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TRAJECTORY MAKER
A Trajectory Simulation Tool
────────────────────────────
USER'S GUIDE
ORBIT HORIZONS
94 Promenade
Irvine, California 92715
U.S.A.
COPYRIGHT NOTICE
This document and software package consisting of the
Trajectory Maker and Trajectory Scape computer programs
are copyrighted (C) 1991, 1992 by H.B. Reynolds,
President, ORBIT HORIZONS. All rights are reserved. No
part of this publication may be reproduced, transmitted,
transcribed, stored in any retrieval system, or translated
into any language by any means without the express written
permission of ORBIT HORIZONS, 94 Promenade, Irvine
California 92715, USA.
SECOND EDITION/FIRST PRINTING
July 1992
ii
LIMITED WARRANTY
THIS USER'S GUIDE, THE PROGRAMS, "TRAJECTORY MAKER" AND
"TRAJECTORY SCAPE" ARE SOLD "AS IS", WITHOUT WARRANTY AS
TO THEIR PERFORMANCE, OR FITNESS FOR ANY PARTICULAR
PURPOSE. THE ENTIRE RISK AS TO THE RESULTS AND
PERFORMANCE OF THE PROGRAMS IS ASSUMED BY THE USER.
HOWEVER, ORBIT HORIZONS WARRANTS THE MAGNETIC DISKETTE(S)
ON WHICH THE PROGRAM IS RECORDED TO BE FREE FROM DEFECTS
IN MATERIALS AND FAULTY WORKMANSHIP UNDER NORMAL USE FOR A
PERIOD OF NINETY DAYS FROM THE DATE OF PURCHASE. IF
DURING THIS NINETY DAY PERIOD THE DISKETTE SHOULD BECOME
DEFECTIVE, IT MAY BE RETURNED TO ORBIT HORIZONS FOR A
REPLACEMENT WITHOUT CHARGE, PROVIDED YOU SEND PROOF OF
PURCHASE OF THE PROGRAM.
YOUR SOLE AND EXCLUSIVE REMEDY IN THE EVENT OF A DEFECT IS
EXPRESSLY LIMITED TO REPLACEMENT OF THE DISKETTE AS
PROVIDED ABOVE. IF FAILURE OF A DISKETTE HAS RESULTED
FROM ACCIDENT OR ABUSE, ORBIT HORIZONS SHALL HAVE NO
RESPONSIBILITY TO REPLACE THE DISKETTE UNDER THE TERMS OF
THIS LIMITED WARRANTY.
IN NO EVENT SHALL ORBIT HORIZONS BE LIABLE TO YOU FOR ANY
DAMAGES, INCLUDING BUT NOT LIMITED TO ANY LOST PROFITS,
LOST SAVINGS, OR OTHER CONSEQUENTIAL DAMAGES ARISING OUT
OF THE USE OR INABILITY TO USE THESE PROGRAMS AND
ACCOMPANYING MATERIALS EVEN IF ORBIT HORIZONS HAS BEEN
ADVISED OF THE POSSIBILITY OF SUCH DAMAGES, OR FOR ANY
CLAIM BY ANY OTHER PARTY.
SOME STATES DO NOT ALLOW THE LIMITATION OR EXCLUSION OF
LIABILITY FOR INCIDENTAL OR CONSEQUENTIAL DAMAGES, SO THE
ABOVE LIMITATION OR EXCLUSION MAY NOT APPLY TO YOU. THIS
WARRANTY GIVES YOU SPECIFIC LEGAL RIGHTS, AND YOU MAY ALSO
HAVE OTHER RIGHTS WHICH VARY FROM STATE TO STATE.
THIS WARRANTY SHALL BE CONSTRUED, INTERPRETED AND GOVERNED
BY LAWS OF THE STATE OF CALIFORNIA.
YOU AGREE TO THESE TERMS BY YOUR DECISION TO USE THIS
SOFTWARE.
iii
TABLE OF CONTENTS
WELCOME ............................................. viii
1 INTRODUCTION ........................................ 1
2 INSTALLATION ........................................ 3
3 TRAJECTORY MAKER INPUT DESCRIPTION .................. 7
3.1 Unit Convention Menu ......................... 7
3.2 Perturbation Model Option Menu ............... 7
3.2.1 Unperturbed Two-Body Option ......... 8
3.2.2 Gravity Harmonic Effects Option ..... 8
3.2.3 Atmospheric Drag Effects Option ..... 8
3.2.4 Gravitational and Drag Effects Option 9
3.3 Initial State Vector Option Menu ............. 9
3.3.1 Orbital Elements .................... 9
3.3.2 Initial Position and Velocity ...... 15
3.4 Time Control Parameters Menu ................ 18
3.5 Output Format Options Menu .................. 21
3.5.1 Geographic Coordinates ............. 21
3.5.2 Common Radar Coordinates ........... 21
3.5.3 Geospherical Inertial Coordinates .. 24
3.5.4 Earth-Centered-Inertial Coordinates 26
3.6 Menu Editing ..................................26
4 TRAJECTORY MAKER OUTPUT DESCRIPTION ................ 28
5 EXAMPLE ORBITS AND TRAJECTORIES .................... 31
5.1 Space Shuttle Discovery ..................... 31
5.2 Repeating Ground Trace ...................... 31
5.3 Sun-synchronous Orbit ....................... 32
5.4 Candidate Surveillance Orbits ............... 33
5.5 An Example Illustrating Orbit Decay ......... 34
5.6 Orbits Near Transition Region ............... 35
iv
5.7 Ballistic Missile Trajectories .............. 35
5.7.1 Western Test Range Trajectories .... 35
5.7.2 Hypothetical Threat Trajectories ... 35
6 TRAJECTORY SCAPE INPUT DESCRIPTION ................. 37
6.1 World Map Projections Options Menu .......... 37
6.1.1 Mercator with stereographic
projection ......................... 37
6.1.2 Hammer-Aitoff Equal Area Projection 38
6.1.3 Mollweide Equal Area Projection .... 38
6.2 Map Resolution Options Menu ................. 38
6.2.1 Fine Option ........................ 39
6.2.2 Medium Option ...................... 39
6.2.3 Coarse Option ...................... 39
6.3 World Map Focus Options Menu ................ 39
6.3.1 Greenwich Meridian ................. 39
6.3.2 American Region .................... 40
6.3.3 Soviet Region ...................... 40
7 TRAJECTORY SCAPE OUTPUT DESCRIPTION ................ 41
8 ADDITIONAL VIEWING CAPABILITY ...................... 45
8.1 Small Circles ............................... 45
8.2 Segmental Arcs and Spherical Polygons ....... 45
APPENDIX A. HISTORICAL NOTES ........................ A-1
APPENDIX B. GLOSSARY ................................ B-1
v
LIST OF MENUS
Menu Page
1 Unit Convention Menu ............................. 7
2 Perturbation Model Option Menu ................... 8
3 Initial State Vector Option Menu ................. 9
4 Initial Orbital Elements ........................ 10
5 Initial Position and Velocity ................... 15
6 Time Control Parameters ......................... 18
7 Output Format Option Menu ....................... 21
8 World Map Projections Menu ...................... 37
9 Map Resolution Options Menu ..................... 38
10 World Map Focus Menu ............................ 39
vi
LIST OF FIGURES
Figure Page
1 Semimajor Axis and Eccentricity ................. 11
2 Orbit Inclination ............................... 13
3 Longitude of the Ascending Node and Argument of
Perigee ......................................... 14
4 Earth-Centered-Inertial Coordinates ............. 16
5 Geographic Coordinates .......................... 22
6 Common Radar Coordinates ........................ 23
7 Geospherical Inertial Coordinates ............... 25
8 Sample Trajectory Maker Graphics Display ........ 30
9 Four Tundra Orbits on Mercator and Stereographic
Projections ..................................... 42
10 Circular, 45 Degree Inclined Orbit on an
Hammer-Aitoff Projection ........................ 43
11 Circular, 45 Degree Inclined Orbit on a
Mollweide Projection ............................ 44
vii
WELCOME
Welcome to Trajectory Maker and its companion program
Trajectory Scape. Whether you are an experienced orbital
mechanic, relatively new to the field or just want to
learn about it, ORBIT HORIZONS is confident that you will
enjoy using these tools.
Trajectory Maker is a professional simulation and design
tool for scientists and engineers involved in mission
analysis and orbital operations.
TRAJECTORY MAKER FEATURES
Trajectory Maker predicts trajectories for earth orbiting
satellites and ballistic missiles. It produces high qual-
ity trajectories and it can accommodate a variety of
applications.
o Trajectory Maker has four optional force models:
Keplerian two-body motion, gravity perturbations,
atmospheric drag perturbations and combined gravity
and atmospheric drag perturbations.
o Graphics displays let you visualize trajectory
ground traces, as their computed, simultaneously,
on three conformal map projections. Altitude trend
is also displayed, and there is a window that
provides numerical time and altitude data.
o Four optional numeric output formats are created:
geographic coordinates, common radar coordinates,
geospherical inertial coordinates and rectangular
Earth-centered-inertial coordinates. In addition
all input data is echoed to a file with various,
auxiliary orbit parameters.
o Flexibility is provided with input time control
options: date and time of injection, mission
duration time, granularity of computations and
output sampling frequency.
o Optional initial state vector input: classical
orbital elements or Earth-centered-inertial
coordinates. Metric or English units may be used.
viii
Trajectory Maker's companion program, Trajectory Scape, is
a multiple trajectory viewing tool. It displays traject-
ories, created with Trajectory Maker, on several optional
map projections, e.g., Mercator, stereographic, Hammer,
Mollweide, etc. It has three optional map resolutions as
well as three central viewing regions.
