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Asking for the moon would be easier. The study of astrodynamics is becoming a common prerequisite for any engineer or scientist who expects to be involved in the aerospace sciences and their many applications. While manned travel in Earth-Moon spa. There are plenty of sites and books with pictures illustrating how to obtain the various curves through sectioning, so I won't bore you with more pictures here. And there are books and entire web sites devoted to the history of conics, the derivation and proofs of their formulas, and their various applications.
An interplanetary spacecraft spends most of its flight time moving under the gravitational influence of a single body – the Sun. Only for brief periods, compared with the total mission duration, is its path shaped by the gravitational field of the departure or arrival planet. The perturbations caused by the other planets while the spacecraft is pursuing its heliocentric course are negligible.
The computation of a precision orbit is a trial-and-error procedure involving numerical integration of the complete equations of motion where all perturbation effects are considered. For preliminary mission analysis and feasibility studies it is sufficient to have an approximate analytical method for determining the total V required to accomplish an interplanetary mission. The best method available for such analysis is called the patched-conic approximation.
The patched-conic method permits us to ignore the gravitational influence of the Sun until the spacecraft is a great distance from the Earth (perhaps a million kilometers). At this point its velocity relative to Earth is very nearly the hyperbolic excess velocity. If we now switch to a heliocentric frame of reference, we can determine both the velocity of the spacecraft relative to the Sun and the subsequent heliocentric orbit. The same procedure is followed in reverse upon arrival at the target planet's sphere of influence.
The first step in designing a successful interplanetary trajectory is to select the heliocentric transfer orbit that takes the spacecraft from the sphere of influence of the departure planet to the sphere of influence of the arrival planet.
If you have not already done so, before continuing it is recommended that you first study the Orbital Mechanics section of this web site. It is also recommended, if you are not already familiar with the subject, that you review our section on Vector Mathematics.
Heliocentric-Ecliptic Coordinate System
Our first requirement for describing an orbit is a suitable inertial reference frame. In the case of orbits around the Sun, such as planets, asteroids, comets and some deep-space probes describe, the heliocentric-ecliptic coordinate system is convenient. As the name implies, the heliocentric-ecliptic system has its origin at the center of the Sun. The X-Y or fundamental plane coincides with the ecliptic, which is the plane of Earth's revolution around the Sun. The line-of-intersection of the ecliptic plane and Earth's equatorial plane defines the direction of the X-axis. On the first day of spring a line joining the center of Earth and the center of the Sun points in the direction of the positive X-axis. This is called the vernal equinox direction. The Y-axis forms a right-handed set of coordinate axes with the X-axis. The Z-axis is perpendicular to the fundamental plane and is positive in the north direction.
It is known that Earth wobbles slightly and its axis of rotation shifts in direction slowly over the centuries. This effect is known as precession and causes the line-of-intersection of Earth's equator and the ecliptic to shift slowly. As a result the heliocentric-ecliptic system is not really an inertial reference frame. Where extreme precision is required, it is necessary to specify that the XYZ coordinates of an object are based on the vernal equinox direction of a particular year or epoch.
From Figure 5.2 we can see that the transfer is the one-tangent burn type, which we examined previously. Selecting a transfer orbit allows the determination of the change in true anomaly and the time-of-flight using equations (4.67) and (4.71).
The target planet will move through an angle of t(t2–t1) while the spacecraft is in flight, where t is the angular velocity of the target planet. Thus, the correct phase angle at departure is,
The requirement that the phase angle at departure be correct severely limits the times when a launch may take place. The heliocentric longitudes of the planets are tabulated in The Astronomical Almanac, and these may be used to determine when the phase angle will be correct. Alternatively, the page Planet Positions provides the data and demonstrates the methods necessary to estimate planet positions without needing to refer to other sources.
Mars Transfer Trajectories
The methods described above provide only a very rough estimate of the phase angle, particularly in the case of Mars. The orbit of Mars is significantly eccentric, meaning its angular velocity changes considerably depending on whether it is near perihelion or aphelion at the time of transfer. For a better estimate we can no longer consider the orbit to be circular.
As can be seen from Figure 5.2, the proper alignment for a transfer to Mars occurs in the months just prior to an opposition. The location of Mars within its orbit at the time of opposition depends on the time of year the opposition occurs. Perihelion oppositions occur in the August-September time period, and aphelion oppositions occur in the February-March time period. We can, therefore, link the phase angle required to the time of year that we initiate the transfer.
