Media Share 9

home *** CD-ROM | disk | FTP | other *** search

/ Media Share 9 / MEDIASHARE_09.ISO / private / trjmkr.zip / USER20.DOC < prev next >

Wrap

Text File | 1992-08-09 | 159KB | 3,595 lines

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 │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ └────────────────────────────────────────────────────┘ 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 ────────────────────────────────────────────────────────── ┌────────────────────────────────────────────────────┐ │ │ │ Figure 2 │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ └────────────────────────────────────────────────────┘ Trajectory Maker User's Guide 14 ────────────────────────────────────────────────────────── ┌────────────────────────────────────────────────────┐ │ │ │ Figure 3 │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ └────────────────────────────────────────────────────┘ 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 ────────────────────────────────────────────────────────── ┌────────────────────────────────────────────────────┐ │ │ │ Figure 4 │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ └────────────────────────────────────────────────────┘ 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 ────────────────────────────────────────────────────────── ┌────────────────────────────────────────────────────┐ │ │ │ Figure 5 │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ └────────────────────────────────────────────────────┘ Trajectory Maker User's Guide 23 ────────────────────────────────────────────────────────── ┌────────────────────────────────────────────────────┐ │ │ │ Figure 6 │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ └────────────────────────────────────────────────────┘ 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 ────────────────────────────────────────────────────────── ┌────────────────────────────────────────────────────┐ │ │ │ Figure 7 │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ └────────────────────────────────────────────────────┘ 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 ────────────────────────────────────────────────────────── ┌────────────────────────────────────────────────────┐ │ │ │ Figure 8 │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ └────────────────────────────────────────────────────┘ 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 ────────────────────────────────────────────────────────── 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 ────────────────────────────────────────────────────────── 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 ────────────────────────────────────────────────────────── ┌────────────────────────────────────────────────────┐ │ │ │ Figure 9 │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ └────────────────────────────────────────────────────┘ Trajectory Maker User's Guide 43 ────────────────────────────────────────────────────────── ┌────────────────────────────────────────────────────┐ │ │ │ Figure 10 │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ └────────────────────────────────────────────────────┘ Trajectory Maker User's Guide 44 ────────────────────────────────────────────────────────── ┌────────────────────────────────────────────────────┐ │ │ │ Figure 11 │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ └────────────────────────────────────────────────────┘ 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.