In addition to trajectory ephemerides, Trajectory Scape
constructs and displays circles, segmental arcs and
spherical polygons on the various map projections.
Both programs are easy to use while providing maximal
flexibility and control for your particular application.
They comprise an excellent introduction to orbital
mechanics, and are outstanding tools for professional
scientists and engineers.
Twenty-four example data sets are included in this User's
Guide for you to get started immediately.
The software requires an IBM PC XT/AT or compatibles with
a hard drive and color card. Graphics visualization
requires EGA or VGA. A math co-processor is optional and
is utilized if present.
ix
Trajectory Maker
User's Guide 1
──────────────────────────────────────────────────────────
1. INTRODUCTION
According to the Lomgman Dictionary of Astronomy &
Astronautics*, a trajectory is "the path of a body in
space or through the Earth's atmosphere. The word is
usually used to describe the paths of rockets and space-
craft rather than that of planets or moons." An orbit is
"the path of one body in space around another." Thus, an
orbit is a trajectory and both terms will be used herein,
depending on the context.
Trajectory Maker does just what its name implies - it
makes trajectories. In particular, it makes trajectories
for objects under the influence of the Earth's gravita-
tional field, whose mass is small relative to the Earth.
Such an object may be a manmade satellite continually
revolving around the Earth, it may be an interplanetary
spacecraft having achieved escape velocity or it may be a
ballistic missile on a destination towards its target.
In any case, the type of trajectory an object attains
depends on the energy manifested by its initial position
and velocity. Trajectory Maker really doesn't care
because it integrates the object's equations of motion
directly, rather than using the classical universal
variables approach implied by two-body theory.
Because the shape of the Earth is more accurately
represented by an ellipsoid of revolution than it is a
sphere, trajectories deviate from idealized conic sections
implied by the universal variables.
Moreover, depending on the objects altitude, the Earth's
atmosphere tends to retard the objects motion, causing it
to decay, ultimately colliding with the Earth. In the
case of ballistic missile trajectories, the retarding
force becomes much greater than the gravitational force as
the object rips through the atmosphere towards its target,
and if its mission is aborted, atmospheric winds may cause
its fragments to drift.
──────────────────────
*Ridpath, Ian, Longman Illustrated Dictionary of Astronomy
& Astronautics, Longman Group UK Limited, Burnt Mill,
Harlow, Essex, 1988.
Trajectory Maker
User's Guide 2
──────────────────────────────────────────────────────────
By integrating the equations of motion directly, these
perturbation effects may be treated without much
difficulty.
Specifically, Trajectory Maker's force model includes both
atmospheric drag and gravitational perturbation models.
The differential equations of motion are integrated with
the Runge-Kutta-Shank's eighth-order, fixed-step numerical
scheme having twelve stages. Its speed is moderate,
depending on the user selected step-size, but it is stable
and very accurate.
Trajectory maker also contains an innovative algorithm for
rapid, high precision altitude computations. All computa-
tions are performed with double precision arithmetic.
However, you don't need to know or be concerned with the
technical details to use the program - that's one reason
for the program !
For the interested reader, some historical background is
given in Appendix A. A glossary of terms used in this
guide is given in Appendix B.
Trajectory Maker
User's Guide 3
──────────────────────────────────────────────────────────
2. INSTALLATION
"Trajectory Maker" and "Trajectory Scape" are designed for
use on IBM PC XT/AT and compatibles with a hard drive.
Numeric data is available on any monitor with a color
card. Graphics visualization requires an EGA or VGA card.
A math co-processor is optional and used if present.
Make a sub-directory and copy all of the files on the
distribution disk(s) from your floppy drive. For example,
C:>MD TMAKER
C:>CD TMAKER
Insert the distribution disk(s) in Drive A: then,
C:>COPY A:*.*
That's it for installation. You should have the following
files in your subdirectory:
TMAKER.EXE ... Trajectory Maker Executable Program.
TSCAPE.EXE ... Trajectory Scape Executable Program.
TMSRB.FON .... Graphics Fonts.
WORLDB.LGE ... Large World Map.
WORLDB.MED ... Medium World Map.
WORLDB.SML ... Small World Map.
Running Trajectory Maker
────────────────────────
Before giving a detailed description of the input options
and data, let's go ahead and run an example to familiarize
you with the operation of the program.
Type TMAKER at the DOS prompt. After the opening banner,
a menu will appear requesting your choice of units.
Select the Nautical Miles option. (You may abort the job
by pressing the Ctrl-Z sequence.)
* Type 1 and press <Enter>
Next, a menu appears requesting your choice of models.
Select the Unperturbed two-body model.
* Type 1 and press <Enter>
Trajectory Maker
User's Guide 4
──────────────────────────────────────────────────────────
Next, a menu appears requesting your choice of input for
the initial conditions. Select the Orbital Elements
Option.
* Type 1 and press <Enter>
The next menu requests the orbital elements of the
satellite. These elements characterize the orbit. For
this example, we will assume the satellite will have a
circular orbit at an altitude of 400 nautical miles (nm),
in a plane inclined 45 degrees with respect to the
equatorial plane.
The radius of the Earth is 3444 nm at the equator, so the
semimajor axis is approximately: a = 3844 nm.
* Type 3844 and press <Enter>
Since the orbit is circular, its eccentricity, e = 0.
* Type 0 and press <Enter>
The satellite's orbit plane has inclination, i = 45°.
* Type 45 and press <Enter>
The next two entries are orientation angles which are
described in Section 3.3.1. Assume Ω = 0°, and w = 0°.
* Type 0 and press <Enter>
* Type 0 and press <Enter>
Notice that the period of the satellite, about 6000
seconds, is displayed. The period is the time it takes
the satellite to make one revolution about the Earth. It
is displayed to aid you in selecting the next parameter,
the time past perigee, when you wish to start the orbit.
We will use τ = 0 seconds.
* Type 0 and press <Enter>
That completes the orbit description.
The next menu is used to define the time parameters for
the orbit. Assume the duration of the mission is one day,
which is about 86400 seconds, starting at time 0. Since,
for this example we shall not be concerned with the date
Trajectory Maker
User's Guide 5
──────────────────────────────────────────────────────────
and time of the orbit injection, enter 0 or just press
<Enter> in the date and time fields. When the cursor is
in the Mission Duration Time field,
* Type 86400 and press <Enter>
The next parameter is the orbit propagation time
increment. We will use one minute, or 60 seconds.
* Type 60 and press <Enter>
The final time parameter is the output sampling frequency
for the data file. We will use the same value as the time
increment.
* Type 60 and press <Enter>
The next menu requests your choice of output formats. We
will select the geographic or geodetic coordinates which
are the most common.
* Type 1 and press <Enter>
You are asked if you want to view the trajectory. We do,
so answer yes.
* Type y and press <Enter>
The program begins by displaying the world on three
projections which are described in Section 4. When the
maps are drawn, you will be prompted to press a key to
start the orbit computations.
When the orbit propagation is complete, your screen should
have a figure resembling Figure 8 in Section 4. You may
select a black and white palette if you want a screen
dump. Then you may press any key to end.
Section 5 contains several examples, for you to try, that
illustrate the various trajectories that you can generate
with Trajectory Maker.
Running Trajectory Scape
────────────────────────
To run Trajectory Scape, type TSCAPE at the DOS prompt.
After the opening banner, a menu appears requesting your
Trajectory Maker
User's Guide 6
──────────────────────────────────────────────────────────
choice of world map projections.
* Type 2 and press <Enter>
Next, a menu appears requesting your choice of map
resolution.
* Type 3 and press <Enter>
Next, a menu appears requesting your choice of world map
focus.
* Type 2 and press <Enter>
You will be prompted, in the command window, for an orbit
file name. The output file that was just created by
Trajectory Maker is HOST.DAT. To display this file,
* Type HOST.DAT and press <Enter>
Your screen should have a figure resembling Figure 10 in
Section 7. You may select a black and white palette if
you want a screen dump. Then you may press any key to
end.
The next section describes the input parameters in detail.
Trajectory Maker
User's Guide 7
──────────────────────────────────────────────────────────
3. TRAJECTORY MAKER INPUT DESCRIPTION
Trajectory Maker is very easy to use. Just go to the
subdirectory where you installed the programs and type
TMAKER at the DOS command line. You will be presented
with a series of five principle menus. These menus and
their input parameters are described in detail in this
section.
3.1 Unit Convention Menu
You are given the option of working in either English or
metric units. In either case all time units are processed
in seconds. Both input and output for angles are measured
in decimal degrees.
If you choose the English Units Option, then length
measurements must be input in nautical miles, and velocity
measurements in nautical miles per second.
If you choose the Metric Units Option, then length
measurements must be input in kilometers, and velocity
measurements in kilometers per second.
These options are illustrated in Menu 1 below.
┌──────────────────────────────────────────────┐
│ Unit Convention │
│ │
│ 1. English ( nm, lb, sec ) │
│ 2. Metric ( km, kg, sec ) │
│ │
│ Type option number and press Enter: 1 │
└──────────────────────────────────────────────┘
Menu 1. Unit Convention.
