison valid, consistency of the prediction
The vectors being used
Tracking Center on two
in the analysis
satellites,
one
at
In order to
different altitudes.
scheme had to be
verified.
were obtained from the Air Force
650 kilometers and one
kilometers, both of which have a high inclination.
The
at
Satellite
approximately 800
vectors, position and velocity
represented in True of Date Cartesian coordinates, are obtained by triangulating the
satellite position
and predicting out to 20:00:00.00
for the evaluation, if the actual observed position
local time.
of the
It
satellites
would have been
better
could have been
obtained, but due to time constraints and logistical problems this could not be achieved.
The accuracy of the
2000
ft
in the
position and velocity vectors
X, Y, and
Z
directions.
It
is
advertised by the
was decided
STC
at the time, that this
to be within
accuracy was
acceptable for the evaluation.
In order to provide consistency, the prediction
as that used by the
STC. This proved
prediction scheme which models
most
scheme developed had to be the same
to be quite a troublesome problem.
if
not
all
47
The STC uses
a
of the known perturbations. Depending on
what
is
requested, certain perturbations can be temporarily
removed from the program.
The vectors obtained from the STC contained only those perturbations which were
inherent to the prediction
STC
previously, the
uses the
placed on placing the
several problems
the
WGS-84
scheme developed, with one exception. As mentioned
WGS-84
same geopotential model
in the
A great
deal of effort
full
is
not a
trivial
developed prediction scheme, but
process.
41 by 41 geopotential matrix
computational time increased greatly.
It
in the
was decided
An
attempt was
made
A six by six matrix representing the first
the
model was eventually used to prevent error build up. Future work on
to include the conversion
reference and incorporation of the
full
of the
to
to incorporate a truncated version of
WGS-84
geopotential model.
into
scheme, but error build up and
the
would have
was
soon became apparent. The conversion of True of Date coordinates
frame of reference
incorporate the
geopotential model.
of
six zonals
this project
TOD coordinates to the WGS
frame of
41 by 41 matrix.
Another way that consistency was achieved was to verify and use astrodynamical
constants consistent with
AIAA
standards.
Depending on which reference one uses
to
obtain values such as the earth's gravitational parameter, earth's equatorial radius, earth
flattening constant
and others, these values would
all
vary by
some
small amount.
found that depending on which constant was used, the error would be different.
decided to use the
in
AIAA
it
It
was
was
published values whenever possible. These values were placed
a subroutine of the prediction
Once
It
scheme
to be used as necessary.
had been determined that the developed prediction scheme was an equal
representation of the STC's prediction scheme, or as close as
allowed, the model evaluation
the time frame
could be gotten
in the
time
was begun. The vectors obtained from the STC spanned
from 21 August 1993 to 22 September 1993
the position and velocity
it
were reported
for 20:00:00.00
48
in
24 hour
forecasts.
That
is,
of each day. These vectors were
then
programmed
800 kilometer
into
60
satellite.
satellite files,
Each of the
30 for the 650 kilometer
satellite files
satellite
contains not only the position and
Ap
velocity vectors, but also the environmental conditions of F10.7 and
the 81 day average F10.7 as reported by the National Geophysical
compared
vectors were propagated over a 24 hour period and
Since the baseline vectors themselves contained error,
results
and 30 for the
it
for that day, plus
Data Center. The
to the next day's vector.
was decided
to
compare the
by using the specific mechanical energy, a conserved quantity, of each of the
satellites
over the 30 day period.
By
converting the position and velocity of the
STC
vectors and those obtained from the developed prediction scheme into specific mechanical
energy using equation 107, a comparison could be made.
E=
— -£
2
A relative comparison of the position,
distance
was then conducted
in
(107)
r
velocity, right ascension, declination,
2000 foot accuracy advertised by the
prediction scheme.