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Figure 5.3 gives the required phase angle for transfers departing on the dates listed across the bottom of the chart. To use Figure 5.3 it is necessary to find the date when the actual phase angle equals the required angle obtained from the chart. For example, let's say we are planning a mission to launch around the October-13 opposition of the year 2020. It's decided we'll use a Type-I trajectory in which the spacecraft's true anomaly change is 150 degrees (magenta curve). In Table 1 we list the actual Mars-Earth phase angle for the months leading up to the October opposition, along with the phase angle read from Figure 5.3. We see that there is a date in July when the two numbers are equal. We can interpolate that the departure date will be sometime in the third week of July and the departure phase angle will be approximately 30 degrees.
Just as phase angle is dependent on Mars' location within its orbit, so is the time of flight. After estimating the departure date from Figure 5.3, we can use Figure 5.4 below to estimate the flight duration. For instance, for the July departure window determined above, the time of flight for a trajectory with a true anomaly change of 150o is found to be about 207 days. The letters superimposed on each curve indicate the departure dates that will result in the spacecraft intercepting Mars at perihelion (P), aphelion (A) or one of the two nodes (N).
Referring again Figure 5.2, we see that the flight path angle of the transfer orbit is positive at the first Mars orbit crossing and negative at the second Mars orbit crossing. Therefore, it may be preferable to use a Type-I trajectory when interception occurs with Mars in the part of its orbit past perihelion and approaching aphelion, when the planet's flight path angle is likewise positive. Conversely, a Type-II trajectory may be preferable when interception occurs with Mars in the part of its orbit past aphelion and approaching perihelion. Having Mars and spacecraft flight path angles both positive or both negative reduces the angle between the velocity vectors, and thus the relative velocity. This can be critical when V is the limiting factor. Of course a Type-I trajectory is always preferable when minimizing flight time is most critical.
Non-coplanar Trajectories
Up to now we have assumed that the planetary orbits all lie in the plane of the ecliptic. However, we know that all planets other than Earth have orbits inclined to the ecliptic. A good procedure to use when the target planet lies above or below the ecliptic at intercept is to launch the spacecraft into a transfer orbit that lies in the ecliptic plane and then make a simple plane change during mid-course when the true anomaly change remaining to intercept is 90-degrees. This minimizes the magnitude of the plane change required and is illustrated in Figure 5.4 below. Since the plane change is made 90o short of intercept, the required inclination is just equal to the ecliptic latitude, , of the target planet at the time of intercept, t2. The V required to produce a plane change was examined previously, and is calculated using equation (4.73).
Alternatively, the injection maneuver that places the spacecraft on its interplanetary trajectory can include a plane change to correctly orient the plane of the transfer orbit to intercept the target planet. Such an orbit between Earth and Mars is pictured in Figure 5.6 below. Since Earth lies in the ecliptic plane, the departure point defines one of the transfer orbit's two nodes, with the other node 180 degrees away on the opposite side of the Sun. Unless the target planet happens to also be passing through one of its nodes at the time of interception, a near 180-degree transfer is not possible without a prohibitively high inclination. Intercepting the target with a true anomaly change several degrees less than 180o (as pictured) or several degrees more than 180o can be achieved with a manageably low inclination.
Click here for example problem #5.4
Selecting a Transfer Orbit
Each time the Gauss problem is solved, the result gives just one of an infinite number of possible transfer orbits. It was previously stated that it generally desirable that the transfer orbit be tangential to Earth's orbit at departure. This is true only in that it minimizes the V required to inject the spacecraft into its transfer orbit; however, it likely results in a less than optimum condition at target intercept. A one-tangent burn produces a trajectory that crosses the orbit of the target planet with a relatively large flight path angle, resulting in a large relative velocity between the spacecraft and planet. This relative velocity can be significantly reduced by selecting a transfer orbit that reduces the angle between the velocity vectors of the spacecraft and target at the moment of intercept. Improving the intercept condition (1) increases the duration of a close flyby encounter, (2) reduces the V required for orbit insertion, or (3) lowers the spacecraft's velocity at atmospheric entry.