3.2 Perturbation Model Option Menu
You are given four force model options for tailoring your
particular application. These options are illustrated in
Menu 2 below.
Trajectory Maker
User's Guide 8
──────────────────────────────────────────────────────────
┌──────────────────────────────────────────────┐
│ Perturbation Model Option │
│ │
│ 1. Unperturbed two-body │
│ 2. Gravitational harmonic effects │
│ 3. Atmospheric drag effects │
│ 4. Gravitational and atmospheric effects │
│ │
│ Type option number and press Enter: 1 │
└──────────────────────────────────────────────┘
Menu 2. Perturbation Model Option.
3.2.1 Unperturbed Two-body Option.
This option assumes that the Earth's gravity field is
modeled with an homogeneous sphere. The orbit motion will
be along a conic section (ellipse, parabola or hyperbola
depending on the initial conditions). However, altitude
is based on the Earth having the shape of an oblate
spheroid. The altitude of a circular orbit will therefore
not normally be a constant.
3.2.2 Gravitational Harmonic Effects Option.
This option assumes that the Earth's gravity field is
modeled with an oblate spheroid (A surface of revolution
generated by rotating an ellipse about its minor axis).
If you choose this option, you will be asked to specify a
positive number between 2 and 22 (inclusive). The larger
the number, the more precise the gravity model. (This
number corresponds to the order of the gravity potential
function.) For example, use order 2 for Sun-synchronous
orbits.
3.2.3 Atmospheric Drag Effects Option.
This option takes into account the drag force created by
the Earth's atmosphere. This force varies with the mass
of the object, its effective cross-sectional area and its
drag coefficient. These three parameters are commonly
lumped together into a single parameter, which you will
ask to supply, if you choose this option. This parameter
Trajectory Maker
User's Guide 9
──────────────────────────────────────────────────────────
is called the ballistic coefficient, ß, and conveniently
has units measured in lbs/sq-ft, or kg/sq-m, sort of like
density.
Heavy masses tend to have larger ß than lighter masses,
but large areas have smaller ß than small areas. A space
based radar antenna may have a ß of 0.5 lb/sq-ft, whereas
a reentry vehicle may have a ß varying from 5 to 1000
lb/sq-ft, for example. The definition used is consistent
with that used at the Western Test Range.
3.3.4 Gravitational and Atmospheric Drag Effects Option.
You may choose both gravity and drag perturbations. For
reentry trajectories, you should consider this option,
with the gravity model having Order 2 or 4.
3.3 Initial State Vector Option Menu
You may input the initial state vector in one of two
forms. Either with Earth-Centered-Inertial (ECI) coordi-
nates or classical orbital elements.
These options are illustrated in Menu 3 below.
┌──────────────────────────────────────────────┐
│ Initial State Vector Option │
│ │
│ 1. Orbital Elements │
│ 2. Earth Centered Inertial │
│ │
│ Type option number and press Enter: 1 │
└──────────────────────────────────────────────┘
Menu 3. Initial State Vector Option.
3.3.1 Orbital Elements
Orbital elements are classical, fundamental parameters
that characterize an orbit. They are defined in almost
any elementary book on astronomy, celestial mechanics or
astrodynamics. The input format for the orbital elements
is illustrated in Menu 4 below.
Trajectory Maker
User's Guide 10
──────────────────────────────────────────────────────────
┌──────────────────────────────────────────────┐
│ Initial Orbital Elements │
│ │
│ a 3844 nm e 0 i 45 deg │
│ │
│ Ω 0 w 0 τ 0 sec │
└──────────────────────────────────────────────┘
Menu 4. Initial Orbital Elements.
a - Semimajor axis
──────────────────
This element defines the size of the orbit. The semimajor
axis of an ellipse or hyperbola is the distance from their
centers to their respective vertices. As used herein, the
semimajor axis of a parabola is the distance from its
focus to its vertex. A circular orbit is a special case
of an ellipse. Its semimajor axis is, simply, its radius.
Input this parameter with the units selected in the Units
Option Menu. This element is illustrated in Figure 1.
e - Eccentricity
────────────────
This element defines the shape of the orbit. It is a
fundamental parameter in the definition of conic sections.
It is dimensionless and can be any nonnegative number. If
it is zero, the orbit is circular; if it is between zero
and one, the orbit is an ellipse; if it is one the orbit
is a parabola; and if it is greater than one the orbit is
an hyperbola. These configurations are also illustrated
in Figure 1.
Remark: If the orbit is an ellipse and the minimum and
maximum distance from the Earth's center is known (i.e.,
its perigee and apogee radius) then its semimajor axis is
given by their mean and the eccentricity is their differ-
ence divided by their sum.
i - Inclination
───────────────
This element defines the orientation of the orbit plane
Trajectory Maker
User's Guide 11
──────────────────────────────────────────────────────────
┌────────────────────────────────────────────────────┐
│ │
│ Figure 1 │
│ │
│ │
│ │
│ │
│ │
│ │
│ │
│ │
│ │
│ │
│ │
│ │
│ │
│ │
│ │
│ │
│ │
│ │
│ │
│ │
│ │
│ │
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└────────────────────────────────────────────────────┘
Trajectory Maker
User's Guide 12
──────────────────────────────────────────────────────────
relative to the equatorial plane. It may be in the range,
0 ≤ i ≤ 180 degrees. If i ≤ 90 degrees, the maximum
latitude of the orbit's ground trace will be equivalent to
i, otherwise the maximum latitude will be its complement,
90 - i. Equatorial orbits have zero inclination, and
polar orbits have 90 degree inclination. This element is
illustrated in Figure 2.
Ω - Longitude of the ascending node
───────────────────────────────────
This element defines the orientation of the orbit plane
relative to the vernal equinox or, optionally, the initial
Greenwich meridian (see Section 3.3.2). Inclined orbits
intersect the equatorial plane in two points. The point
where the orbiting object intersects the equatorial plane
on its journey from south to north is the orbit's
ascending node. The other point is its descending node.
The line connecting the two nodes is the line of nodes.
The longitude of the ascending node is the angle, measured
in the equatorial plane, between the vernal equinox or the
initial Greenwich meridian and the line of nodes. It may
be in the range, 0 ≤ Ω < 360 degrees. This element is
illustrated in Figure 3.
w - Argument of perigee
───────────────────────
This element defines the orientation of the object's
perigee in the orbit plane relative to the ascending node.
It is the angle measured in the orbit plane between the
line of nodes and the line connecting the center of the
Earth and the object's radius of perigee. It may be in
the range, 0 ≤ w < 360 degrees. This element is also
illustrated in Figure 3.
τ - Time past perigee
─────────────────────
This element defines the object's initial location in the
orbit plane relative to the argument of perigee. It is
measured in seconds, and for elliptical orbits is normally
not greater than the orbital period. If the orbit is,
indeed, an ellipse, the program computes the orbital
Trajectory Maker
User's Guide 13
──────────────────────────────────────────────────────────
┌────────────────────────────────────────────────────┐
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└────────────────────────────────────────────────────┘
Trajectory Maker
User's Guide 14
──────────────────────────────────────────────────────────
┌────────────────────────────────────────────────────┐
│ │
│ Figure 3 │
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└────────────────────────────────────────────────────┘
Trajectory Maker
User's Guide 15
──────────────────────────────────────────────────────────
period and displays the value on your screen. This is
useful in making a choice regarding when to start the
orbit, and in the Time Options Menu, when to end the
orbit.
3.3.2 Initial Position and Velocity
Rectangular, Earth-Centered-Inertial (ECI) coordinates are
the fundamental coordinates used in Trajectory Maker for
measuring an objects motion, i.e., its position and
velocity. Indeed, the equations that govern the motion
are propagated relative to this reference frame. If the
orbital elements option is selected, they are transformed
into ECI coordinates prior to propagating the orbit. The
input format for these coordinates is illustrated in Menu
5 below.
┌──────────────────────────────────────────────┐
│ Initial Position and Velocity │
│ │
│ x 1404.72 y 845.78 z 3242.26 (nm) │
│ │
│ xd -0.959 yd -2.244 zd 2.436 (nm/sec) │
└──────────────────────────────────────────────┘
Menu 5. Initial Position and Velocity.
There are two options for the ECI coordinate system. One
option assumes that the principle x-axis lies in the
equatorial plane and is in the direction of the vernal
equinox, as illustrated in Figure 4.
The other option assumes that the principle x-axis lies in
the equatorial plane and is in the direction of the
Greenwich meridian at the time of orbit insertion and
remains fixed in that direction in space for the duration
of the trajectory. The Greenwich meridian, of course,
rotates relative to this frame. This option is useful for
positioning satellites for geographical viewing regions.
Your selection may be obtained with the Time Control
Parameters menu described in Section 3.4 below.
The ECI coordinate definitions are as follows:
Trajectory Maker
User's Guide 16
──────────────────────────────────────────────────────────
┌────────────────────────────────────────────────────┐
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Trajectory Maker
User's Guide 17
──────────────────────────────────────────────────────────
x - coordinate
──────────────
Position coordinate in the equatorial plane, positive in
the direction of the vernal equinox or, optionally,
initially aligned in the direction of the Greenwich
meridian.
y - coordinate
──────────────
Position coordinate in the equatorial plane orthogonal to
the x axis, and directed 90 degrees to the east so that
the system is right-handed.
z - coordinate
──────────────
Position coordinate coincident with the Earth's spin axis,
positive northwards.
xd - velocity component
───────────────────────
Velocity component in the x-direction.
yd - velocity component
───────────────────────
Velocity component in the y-direction.
zd - velocity component
───────────────────────
Velocity component in the z-direction.