As mentioned,
for a
radial
order to determine which model provided the most
accurate results, the object being to meet or beat the
STC
and
the evaluation procedure consisted of propagating the baseline vector
24 hour integration period for each of the
running the routine again.
were run with
relatively
quite the opposite.
that the
Due
then switching atmospheres and
must be noted that the Jacchia 60 and Jacchia 7 1 models
little difficulty.
The MSIS 86 atmospheric model proved
to an unexpected
atmospheric model
the evaluation
It
satellites,
hardware
failure late in the project,
itself is quite difficult to integrate into
of this model could not be completed
to correct this situation.
49
in time.
to be
and the
fact
the prediction scheme,
Work
is
currently under
way
A.
JACCHIA
The
first
first
MODEL EVALUATION
60
evaluation
was conducted on the Jacchia 60
earth atmospheric model.
The
procedure was to determine the values for the solar flux and geomagnetic activity
encountered during the time period between August 21 to September 22, 1993. Figure 13
shows the reported values of F10.7 and Ap
it
for this time period.
As compared
to Figure 6,
can be seen that the solar flux values are on the low side of the solar cycle, and the
geomagnetic
activity
is
relatively low.
perturbed atmosphere would have
This allows for an easier comparison, because a
made some of the
data points obtained during an active
period out of range of those obtained during a quiet period.
The
first
evaluation
illustrates the specific
vectors.
It
was conducted on the
mechanical energy of the
satellite at
STC
650 kilometers. Figure 14
reported vectors versus the predicted
can be seen that the predicted vectors are extremely close to that of the
reported values. The two error bars located on the graph illustrate the 2000
advertised by the
STC
prediction scheme.
In
one case
it
was found
that the
ft
STC
error range
STC
obtained
vector on Julian date 242 was actually reported incorrectly. For the most part, however,
the predicted values
errors
fell
degrees
degrees.
showed only small variations with the
within the range of 1700 to
in declination.
9000
fell
had some small error
In-track
and 0.01 to
0.
15
within the range of 0.01 to
0.
15
feet in radial distance,
Cross-track errors also
Satellite velocities
baseline vectors.
in the
range of 0.2 to ten feet per
second.
The same procedure was then conducted on
Once again
the 800 kilometer satellite vectors.
the prediction scheme provided vectors very similar to those obtained from the
STC. Figure 15
illustrates the results.
It
must be noted
at this point, that the
the prediction scheme suffered at the higher altitude. This
model.
As mentioned
previously, accuracy
is
accuracy of
due to the atmospheric
of the atmospheric model deteriorates the
50
higher the altitude. In-track errors
were found
to range
distance, and 0.01 to 0.2 degrees in declination.
0.01 to 0.8 degrees. Velocity
B.
JACCHIA
Once again
71
was
in the
from 2500 to 9000
Cross-track errors were
The
track errors were
250
to
range of
MODEL EVALUATION
the procedure
prediction
in the
range of one to eight feet per second.
was
run, this time with the Jacchia 7
1
model inserted
prediction scheme. Figure 16 illustrates the results of the prediction runs
satellite.
feet in radial
scheme provided very
8000
little
feet in radial distance
into the
on the 650 kilometer
variation with the baseline vectors.
and 0.001 to
0.
The
in-
14 degrees in declination.
Cross-track errors were also 0.001 to 0. 14 degrees in right ascension. Errors in velocity predicted
were
0.
1
to ten feet per second.
The
results
of this atmosphere
accurate than those of the Jacchia 60 model. In order to confirm
run again with the vectors of the 850 kilometer
8000
feet in radial
range and 0.001 to
0.
satellite.
650 kilometers were much more
at
this,
scheme was
the prediction
In-track errors
were
in the
range of 200 to
10 degrees in declination. Cross-track errors were also
comparable with the error range being 0.001 to 0.10 degrees of right ascension. This confirmed
the fact that the Jacchia 71
that the Jacchia 71
model provided better accuracy
at both altitudes.