Once the initial state vector has been entered, a message
is displayed relating the period of the orbit, an escape
condition or the Kepler flight time (no drag or oblate-
ness).
This latter condition occurs if the trajectory is likely
Trajectory Maker
User's Guide 18
──────────────────────────────────────────────────────────
to impact with the Earth. These parameters are provided
to aid you in your choice of the time parameters below.
3.4 Time Control Parameters
These parameters control the time span and precision of
the trajectory. The input format for these parameters is
illustrated in Menu 6.
┌──────────────────────────────────────────────┐
│ Time Control Parameters │
│ │
│ Date and time of injection: │
│ Year 1992 Month 2 Day 3 │
│ Hours 22 Minutes 27 Seconds 30 │
│ │
│ Mission duration time 86400 Seconds │
│ Propagation increment 300 Seconds │
│ Output frequency 60 Seconds │
└──────────────────────────────────────────────┘
Menu 6. Time Control Parameters.
The date and time of injection are used to define the
initial Greenwich sidereal time when the orbit begins.
The time must be input in Universal Time units (i.e.,
Greenwich Mean Time, or Zulu Time).
If these data are entered, the Greenwich sidereal time is
computed and the initial state vector is assumed to be
referenced relative to the vernal equinox.
If these data are not entered, the initial Greenwich
sidereal time is set to zero and the state vector is
assumed to be referenced relative to the Greenwich
meridian.
Year
────
Year the orbit begins. This item must be a four digit
number such as 1992.
If you enter zero or hit the enter key, the cursor skips
Trajectory Maker
User's Guide 19
──────────────────────────────────────────────────────────
to the hours' field, and the inertial reference frame is
assumed to be the aligned with the Greenwich meridian at
the initial time, t0, defined with the hours, minutes,
seconds data fields.
Month
─────
Month the orbit begins. This item must be a number with
no more than two digits, such as 1 or 12.
Day
───
Day the orbit begins. This item must be a number with no
more than two digits, such as 1 or 31.
Hours
─────
Hours past midnight when the orbit begins. This item must
be a number with no more than two digits such as 9 which
means 9:00 am, or 15 which means 3:00 pm.
Minutes
───────
Minutes past the hour when the orbit begins. This item
must be a number with no more than two digits such as 3 or
15.
Seconds
───────
Seconds past minutes when the orbit begins. This item
must be a number with no more than two digits such as 3 or
30.
Mission duration time (tf)
───────────────────────────
This parameter defines the time span of the orbit, and
Trajectory Maker
User's Guide 20
──────────────────────────────────────────────────────────
must be input in seconds.
If your application is a reentry trajectory and you are
using a perturbation model, the input value for the
mission duration should be larger than the Kepler flight
time. The Kepler flight time is shown to aid you in your
selection.
Propagation increment (dt)
───────────────────────────
This parameter is the propagation step-size (seconds). It
controls the precision of the computations, and should be
a multiple of the duration time above, and must also be in
seconds.
Output frequency (pt)
──────────────────────
You may want to propagate the trajectory at one step-size,
dt, to maintain accuracy, but output the data at some
larger multiple of dt. If you just press the return key,
with no data entry, the program will default this
parameter to the propagation increment above.
Numerical experience has shown that for non-drag perturbed
trajectories, a step size of 300 sec will give seven
significant figures accuracy for tf = 86400 sec (i.e. one
day), six figures for 14 days and five figures for 28
days. This latter case means that your results will be
accurate to five decimal digits after 8064 integration
steps.
To maintain stability and numerical integrity for reentry
trajectories, you will have to cut the step-size down to
10, or even 5, seconds, depending on the ballistic coeffi-
cient. You should not have to go below 1 second, unless
you are dealing with very light masses with ballistic
coefficients of the order of 1 lb/sq-ft or less.
Program execution time will vary depending on your choice
of the above time parameters. On a 12 MHz AT having a
math co-processor, for unperturbed orbits, assume 77 msec
per step, add 20 msec for each gravity harmonic, and add
62 msec per step for drag effects.
Trajectory Maker
User's Guide 21
──────────────────────────────────────────────────────────
3.5 Output Format Options Menu
You may choose one of four output format options from the
menu illustrated in Menu 7.
┌──────────────────────────────────────────────┐
│ Output Format Option │
│ │
│ 1. Geographic coordinates │
│ 2. Common radar coordinates │
│ 3. Geospherical inertial coordinates │
│ 4. Earth centered inertial coordinates │
│ │
│ Type option number and press Enter: _ │
└──────────────────────────────────────────────┘
Menu 7. Output Format Option.
3.5.1 Geographic Coordinates.
These coordinates are standard geographic coordinates that
are used for maps: latitude, longitude and altitude.
Geodetic altitude, h, and latitude, φ, are illustrated in
Figure 5.
The altitude of an object is its instantaneous height
above the Earth's surface. Since the Earth is modeled as
an oblate spheroid, the polar and equatorial radii differ
by about 10 nautical miles, thus, circular orbits usually
will not have constant altitude.
Latitude is the instantaneous angle between the equatorial
plane, and the line that contains the object and is normal
to the Earth's surface.
Longitude is the instantaneous angle between the Greenwich
meridian and the projection of the line connecting the
object with the Earth's center.
3.5.2 Common Radar Coordinates.
These coordinates, illustrated in Figure 6, are commonly
used for measurements at radar sites. They consist of the
object's range, azimuth and elevation relative to the
Trajectory Maker
User's Guide 22
──────────────────────────────────────────────────────────
┌────────────────────────────────────────────────────┐
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│ Figure 5 │
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└────────────────────────────────────────────────────┘
Trajectory Maker
User's Guide 23
──────────────────────────────────────────────────────────
┌────────────────────────────────────────────────────┐
│ │
│ Figure 6 │
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└────────────────────────────────────────────────────┘
Trajectory Maker
User's Guide 24
──────────────────────────────────────────────────────────
radar site coordinates.
Range, R, is the distance from the site to the object.
Azimuth, A, is the angle between the object and the radar
site's meridian, measured clockwise from the north.
Elevation, E, is the angle above or below the horizon.
.
Range rate, R, is the rate of change of range with respect
to time.
If you select this option, a menu appears requesting the
geographic coordinates of the radar site. Namely, its
geodetic latitude, longitude and altitude. It is tempting
to input altitude in feet or meters, but it must be in the
same units selected in the Unit Options Menu.
3.5.3 Geospherical Inertial Coordinates.
This output format contains the magnitude of the object's
radius vector, its right ascension and declination. It
also contains its inertial velocity magnitude, velocity
azimuth and flight-path angle. These parameters are
illustrated in Figure 7.
The magnitude of the radius vector is, r, is the distance
from the Earth's center to the orbiting object.
As used herein, the right ascension of the object is the
angle, α, measured in the equatorial plane between the
vernal equinox or the initial Greenwich meridian and the
projection of the radius vector onto the equatorial plane.
The declination of the object, δ, is the angle between the
equatorial plane and the radius vector. It is equivalent
to geocentric latitude.
The magnitude of the velocity vector, v, is the speed of
the object.
The velocity azimuth angle, σ, is the angle measured
clockwise from the northern direction to the projection of
the velocity vector on the local horizontal plane. (The
local horizontal plane is the plane normal to the radius
vector.)
Trajectory Maker
User's Guide 25
──────────────────────────────────────────────────────────
┌────────────────────────────────────────────────────┐
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└────────────────────────────────────────────────────┘
Trajectory Maker
User's Guide 26
──────────────────────────────────────────────────────────
The flight-path angle, Γ, is the angle between the
velocity vector and the local horizontal plane.
3.5.4 Earth-Centered-Inertial Coordinates.
This output format contains the object's rectangular,
Earth-centered-inertial position, velocity and accel-
eration components in double precision. Their definitions
are given in Section 3.3.
Once you have made your choice for output format, you are
given the choice of either text or graphics output. If
you just want the data file, choose the text format. If
you choose graphics format, you will get both text and
graphics output. The graphics display is in geographic
format regardless of your choice of output format.
3.6 Menu Editing
You may change the value of a data item in any data entry
field without exiting the program with the Tab key or
Shift-Tab key sequence.
Pressing the Tab key moves the cursor to the next data
field. Pressing the Shift-Tab sequence (or the Esc key)
moves the cursor to the previous field.
For example, if you entered a data item that you wish to
change, simply move to the field that contains the data
item, reenter the new number and press <Enter>. Do not
try to edit the data item with the Arrow, Ins or Del keys.
The data in a data field must be reentered. You can, of
course, use the Backspace key.
If you move to a field that contains data that you wish to
keep, skip over it with the Tab key or Shift-Tab key. Do
not press <Enter> if you want to keep a previously entered
data item, as this has the effect of setting the data item
to zero. Press <Enter> only when you enter a data item or
you want a data item to have the value zero.
You may return to a previous menu by using the Esc key
whenever your cursor is in the first data field of the
current menu. If you press this key when the current menu
is the Units Option Menu, the job is aborted. You can
always abort a job with the Ctrl-Z sequence.
Trajectory Maker
User's Guide 27
──────────────────────────────────────────────────────────
If you are satisfied with your entries you can move to the
next menu with the Tab key.
Trajectory Maker
User's Guide 28
──────────────────────────────────────────────────────────
4. TRAJECTORY MAKER OUTPUT DESCRIPTION
The output data is in ASCII format and is contained in two
files. For convenience, one file, HOST.HED, echoes your
input data. The other file, HOST.DAT, contains your
selected output format from the Output Format Options
menu. You may wish to rename these files, and save them
for use in the companion program, Trajectory Scape.
The first record in HOST.DAT contains information used by
Trajectory Scape. Just delete this record, and you can
plot any desired data items with your favorite plotting
package.
If you selected Geographic Coordinates, HOST.DAT contains
four data columns: time, geodetic latitude, longitude and
altitude, in that order. These parameters are described
in Section 3.5.1.
If you selected Common Radar Coordinates, HOST.DAT
contains five data columns: time, range, azimuth,
elevation and range rate, in that order. These parameters
are described in Section 3.5.2.
If you selected Geospherical Inertial Coordinates,
HOST.DAT contains seven data columns: time, radius, right
ascension, declination, speed, velocity azimuth and flight
path angle, in that order. These parameters are described
in Section 3.5.3.
If you selected Earth-Centered-Inertial coordinates,
HOST.DAT contains ten data columns: time, three position,
three velocity and three acceleration components, in that
order. The three components represent the x, y, z
directions. These parameters are described in Section
3.3.2.
In addition to the numeric data, you have the option of
displaying the trajectory's ground trace on your monitor
while it is being computed. The graphics format displays
three conformal views of the trajectory.
(1) A Mercator projection.
(2) A stereographic projection of the northern
hemisphere.
Trajectory Maker
User's Guide 29
──────────────────────────────────────────────────────────
(3) A stereographic projection of the southern
hemisphere.
Conformal projections are useful because angles are
preserved under the mapping. Therefore, the relative
shapes of objects are preserved. Other projections may be
viewed with Trajectory Scape (see Section 6). A sample of
the graphics display is shown in Figure 8.
At the top of the screen there is a chart that displays
the altitude trend as it is being computed. This is a
very useful display, for you can immediately correlate
altitude with location.
The actual numeric altitude values that are being computed
are displayed in a text window at the bottom of the
screen. Time is also displayed in this window in sexa-
gesimal format, that is, days:hrs:mins:secs:
You may pause the visual display by pressing any key
except the Esc or the Ctrl-Z sequence. Either of these
keys will abort the job and return the system to DOS.
When the computations have been completed you may change
the color palette to black and white. This option is
included so that a readable screen dump may be obtained on
a black and white printer. It should be noted that
standard screen dumps give a distorted image on some
printers. However, there are several commercial products
on the market that will provide color and distortion-free
prints.
Trajectory Maker
User's Guide 30
──────────────────────────────────────────────────────────
┌────────────────────────────────────────────────────┐
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│ Figure 8 │
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└────────────────────────────────────────────────────┘
Trajectory Maker
User's Guide 31
──────────────────────────────────────────────────────────
5. EXAMPLE ORBITS AND TRAJECTORIES
This section provides initial conditions for various
trajectories that Trajectory Maker is capable of predict-
ing.
5.1 Space Shuttle Discovery Mission STS-42
The space shuttle Discovery was launched on Wednesday,
January 22, 1992. It was inserted into orbit at
18:52:32.960 Greenwich Mean Time with the following
orbital elements, obtained from L. B. J. Space Center in
Houston, Texas.
a = 3603.1925 nautical miles (nm)
e = 0.000626
i = 56.74595°
Ω = 242.95604°
w = 65.48579°
τ = 2781.72794 sec
The following drag related parameters were also given:
Weight = 231448.0 lb
Drag Coefficient = 2.0
Effective drag area = 2750.0 ft²
The ballistic coefficient is, therefore,
ß = 231448/(2 X 2750) = 42.08 lb/ft²
Try a gravity model of Order 2 with atmospheric drag
effects (Model 4) with the above parameters and the
following time parameters:
Year: 1992 Month: 1 Day: 22
Hour: 18 Minutes: 52 Seconds: 32.96
Mission duration (tf): 86160 sec (one sidereal day)
Propagation increment (dt): 60 sec
Output frequency (pt): 60 sec
5.2 Repeating Ground Trace
If a satellite's period is a multiple of the time it takes
for the Earth to make a complete rotation on its axis (23
Trajectory Maker
User's Guide 32
──────────────────────────────────────────────────────────
hrs 56 min or 86160 seconds), then the satellite will
retrace the same path over the Earth as it did initially.
The following parameters define an orbit whose ground
trace repeats after 15 revolutions. The period is 5743.84
seconds.
a = 3743 nm
e = 0
i = 60°
Ω = 0°
w = 0°
τ = 0 sec
Try an unperturbed gravity model (Model 1) with these
elements and the following time parameters:
t0 = 0 sec
tf = 172320 sec (two sidereal days)
dt = 60 sec
pt = 60 sec
Repeat this case with gravity model of order two (Model
2). The ground trace will be displaced a few degrees from
the unperturbed case, demonstrating the effect of a non-
spherical Earth.
5.3 Sun-synchronous Orbit
An orbit in which a satellite passes over the same part of
the Earth at the same time each day is a Sun-synchronous
orbit. The orientation of the orbit plane will remain
nearly fixed relative to the Sun as the Earth moves in its
orbit.
Such an orbit is achieved by adjusting its orbital
elements so that the regression rate of the line of nodes
is just that of the Earth's angular rate about the Sun,
which is about 1° each day (0.9856 deg/day).
The above orbit becomes Sun-synchronous if the inclination
is elevated from 60° to 97.61°. The regression rate may
be inspected in the output file, HOST.HED. Also note that
the orbit is a retrograde orbit.
Trajectory Maker
User's Guide 33
──────────────────────────────────────────────────────────
5.4 Candidate Surveillance Orbits
The next few examples generate repeating ground traces
that are especially good candidates for surveillance
and/or communication. The orbital elements are adjusted
such that apogee is in the northern hemisphere, where
they will dwell the longest.
Their inclination is critical (63.44°) so that gravity
will not cause their line of apsides to rotate (so that
apogee will stay put). This can be verified by inspecting
the file HOST.HED. A value of zero, or nearly so, should
be given for the argument of perigee rate.
Also, the longitude of the ascending node and time past
perigee can be adjusted to synchronize multiple orbits of
these types for various satellite constellation considera-
tions. For example, you could try Ω = 0°, 90°, 180°, 270°.
Tundra Orbits (24 hour period)
──────────────────────────────
a = 22766 22766 22766 22766 nm
e = 0.33 0.69 0.69 0.69
i = 63.442° 63.44° 63.44° 63.44°
Ω = 0.00° 0.00° 0.00° 0.00°
w = 270.00° 270.00° 290.00° 300.00°
τ = 0 0 0 0 sec
Model 1: Unperturbed Gravity Model
t0 = 0, tf = 86160, dt = 300, pt = 300 (sec)
Molniya Orbits and Hookers (12 hour period)
───────────────────────────────────────────
a = 14342 14342 14342 14342 nm
e = 0.69 0.69 0.69 0.69
i = 63.44° 63.44° 63.44° 63.44°
Ω = 0.0° 0.0° 0.0° 0.0°
w = 270° 90° ± 290° ± 310° (etc)
τ = 0.0 0.0 0.0 0.0 sec
Model 1: Unperturbed Gravity Model
t0 = 0, tf = 86160, dt = 120, pt = 120 (sec)
Trajectory Maker
User's Guide 34
──────────────────────────────────────────────────────────
An Eight-hour orbit
───────────────────
a = 10945 nm
e = 0.594
i = 63.44°
Ω = 0°
w = 270°
τ = 0 sec
Model 1: Unperturbed Gravity Model
t0 = 0, tf = 86160, dt = 120 pt = 120 (sec)
5.5 An Example Illustrating Orbit Decay
a = 6478 km
e = 0
i = 45°
Ω = 0°
w = 0°
τ = 0 sec
Model 3: Drag perturbation model with ß = 100 kg/m²
t0 = 0, tf = 2000, dt= 5, pt= 5 (sec)
5.6 Orbits Near Transition Region
The next three examples illustrate orbits all having the
same radius of perigee near the transition region from a
closed to an open orbit.
Elliptical Orbit Parabolic Orbit Hyperbolic Orbit
──────────────── ─────────────── ────────────────
a = 40000 nm 4000 nm 40000 nm
e = 0.9 1.0 1.1
i = 80° 80° 80°
Ω = 180° 180° 180°
w = 0° 0° 0°
τ = 0 sec 0 sec 0 sec
Model 1: Unperturbed Gravity Model
t0 = 0, tf = 200660, dt = 120 pt = 120 (sec)
Trajectory Maker
User's Guide 35
──────────────────────────────────────────────────────────
5.7 Ballistic Missile Trajectories
The remaining set of trajectories are defined with initial
state vectors given in ECI coordinates (Initial State
Vector Option 2). These trajectories exit and reenter the
Earth's atmosphere on a predefined target destination.
5.7.1 Western Test Range Examples
These two hypothetical trajectories are launched from
Vandenberg Air Force Base with destination in the
Kwajalein Atoll.