This
is
due
to the fact
model provides a more accurate modeling of the polar regions, whereas the
Jacchia 60 does not model the polar regions as well.
extremely important for the
satellite
The modeling of the polar regions
program being studied due
51
to
its
high inclination.
is
IV.
SUMMARY
This thesis has attempted to provide the underlying principles used in orbital
The
prediction and a comparison of atmospheric models.
created
was used
was found
60.
prediction scheme that has been
compare the Jacchia 60 and Jacchia 71 earth atmospheric models.
to
that the Jacchia 71
model provided much more accurate
It
results than the Jacchia
This would stand to reason due to the fact that the J71 model had a bigger data base
with which to
work from. The J71 model provides
both on solar activity as well as geomagnetic
activity.
The J71
also
being investigated.
is
earth atmospheric
whereas the J60 only
is
is
dependent
relies
essential to the satellite
As discussed
earlier, the
made
to the prediction
scheme
in
the complete 41 by 41 geopotential matrix incorporated.
Doing
to
scheme would be to vary the B-factor as the
orbital pass.
this
would
increase
satellite cross-sectional
Currently the B-factor
is
The
frame of reference, and
computational time, but accuracy would greatly improve. Another improvement
its
program
order to improve
geopotential model needs to be incorporated.
WGS-84
changes through
solar
model could not be compared due
True of Date coordinates need to be transferred into the
prediction
on
currently being integrated into the prediction scheme.
Several improvements can be
accuracy.
activity,
models the polar regions, which
The MSIS 86
time constraints, but
density information that
in
the
area
held at a constant value.
By
changing the B-factor as a function of latitude or declination, error can be reduced.
Follow on work on
this project
for comparison, such as the
be better
if
MSIS 86
would include
and
MSIS
the satellite vectors being used
themselves. This
would provide a
were
90.
inserting other atmospheric
Before doing
however,
it
would
actual observations and not predictions
better realization
of the accuracy of the program, and
hence the atmospheric models. Another improvement
to convert
this,
models
in the
program's accuracy would be
from True of Date Cartesian coordinates to normalized spherical coordinates
and then integrate. Doing
this
would
greatly decrease the error build up.
52
As
the
prediction
scheme stands now, the error build-up causes accuracy problems
are integrated for
more than
Orbital prediction
program of interest,
is
if the
if the
vectors
a twenty four hour period.
an essential tool
in today's
space age. In the case of the
satellite
accuracy of the orbital prediction can be improved, then the
mission analysis can be improved. The interesting point of this project, was that the Bfactor
was no longer being used
as the error catch-all.
The B-factor
for any given satellite
can be correctly used, and the position and velocity as well as any other of the
ephemirides can be calculated.
By
incorporating
more subroutines
that
satellite
model other
perturbing forces, this prediction scheme can be used for other satellite programs.
53
APPENDIX A
FIGURES
U.V.
«"
X
VB.
'"
I
DOMXAW
A.
coNsrmje<T
-OHq x
t-u eapttu
a»i>ii
WYDBOQOI
KM
Figure
1
.
Atmosphere Breakdown
54
Exosphere
Turbopause
11
200
400
600
800
1000
Turbopause
1200
Temperature (K)
Figure
2.
Temperature
55
vs.
Altitude
1600
10
10
2
10
J
ALTITUOE (km)
Figure
3.
Temperature Profile of Earth's Atmosphere
56
10
:
280
240-
E
J2
'S3
X
10»
Number
Figure
4.
1011
density (cm
10 ,s
-3
)
Height Variation of Number Density
57
E
o
2
-16
Figure
5.
Diurnal Variation in Density
58
1945
1950
1955
1960
1970
19S5
1975
1990
Ymt
Figure
6.
Solar Flux History
59
1985
1990
1996
10- ,5
10
H
-16.
100
200
300
400
500
600
700
800
Altitude (km)
Figure
7.