X = -1705.16 nm
Y = -2509.17 nm
Z = 1926.63 nm
XD= -3.102 nm/sec
YD= -0.525 nm/sec
ZD= -0.483 nm/sec
Model 4: Gravity model order 2 and ß = 25 #/sq-ft.
t0 = 200, tf = 2000, dt = 5, pt = 5 (sec)
X = -2525.03 nm
Y = -2146.25 nm
Z = 2012.86 nm
XD= -2.608 nm/sec
YD= 1.351 nm/sec
ZD= -0.318 nm/sec
Model 4: Gravity model order 2 with ß = 75 #/sq-ft.
t0 = 475, tf = 2000, dt = 5, pt = 5 (sec)
5.7.2 Hypothetical Threat Trajectories
The next three trajectories are hypothetical Soviet
launched ICBMs with Continental USA destination targets.
Trajectory Maker
User's Guide 36
──────────────────────────────────────────────────────────
A Nominal Trajectory (Reentry angle 24 degrees)
───────────────────────────────────────────────
X = 1404.72 nm
Y = 845.78 nm
Z = 3242.26 nm
XD = -0.959 nm/sec
YD = -2.244 nm/sec
ZD = 2.436 nm/sec
Model 4: Gravity model order 2 and ß = 1200 #/sq-ft.
t0 = 0, tf = 1600, dt = 10, pt = 10 (sec)
A Depressed Trajectory (Reentry angle 20.17 degrees)
────────────────────────────────────────────────────
X = 1149.67 nm
Y = 1075.78 nm
Z = 3348.70 nm
XD = 1.007 nm/sec
YD = -2.889 nm/sec
ZD = 1.719 nm/sec
Model 4: Gravity model order 2 and ß = 1200 #/sq-ft.
t0 = 0, tf = 1600, dt = 10, pt = 10 (sec)
A Lofted Trajectory (Reentry angle 34.91 degrees)
─────────────────────────────────────────────────
X = 980.98 nm
Y = 1218.02 nm
Z = 3523.47 nm
XD = -0.393 nm/sec
YD = -1.939 nm/sec
ZD = 2.787 nm/sec
Model 4: Gravity model order 2 and ß = 1200 #/sq-ft.
t0 = 0, tf = 2400, dt = 5, pt = 5 (sec)
Trajectory Maker
User's Guide 37
──────────────────────────────────────────────────────────
6. TRAJECTORY SCAPE INPUT DESCRIPTION
Trajectory Scape is a multiple trajectory viewing tool.
Once you have created one or more geographic output files
with trajectory Maker (Output option 1), you may view any
or all of them with Trajectory Scape. At the DOS prompt
type TSCAPE. You will be presented with a series of three
menus that simply request choices for map projections,
their resolution and central focus region.
6.1 Word Map Projections Options Menu
You are given the option of three world map projections.
Actually, there are five, but three projections are
combined in one display. These options are illustrated in
Menu 8 below.
┌──────────────────────────────────────────────┐
│ World Map Projections │
│ │
│ 1. Mercator with Stereographic │
│ 2. Hammer-Aitoff Equal Area │
│ 3. Mollweide Equal Area │
│ │
│ Type option number and press Enter: _ │
└──────────────────────────────────────────────┘
Menu 8. World Map Projections.
6.1.1 Mercator with Stereographic Projections.
A Mercator projection is displayed with stereographic
projections of both the northern and southern hemisphere.
These projections are perhaps the most useful of all map
projections for navigation. They have borne the test of
time.
They are conformal projections, that is they preserve
angles, so that relative shapes remain the same. Rhumb
lines are straight on the Mercator projection. Distor-
tions are maximal in the polar regions of the Mercator
projection, but minimal with the stereographic projec-
tions. The stereographic projections also preserve
circles. The combination compliments one another.
Trajectory Maker
User's Guide 38
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Examples of these projections are illustrated in Figure 9,
Section 7.
6.1.2 Hammer-Aitoff Equal Area Projection.
This projection preserves areas and is best near the
center of the map. It is similar to the Mollweide
projection described next, but it has less distortion in
the polar regions. Far east and west are not as
distorted. This projection is useful for atlases on
physical geography or for statistical distribution
purposes. It is used in astronomical work for showing the
distribution of stars. An example of this projection is
illustrated in Figure 10, Section 7.
6.1.3 Mollweide equal area projection.
This projection also preserves areas. It is also known as
Babinet's equal surface projection. The distortions
increase as you move away from the center, horizontally or
vertically. Its major use is for geographic illustrations
relating to area such as density and populations or extent
of forests, etc. It serves somewhat the same purpose as
the Hammer-Aitoff projection. An example of this projec-
tion is illustrated in Figure 11, Section 7.
6.2 Map Resolution Options Menu
You are given three map resolution options illustrated in
Menu 9 below. Your trade-off is between speed and detail.
┌──────────────────────────────────────────────┐
│ Map Resolution Options │
│ │
│ 1. Fine ..... Spiffy but slow. │
│ 2. Medium ... Good and faster. │
│ 3. Course ... Fair and fastest. │
│ │
│ Type option number and press Enter: _ │
└──────────────────────────────────────────────┘
Menu 9. Map Resolution Options.
Trajectory Maker
User's Guide 39
──────────────────────────────────────────────────────────
6.2.1 Fine Option.
This option gives the best detail but is the slowest of
the three. The data file contains approximately 4500
latitude longitude pairs.
6.2.2 Medium Option.
This option gives a good presentation and is about twice
as fast as the fine option. The data file contains
approximately 2000 latitude/longitude pairs. This option
is used in Trajectory Maker.
6.2.3 Coarse Option.
This option gives fair presentation and is the fastest -
about twice as fast as the medium option. The data file
contains approximately 1000 latitude/longitude pairs.
6.3 World Map Focus Options Menu
The final menu, illustrated in Menu 10, allows you to
focus on one of three regions of the world, provided that
you selected one of the equal area projections.
┌──────────────────────────────────────────────┐
│ Word Map Focus │
│ │
│ 1. Greenwich Meridian │
│ 2. American Region │
│ 3. Soviet Region │
│ │
│ Type option number and press Enter: _ │
└──────────────────────────────────────────────┘
Menu 10. World Map Focus.
6.3.1 Greenwich Meridian.
The central meridian is the Greenwich meridian. The
center of the map is at 0 degrees longitude.
Trajectory Maker
User's Guide 40
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6.3.2 American Region.
The central meridian is at 80 degrees west longitude which
puts the focus of the map between North and South America.
6.3.3 Soviet Region.
The central meridian is at 80 degrees east longitude which
puts the focus of the map in central Russia.
Trajectory Maker
User's Guide 41
──────────────────────────────────────────────────────────
7. TRAJECTORY SCAPE OUTPUT DESCRIPTION
If either the American or Soviet region is selected, the
displayed map will have a yellow and a red meridian. The
yellow meridian represents the Greenwich meridian and the
red meridian represents the international date line.
These two meridians provide a reference base. The
graticule is 20 by 20 degrees and is not labeled to avoid
clutter.
Once the map has been generated, the command window at the
bottom of the display asks for the name of a geographic
trajectory file to display. Simply type the name of your
file, and the program will display the orbit or
trajectory. If you store your data files in another
directory, give the complete path specification. Once
plotted, you will be asked if you would like to view
another trajectory. The program will continue until you
answer no.
To aid in discrimination, multiple trajectories are
displayed in different colors, a maximum of five. You may
display more trajectories, however, you are limited to
five colors.
Examples of the various projections are illustrated in
Figures 9, 10 and 11. Figure 9 illustrates four "tundra"
orbits (Section 5.4) on the conformal projections.
Figures 10 and 11 illustrate the circular, 45 degree orbit
that you generated in Section 2, on Hammer-Aitoff and
Mollweide projections, respectively.
Trajectory Maker
User's Guide 42
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┌────────────────────────────────────────────────────┐
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Trajectory Maker
User's Guide 43
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┌────────────────────────────────────────────────────┐
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└────────────────────────────────────────────────────┘
Trajectory Maker
User's Guide 44
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┌────────────────────────────────────────────────────┐
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Trajectory Maker
User's Guide 45
──────────────────────────────────────────────────────────
8. ADDITIONAL VIEWING CAPABILITY
In addition to trajectory ephemerides, you may construct
and view circles, segmental arcs and spherical polygons.
Because of their area preserving property, the Mollweide
and Hammer-Aitoff projections come in handy for these
kinds of figures. Also, recall that stereographic projec-
tions preserve circles.
8.1 Small Circles
Any plane intersecting the Earth but not passing through
its center will cut its surface in a small circle. Small
circles are useful for marking regions of interest such as
communication regions or satellite sensor coverage
regions.
To construct and view small circles on the various map
projections, when prompted for the file name in the
command window, press the F7 key.
Data fields are presented for the geographic coordinates
of the circle's center and its radius. Enter the circle's
latitude (Lat), longitude (Lon) and its radius (Rad) in
degrees. Once the circle is displayed, you will be asked
if you wish to view another file. If you do, type y for
yes and repeat the above process.
8.2 Segmental Arcs and Spherical polygons
Any plane passing through the center of the Earth cuts its
surface in a great circle. A great circle arc or arc is
defined by two points on a great circle. A segmental arc
is constructed from finite number arcs joined in order in
such a manner that the initial point of each coincides
with the terminal point of the preceding arc. If the
initial and terminal points of a segmental arc coincide,
it is said to be a spherical polygon (e.g. a spherical
triangle).