Density
vs.
Altitude for Various F10.7 Values
60
900
1000
Figure
8.
Solar Radiation Spectrum
61
S&n Juan
irr" n »-or *
08
12
13
24
18
18
12
January 1967
14
January 1967
Hours U.T
Figure
9.
Geomagnetic Storm Recording
62
24
o
T
10
12
14
16
18
Date: Nov. i960
Figure 10. Density
vs. Altitude for
63
Geomagnetic Storm
2
•c
4)
-
eo
O
—
o
O
urn
i
(I
r iCjJAEj2
jaASJ B3S) SSB^j \]U[\ Jad
90JOJ
Figure 11. Comparison of Perturbation Magnitudes
64
YEAR
PRIMARILY TOTAL DENSITY
DATA FROM SATELLITE DRAG
ICAO
AR DC
1960
.
\0ENSEL.
—
*0
PRIMARILY TEMPERATURE AND
COMPOSITON DATA FROM GROUND AND
IN-SITU INSTRUMENTS
CIRA61
LOCKHEEDUS&A62
HA RRis.
PRIESTER
JACCHIA
1965
JACCHIAL6SASLFP)
GENERAL CIRCULATION
MODELS
WALKER
8RUCE
-
/
1970
J65
CIRA65
J70
J71
CIRA72
GRAM
OG06
VOAC
1975
USSA76
i
MSIS77
UCL
NCAR
GTM
TGCM
CTIM
TIGCM
J77
MSIS
1980
UT/LONG
GRAM3
I
MSIS83
1985
MSIS86
TIEGCM
1990
GRAM90
MSIS90
CIRA90
Figure 12. Atmospheric
65
Model History
Daily
Measured Values
of F1
0.7cm and Ap
'
i
i
100-
80C/5
0)
03
>
60
line
= F1 0.7cm
k_
--Ap
CO
03
03
\
40
t
/
20
'
'
/
*
V
l
1
\
/
j
v.
**
235
/
\
/
^
^
*
\
240
>.
^
/
y
245
255
250
260
265
Julian Date
Figure 13
Observed values of F10.7 and
Ap
66
for 21
August to 22 September 1993
Specific Mechanical Energy Jacchia
100.6
60
i
I
100.4-
100.2
I
1
C
CD
%....%
J {..} [
^yV V V
;
_...... ^~
C
w
^
M ...IS...
..
VVVV
i
LU
8
99.
E
...
o 99.6
99.4
*
= baseline
650km
99.2
--
= Jacchia 6 Otes it
99
235
240
245
250
255
260
Julian Date
Figure
14.
Jacchia 60 Evaluation at
67
650
km
265
Specific Mechanical
Energy Jacchia 60
1
100.4-
100.2/
S>
\
/
*
100
C
LU
"S
N
99.
76
E
o 99.6
99.4
*
= baseline
800km
99.2
=
Jacchia 6 Otest
99
235
240
245
250
255
260
Julian Date
Figure 15. Jacchia 60 Evaluation at 800
68
km
265
Specific Mechanical
100. 6i
Energy Jacchia 71
r
100.4
100.2
CD
k—
Q)
100
C
LU
T3
0)
99.8
05
E
k—
o 99.6
Z
99.4
*
= baseline
650km
99.2
--
99
= Jacchia 71 test
235
240
245
250
255
260
Julian Date
Figure
16.
Jacchia 71 Evaluation at 650
69
km
265
Specific Mechanical Energy Jacchia 71
100.6
i
100.4-
.*....
100.2-
%
100-
%
...
A/V
vy V^"A'ft
/
\/
*
<x>
c
111
"8
N
E
o 99.6
...
99.4
*
= baseline
800km
99.2
Jacchia 7
99
235
1
test
240
245
250
260
255
Julian Date
Figure 17. Jacchia 71 Evaluation at 800
70
km
265
,
APPENDIX B
TABLES
TABLE
1:
PERTURBATION ACCELERATIONS
Vvfleraiiom (cm
lifos) rKhronous
s
)
Starlette
CCS
(in
1
1
Earth
GW_
s
monopole
cm
:
986
n
km
:
|
2-0
-MOO
00"
'
:)
-"100
o oi
o
:
3
.
1
io-"
•
Us
I'V
4: iro
*
V »MU,-m
unitsi
= 3
.
«
»v»i
•>!
satellite
Parameter!
; : >
io
4
.
10
'
3
x
10
*
r
CAI
Eur:h\
R
I
.
J :a
ohlatenfN
ft
84
- 4
.
6 3"8
=
.
SI
'
10
•
10'
« 10
*
4
harmonic
I
» 10
6
«
'
8 3 «
4
10
9
10
10
'
4 8 x 10
4 x
10
"
•
10
3
9x
10
'
3
'
eg •=:.
/
(21
= 6.
m
- 6
A.
geopotential
harmonics
[3)
e g
/=
m
18
=
K"
CU
— '" —
-
19
y,t,a
.18.10
j ltu
4: « io
"
4 5 « 10
8»IO*
8
5
6
«
10
-
6.9 * IO
'
2 2 x 10
'
Perturbation
V.
C.\t-
Perturbation
CW
due to the
«T
r- =
Sun
W. 813
=
=
r-
Moon
(31
:
18.
due to the
(31
/M =
High-order
8x
3
=3.19 X 10'V:
|.J x 10"
M
)» 10"
10'°
3
3 x
10"
»
10"
9 6 x 10
'
I
3 x
5 7
10"
x 10"'
3
I
x
10"
5.6 x 10
Perturbation
due to other
planets
(eg
,
M, =0
Gsf
*
82.W;
4x 10""
r- 5
r\
4 3 x
10
'
4 x 10
*
1
3
x 10"'
7 5 x
10"'
7 3 x
*
10
Venusl
(4i
Indirect
oblation
(5
1
General
relansislu
"»~(-m
CU
CW:
G\t
:
I
l.4x 10"'
9
5 x
10"'
14
x 10
4
x 10
5
'
4 x 10
I
4
9x
10
•'
correction
(6)
Atmospheric
I
-
drag
("I
V
—
n b>
_
Co
Co
,
;
'
</
presure
3
x 10"'°
7 x
10*'
2x
10"
4>
:
.
U
=
1
38
.
6x
10"'
92x
10"
Ox
10'
9x
10'
4 6 x 10
3
2 x 10"'
4
4 2 x 10"'
3
4 x 10"'
1.4 x 10"'
3.
I
9x
2.7 x 10"'"
I
i
Earth's albedo
radiation
pressure
|9)
«?)
Solar
radiation
(8)
2-4
0-10 "*
=
a =
Thermal
emission
-.r<*m
4
V
9 .0
+
c
:
i/
So
,4 r
a
=
4
-0 4-0
:/=
7
9
1-20
71
5
x
IO"
10"'°
TABLE
2:
TYPICAL BALLISTIC COEFFICIENTS FOR LEO SATELLITES
mass
Satellite
Oscar-
Intercosmos
(kg)
Shape
Drag
Drag
Max.
Min.
coeffi-
coeffi-
cross-
cross-
cient
cient
Max.
MIn.
sectio-
sectio
for
for
ballistic
ballistic
coeffi-
nal
-nal
max.
min.