Segmental arcs and polygons are useful for illustrating
minimal distances or for bounding certain geographic
regions such as a missile threat complex, missile
corridors and abort regions.
To construct and view arcs, segmental arcs or spherical
Trajectory Maker
User's Guide 46
──────────────────────────────────────────────────────────
polygons (arc segments of parallels of latitude may also
drawn), when prompted for the file name in the command
window, press the F8 key.
Data fields are presented for the geographic coordinates
of the first point. Input the point's latitude and
longitude in degrees.
Data fields are then presented for the next point. After
you have entered the next point, the arc connecting the
two points is drawn. You may continue constructing arcs
in this manner, i.e. by entering the coordinates of
successive points, to form a segmental arc or spherical
polygon. When your figure is complete, press the F8 key
or the Esc key.
You will be asked if you wish to view another file. At
this time you may again press the F7 key for a circle, or
the F8 key for a segmental arc or simply type the name of
an orbit file, e.g. HOST.DAT, that you have previously
constructed with Trajectory Maker.
Trajectory Maker Historical Notes
User's Guide A-1
──────────────────────────────────────────────────────────
APPENDIX A. HISTORICAL NOTES
In early 1976 while employed under the auspices of the
Western Test Range at Vandenberg Air Force Base, Calif-
ornia, the author was assigned the task of researching the
existing trajectory related computer programs in use on
the range, and assess the relative accuracy of their
predicted impact points based on the integrity of their
numerical integration schemes. Initially, the study was
not to be concerned with the force model, but limited to
the integration process itself.
At that time, the author found that there existed at least
eleven distinct programs lying around for propagating
reentry trajectories. Namely, VPAR/BIPS, MIPS/AMARKIN,
FFTRG2, RIP, F&G, ARMS, OASYS, FIT500, TGP, NITE and TAPP
(FIAT). These programs were written in FORTRAN for the
following computers: IBM 7094, IBM 360/65, SIGMA 5/7, CDC
6400/6600, UNIVAC 1108, GE 635.
Their numerical integration schemes covered a fair gamut:
trapezoidal, Taylor series, fixed and variable step
fourth-order Runge-Kutta and Runge-Kutta-Gill, fourth-
order Adams-Moulton, eighth-order Gauss-Jackson, variation
of parameters and f & g series. Moreover, the force
models of the various programs varied.
For example, all of the models included at least two zonal
harmonic coefficients, in either Jeffrey's or Vinti's,
potential function. One potential function included six
zonal harmonics, and another included nine zonal harmonics
and thirty-six tesseral harmonics. All models included
atmospheric drag but one used the U.S. 1962 Standard
Atmosphere, while another used ARDC 1959. There also
existed nonuniformity in the constant parameters.
All of the models have their merits and limitations, but
when it comes to computational precision, each of the
things above must be considered. A tenth of a foot
difference in the Earth's equatorial radius may cause
unknown output differences. For example, when comparing
the output of two models, one may observe differences in
the ninth or tenth decimal place, which could be due to
computational precision, or some parameter difference,
among other things. One must be able to isolate the
source of any discrepancy.
But the primary thrust of the study was to evaluate the
Trajectory Maker Historical Notes
User's Guide A-2
──────────────────────────────────────────────────────────
numerical integration process as applied to reentry
trajectories. Consequently, only a representative force
model was required. One that would exhibit the underlying
behavior, when the differential equations of motion were
integrated. This model should at least contain a few
gravity harmonics, and more importantly, a representative
atmospheric drag model, which is what really taxes the
integrator.
The problem then, became one of isolation. A baseline
integrator was required, whose precision was known with
certainty, in order to compare the integrators. Also, a
representative force model was required that could,
without too much difficulty, be substituted into the
various trajectory programs, including the baseline model.
The force model didn't need to be all that exotic, just
representative of the kinds of forces the differential
equations would experience.
The gravity model chosen for the baseline model was the
one developed by R.M.L. Baker at Computer Sciences
Corporation for use in their f & g series approach. The
reason for this was that their program was one of interest
to some range people, and their f & g series approach was
based on high order derivatives of the gravity potential
function, which were embedded in the program code. This
model included the zonal gravity harmonics of the
Vinti-potential of order six.
The drag force model selected was based on the abbre-
viated, preliminary tables for the U. S. 1976 Standard
Earth Atmosphere. The reason for this was that most of
the atmospheres used in the various models were lengthy
tables. The abbreviated tables contained just 21 pairs of
altitude versus density, which were more manageable, and
could be accurately modeled with cubic spline functions.
In addition, the model was the latest and greatest.
The numerical integration scheme chosen for the baseline
integrator was the eighth-order, fixed-step, Runge-Kutta-
-Shanks' technique having twelve stages. This scheme is
stable, and was the most accurate scheme that the author
knew of at the time.
A fixed-step scheme was desirable, because one could
determine precisely the accuracy of the computations.
This could be done by selecting a step-size, integrating
the differential equations to a predefined time, cutting
Trajectory Maker Historical Notes
User's Guide A-3
──────────────────────────────────────────────────────────
the step-size in half, repeating the integration and
comparing the two trajectories. This process could be
continued until two successive trajectories matched in
their fifteenth or sixteenth significant figures (the pre-
cision of double precision arithmetic on the IBM 360/65).
Clearly, computation was of no concern - just accuracy.
Once the baseline force model together with all the
relevant constants was substituted into the other
trajectory programs, the accuracy of their integrators
could be determined with a fair amount of certainty.
Certainty could be claimed if it wasn't for the fact that
the various hardware on which the software resided had
some differences in precision.
It was soon discovered that the baseline model was
producing rather accurate trajectories. In fact, by
shifting density models it was proven that the USAF
"certified" program, FFTRG2, could miss impact points by
as much as two nautical miles, just because of the manner
in which it was interpolating the density profile.* More-
over, its interpolation scheme, used presumably for compu-
tation speed, actually slowed the integrator down by
causing the step-control to artificially decrease the
step-size.
Considerations such as these led to many enhancements of
the force model in the baseline program. The final
version included the following:
o Six zonal gravity harmonic coefficients.
(Smithsonian 1973 II values)
o U. S. 1976 Standard Atmosphere.
Fitted with cubic splines to the logarithm of
the density profile.
o Drag coefficient as a function of either
altitude, or Mach number. Fitted with cubic
splines or linear depending on the profile.
─────────────────────
*Ivy, L. H. and H. B. Reynolds, On the Sensitivity of
Impact Prediction to Methods of Interpolating Atmospheric
Density, ITT Federal Electric Corporation, Technical Note
TN-76-1463, Vandenberg Air Force Base, CA, Dec 22, 1976.
Trajectory Maker Historical Notes
User's Guide A-4
──────────────────────────────────────────────────────────
o Accurate speed of sound computation based on the
1976 temperature profile.
o Atmospheric wind profiles. Fitted with cubic
splines.
o Innovative altitude computation which produces
trigonometric functions for wind effects, and
the topocentric rotation matrix as a by product.
The baseline integrator, i.e. the Runge-Kutta-Shanks'
fixed-step scheme, proved to be very accurate. To speed
things up, a dual mesh step control option was added.
What started out to be an accuracy study for numerical
integration schemes of various Western Test Range
trajectory programs ended up being another trajectory
program - a rapid, high precision reentry trajectory
generator. And that program is the basis of Trajectory
Maker.
Later, the author joined another aerospace company, and
shortly thereafter, an assignment required a trajectory
program. The company's program (actually, their program,
RIPR, a variation of RIP, was bootlegged from Vandenberg)
would go unstable for certain trajectories when they
entered the atmosphere. This led to a conversion of the
baseline program to run on their in-house PDP-11. The
baseline program was then used for validating Space
Shuttle ephemerides, and with its use an error in launch
azimuth was uncovered in a NASA simulation program.
Another aerospace company had spacecraft computers
in-house for use in verifying, validating, and testing of
the flight software for a particular USAF satellite
system. In addition they had a separate computer for
simulating the spacecraft's environment.
Several problems were manifested that required the use of
an independent trajectory simulation. The baseline
program was again modified for use on their in-house
Perkin Elmer 8/32 computer. This time, all of the drag
stuff was stripped out of the program, and emphasis was
placed on the integrator and gravity model. Clever closed
form expressions were developed and implemented for the
gravity model so that it would include an arbitrary number
of harmonics with rapid computation times.
Trajectory Maker Historical Notes
User's Guide A-5
──────────────────────────────────────────────────────────
The program was used for several applications including
another satellite program. The baseline program was
further vindicated when it matched orbits generated with
the Advanced Orbit Ephemeris Subsystem in use at the
Satellite Control Facility at Sunnyvale, California.
The program was also used for Strategic Defense Initiative
applications. In particular, it was used to propagate
thousands of missile threat trajectories into the field of
view of special sensors. It was also used to study the
following:
o A Space based kinetic energy weapon interceptor
system.
o Evasive enemy spacecraft orbit maneuvers in the
Earth's shadow.
o Multi-satellite constellation design for a
special sensor.
o Drag effects on a space based rotating radar
antenna system.
o Relative motion for a space laser targeting
experiment.
Having reviewed the merits and limitations of the baseline
program relative to the many applications in which it has
been involved, and modifications it has undergone, the
author decided to redo the entire program from a different
perspective, and in a different language.
There exists a number of software tools available in the
market today that are neat programs for orbital mechanics
and astrodynamics applications, and several were purchased
at one of the recent companies where the author was an
employee. But the fact of the matter is, all of them were
looked at briefly, and put on a shelf, for none of them
did what was needed to be done.
Invariably, it was the case, that a special tool needed to
be developed to solve the problem at hand. Moreover, it
was usually the case that individual trajectories needed
to be generated and plotted (usually plotted by hand) in
order to shape the underlying problem. The most useful
tool has been the capability to generate a trajectory, of
varying degrees of complexity, depending in the applica-
Trajectory Maker Historical Notes
User's Guide A-6
──────────────────────────────────────────────────────────
tion, and see what it looked like.
Trajectory Maker is the result of selecting and adding
those features that the author has found to be the most
useful in many aerospace applications requiring the use of
a trajectory simulation.
Trajectory Maker Glossary
User's Guide B-1
──────────────────────────────────────────────────────────
APPENDIX B. GLOSSARY
This appendix contains definitions of terms used in this
User's Guide. Although many of these terms are applicable
for other celestial bodies, the definitions herein are
given in reference to planet Earth. For example, we use
perigee rather than periapsis.
Altitude The height of an object above the surface of the
Earth.
Apogee The point on an elliptical orbit farthest from the
Earth's center.
Argument of perigee The angular distance measured in the
orbit plane in the direction of motion from the line of
nodes to the line of apsides.
Ascending node The node within the equatorial plane
through which an object passes from South to North.
Azimuth The angular measure in the plane of the horizon
from the north point on the horizon clockwise to the
object.
Ballistic coefficient The mass of an object divided by
the product of its drag coefficient with its effective
cross sectional area.
Ballistic trajectory A trajectory that is in free-flight,
having no power applied to it, that intersects the surface
of the Earth.
Celestial sphere An imaginary sphere with the Earth at
its center, on which all objects in the sky seem to lie.
Conformal projection A projection of a sphere onto a
plane that preserves angles.
Conic section One of three curves formed by the inter-
section of a plane with a cone. They are: (1) an ellipse;
(2) a parabola; (3) an hyperbola.
Critical inclination The inclination of an orbit where
the perigee point neither advances or regresses. This
occurs at 63.44 degrees.
Trajectory Maker Glossary
User's Guide B-2
──────────────────────────────────────────────────────────
Declination The angular distance of an object North or
South of the equator.
Descending node The node within the equatorial plane
through which an object passes from North to South.
Differential equation A mathematical equation involving
derivatives, e.g. the equations of motion.
Drag The force acting on an object due to the Earth's
atmosphere. Drag acts in a direction opposite to that of
the objects motion relative to the atmosphere.
Drag coefficient A dimensionless coefficient used in
characterizing the atmospheric drag force. A good number
is 2.2.
Eccentricity A parameter that characterizes a conic
section: ellipse, parabola, hyperbola.
Ecliptic The path that the Sun appears to follow around
the celestial sphere each year. The ecliptic approximates
a great circle inclined about 23.5 degrees to the celes-
tial equator.
Elevation The angle an object is above or below the
horizon.
Ellipsoid A surface of revolution generated by an
ellipse.
Ephemeris A table of predicted positions for a celestial
body. As used herein, the table may include velocities
and accelerations as well.
Escape velocity The speed at which an object will just
break free from the gravitational pull of the Earth. The
value is about 11 km/sec. The path is a parabola or
hyperbola depending on the precise value.
Flight path angle The angle between the inertial velocity
vector an the local horizontal plane.
Geocentric latitude The angle at the center of the Earth
between the radius through a given place or object and the
equatorial plane.
Trajectory Maker Glossary
User's Guide B-3
──────────────────────────────────────────────────────────
Geodetic latitude The angle between a normal to the
reference ellipsoid at the point and the equatorial plane.
Graticule The network of lines of latitude and longitude
upon which a map is drawn.
Gravitational Harmonics Terms appearing in the gravita-
tional potential function (zonal, sectorial and tesseral)
Great circle Any circle on the celestial sphere that has
the Earth at its center.
Greenwich Mean Time (GMT) Mean solar time on the
Greenwich meridian. Also known as Universal Time (UT) or
Zulu time (Z). It may be determined from local zone time
by adding (when West) or subtracting (when East)
longitude.
Greenwich meridian The zero meridian from which geograph-
ical longitude is measured, passing through the Greenwich
observatory, England ( also called the prime meridian).
Ground trace The path on the surface of the Earth traced
by an orbiting object's subvehicle point, i.e., the point
directly below the object. That is, the path along which
an object passes directly overhead as seen from the sur-
face of the Earth.
Hammer-Aitoff projection A map projection that preserves
area. Its graticule is constructed from curved parallels
and meridians which reduce distortions near the edge of
the map. First described by Hammer, a professor of
surveying at Stuttgart, in 1892 following an idea of
Aitoff's.
Hour angle The angle between the observer's meridian and
the hour circle that passes through the object.
Hour circle A great circle that passes through an object
on the celestial sphere and the celestial pole.
Inclination The angle between the orbit plane and the
equatorial plane.
Injection The act of placing an object on a calculated
trajectory.
Trajectory Maker Glossary
User's Guide B-4
──────────────────────────────────────────────────────────
Insertion The act of placing an object into orbit around
the Earth.
Kepler flight time The time of flight for a ballistic
missile, describing an elliptical orbit, to impact with
the Earth's surface.
Keplerian motion Motion that obeys Kepler three laws: (1)
The orbit of each planet is an ellipse with the Sun at one
focus. (2) The radius vector joining the Sun to a planet
sweeps over equal areas in equal intervals of time. (3)
The ratio of the squares of the periods of any two planets
is equal to the ratio of the cubes of their mean distances
from the Sun.
Latitude The angular distance north or south of the
equator. See geodetic latitude above.
Line of apsides A line connecting the perigee and apogee
points.
Line of nodes The intersection of the equatorial plane
and the orbit plane.
Local horizontal The plane normal to the radius vector
that contains the object.
Longitude The angle between the Greenwich meridian and
the meridian through a place or object, measured East or
West. Geocentric longitude is the same as geodetic
longitude.
Longitude of the ascending node The angular distance from
the vernal equinox measured eastwards in the equatorial
plane to the point of intersection of the orbit plane
where the object crosses from South to North.
Mercator projection A conformal map projection introduced
by Gerhard Mercator in 1569. Loxodromes ( rhumb lines) on
the sphere are mapped into straight lines on the map.
Mercator's projection is the only one that does this.
Meridian A great circle passing through the North and
South poles.
Mollweide projection A map projection that preserves
area. The graticule is constructed from parallels that
are unequally spaced straight lines and elliptical
Trajectory Maker Glossary
User's Guide B-5
──────────────────────────────────────────────────────────
meridians. The central meridian is a special case where
the ellipse degenerates into a straight line. The 90 th
meridian is the other special case where the ellipse
becomes a circle. First described by Karl Mollweide, a
Mathematician and Astronomer, in 1805.
Nodes The points of intersection of an orbit with and the
equatorial plane.
Object A body, satellite, spacecraft, missile, etc.
Orbit The path of one body in space around another.
Orbital element Six quantities that describe the orbit of
a body in space. They are: (1) the semimajor axis; (2)
eccentricity; (3) inclination; (4) longitude of the
ascending node; (5) argument of perigee; (6) the time of
perigee passage.
Oblate spheroid A surface generated by rotating an
ellipse about its semiminor axis.
Perigee The point in an orbit nearest the Earth's center.
Period The time required for one complete circuit of an
orbit.
Perturbation Deviation from exact reference motion (e.g.
conic section) caused by gravity anomalies or other
forces.
Potential function At a point, the work required to
remove a unit mass from that point to infinity.
Radius vector The vector emanating from the center of the
Earth to the orbiting object.
Range The distance from a radar site to an object.
Range rate The time rate of change of radar range.
Right ascension Angular distance of an object measured
eastwards along the celestial equator from the vernal
equinox to the great circle passing through the north
celestial pole and the object.
Trajectory Maker Glossary
User's Guide B-6
──────────────────────────────────────────────────────────
Sidereal time The hour angle of the vernal equinox.
Spheroid An oblate spheroid which closely approximates
the mean sea-level figure of the Earth's geoid.
Stereographic projection A circle preserving, conformal
map projection obtained by projecting the points of the
sphere from the North pole onto the tangent plane at the
South pole (or vice versa). Meridians pass into straight
lines, parallels into concentric circles. It may be to
Hipparch (c. 150 B.C.).
Sun synchronous An orbit in which the angle between the
orbit plane and the line joining the Sun and Earth is
constant. An object in such an orbit enjoys passing over
the same part of the Earth at the same time each day.
Trajectory The path of a body in space or through the
Earth's atmosphere.
Universal time Mean solar time referenced to the
Greenwich meridian.
Universal variable A variable that bridges the transition
from closed to open orbits in the two-body theory of
motion. Universal because one variable replaces the
awkwardness of using three distinct formulations for each
kind of conic section.
Velocity azimuth angle The angle measured clockwise from
the North to the projection of the inertial velocity
vector onto the local horizontal plane.
Vernal equinox The point of intersection of the ecliptic
and celestial equator where the Sun crosses the equator
from South to North in its apparent annual motion along
the ecliptic. This event occurs near March 21.
Zulu time Greenwich Mean Time or Universal Time.