coeffi-
area
area
cross,
cross,
cient
cient
(m 2 )
(m 2 )
area
area
(kg/m 2 )
(kg/m a )
2
42 8
167
comm
82 9
76 3
scientific
075
0584
Type
of
Mission
5
box
550
cylin
2.7
316
2.67
277
octag
2 25
0833
4
26
128
30 8
scientific
4
2
1
-16
Viking
Explorer- 11
37
octag
18
07
283
2.6
203
72 6
astronomy
Explorer- 17
1882
sphere
621
0621
2
2
152
152
scientific
Space Tele-
11000 cylmd w/
143
3 33
4
192
29 5
astronomy
05
367
29
437
165
solar
phys
81
376
4
147
472
solar
phys
scope
112
arrays
OSO-7
634
9-sided
OSO-8
1063
cyhnd w/
1
05
5 99
1
264
145
3.3
4
181
12
104
1.81
34
4
123
252
arrays
Pegasus-3
10500 cylmd. w/
1
scientific
arrays
Landsat-1
891
cylind w/
ERS-1
2160
box w/
45
1
4
4
4
135
120
9695
12-face
39
267
14 3
4
169
93
1
3150
hexag
exper
platform
cylind.
HEAO-2
remote
sensing
arrays
LDEF-1
remote
sensing
arrays
139
4 52
2 83
4
174
80
1
astronomy
prism
Vanguard-2
SkyLab
9 39
sphere
02
0.2
2
2
235
235
scientific
76136
cylind w/
462
464
35
4
410
47
scientific
731
731
2
2
0515
0515
437
0515
1
arrays
Echo-1
75 3
sphere
Extrema
72
comm.
LIST
Adler,
J. J.,
OF REFERENCES
Thermospheric Modeling Accuracies Using F10.7 and Ap, M.S.
Thesis, Naval Postgraduate School,
Monterey
California,
December 1993.
Akasofu,S., Chapman, S., Solar Terrestrial Physics, Oxford University Press, 1972.
Bate, R.R., Mueller, D.D., White, J.E.,
Fundamentals of Astrodynamics, Dover
Publications, Inc., 1971.
Burns,
AAS
AG.,
Killeen, T.L.,
Changes of Neutral Composition
in the
Thermosphere,
Publications Office, 1991.
Fleagle, R.G., Businger,
J. A.,
An Introduction
to
Atmospheric Physics, Academic Press,
1963.
A Global Thermospheric Model Based on Mass Spectrometer and
Incoherent Scatter Data; MSIS I. N2 Density and Temperature, Journal of Geophysical
Hedin, A.E.,
et.al.,
Research, Vol. 82, No. 16, 1977.
Global Model of Longitude/UT Variations in Thermospheric
Based on Mass Spectrometer Data, Journal of
Temperature
Composition and
Hedin, A.E.,
et. al.,
Geophysical Research, Vol. 84, No. Al, 1979.
Hedin, A.E., MSIS-86 Thermospheric Model, Journal of Geophysical Research, Vol. 92,
No. A5, 1987.
Hess, W.N., Introduction to Space Science,
Gordon and Breach, Science
Publishers,
1968.
Jacchia, L.G.,
New Static Models of the Thermosphere and Exosphere
with Emperical
Temperature Profiles, Smithsonian Astrophysical Observatory, 1970.
Jacchia, L.G., Variations in the Earth's
Upper Atmosphere as Revealed by
Satellite
Drag,
Smithsonian Astrophysical Observatory, 1963.
Larson, W.J., Wertz, JR., Space Mission Analysis
1992.
Lockheed, Lockheed Working Papers, 1992.
73
and Design, 2nd
Ed.,
Microcosm,
Inc.,
Marcos, FA.,
et. al.,
Satellite
Drag Models: Current Status and Prospects, AIAA, AAS
Publications, 1993.
Milani, A., Nobili, A.M., Farinella, P., Non-Gravitational Perurbations
Geodesy,
Adam
Ratcliffe,
J.
A.,
and Satellite
Hilger, 1987.
An Introduction
to the Ionosphere
and Magnetosphere, Cambridge
University Press, 1972.
Ross,
I.
M. AA4850:
U.S. Air Force,
1994.
Handbook of Geophysics, The Macmillan Company, 1960.
74
.
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4.
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5.
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stgrae
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•
-
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t/t
IU 'w''