Tables 2 and 3 below provide sample data for a hypothetical mission to Mars in the year 2020. Table 2 gives the V required for Trans-Mars Injection (TMI) for a variety of different departure dates and times of flight. TMI is the maneuver that places the spacecraft into a trajectory that will intercept Mars at the desired place and time. In this sample, it is assumed that TMI is performed from an Earth parking orbit with an altitude of 200 km. Table 3 gives the V required for Mars-Orbit Insertion (MOI) for the same departure dates and times of flight found in Table 2. MOI, as it names implies, is the maneuver that slows the spacecraft to a velocity that places it into the desired orbit around Mars. In this sample, it is assumed that MOI is performed at periapsis of a insertion orbit with a periapsis altitude of 1,000 km and an apoapsis altitude of 33,000 km. Placing a spacecraft into a high eccentricity orbit such as this is common, as it provides for a MOI burn with a relatively low V.
| Table 2 | |||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|
| Trans-Mars InjectionDV (m/s), launch altitude = 200 km | |||||||||||
| Departure Date, 2020 | Time of Flight (days) | ||||||||||
| 180 | 185 | 190 | 195 | 200 | 205 | 210 | 215 | 220 | 225 | 230 | |
| 7/7 | 3876 | 3862 | 3854 | 3851 | 3853 | 3863 | 3881 | 3912 | 3962 | 4043 | 4180 |
| 7/12 | 3841 | 3830 | 3824 | 3823 | 3826 | 3835 | 3851 | 3877 | 3917 | 3978 | 4074 |
| 7/19 | 3819 | 3812 | 3808 | 3808 | 3811 | 3819 | 3833 | 3853 | 3882 | 3925 | 3988 |
| 7/26 | 3834 | 3829 | 3826 | 3826 | 3829 | 3836 | 3846 | 3862 | 3883 | 3913 | 3956 |
| 8/2 | 3892 | 3887 | 3885 | 3884 | 3886 | 3890 | 3897 | 3908 | 3923 | 3943 | 3972 |
| 8/9 | 3999 | 3994 | 3990 | 3987 | 3987 | 3987 | 3991 | 3996 | 4005 | 4017 | 4034 |
| 8/16 | 4162 | 4154 | 4147 | 4141 | 4137 | 4133 | 4131 | 4131 | 4133 | 4138 | 4146 |
| 8/23 | 4386 | 4373 | 4362 | 4351 | 4341 | 4332 | 4325 | 4318 | 4313 | 4310 | 4309 |
| Table 3 | |||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|
| Mars Orbit InsertionDV (m/s), insertion orbit = 1000 × 33000 km | |||||||||||
| Departure Date, 2020 | Time of Flight (days) | ||||||||||
| 180 | 185 | 190 | 195 | 200 | 205 | 210 | 215 | 220 | 225 | 230 | |
| 7/7 | 1371 | 1258 | 1163 | 1086 | 1025 | 982 | 959 | 957 | 984 | 1052 | 1187 |
| 7/12 | 1290 | 1188 | 1102 | 1033 | 979 | 940 | 918 | 915 | 933 | 982 | 1074 |
| 7/19 | 1186 | 1097 | 1024 | 965 | 920 | 888 | 870 | 866 | 879 | 911 | 970 |
| 7/26 | 1093 | 1019 | 957 | 909 | 872 | 847 | 833 | 830 | 840 | 864 | 905 |
| 8/2 | 1016 | 954 | 904 | 865 | 837 | 818 | 808 | 808 | 817 | 836 | 867 |
| 8/9 | 957 | 907 | 868 | 838 | 817 | 804 | 799 | 801 | 811 | 828 | 853 |
| 8/16 | 920 | 881 | 852 | 830 | 816 | 809 | 808 | 813 | 823 | 839 | 862 |
| 8/23 | 910 | 881 | 860 | 846 | 838 | 836 | 839 | 846 | 857 | 873 | 893 |
As can be seen from Tables 2 and 3, in most instances, TMI V and MOI V are inversely proportional. That is, trying to optimize one increases the other, and vice versa. Selecting the 'best' transfer orbit therefore comes down to making a compromise. The size of the launch window is also often limited by the V budget. For example, suppose our launch vehicle can deliver no more than 3,900 m/s for TMI, and our spacecraft's MOI budget is 900 m/s. Our potential launch opportunities are limited to those in which both of these conditions are met, which we see represented by the launch dates and flight durations highlighted above.
Solving the Gauss problem gives us the position and velocity vectors, r and v, of a spacecraft in a heliocentric-ecliptic orbit. From these vectors we can determine the six orbital elements that describe the motion of the satellite. The first step is to form the three vectors, h, n and e, illustrated in Figure 5.08.
The specific angular momentum, h, of a satellite is obtained from
It is important to note that h is a vector perpendicular to the plane of the orbit.
The node vector, n, is defined as
From the definition of a vector cross product, n must be perpendicular to both z and h. To be perpendicular to z, n would have to lie in the ecliptic plane. To be perpendicular to h, n would have to lie in the orbital plane. Therefore, n must lie in both the ecliptic and orbital planes, or in their intersection, which is called the 'line of nodes.' Specifically, n is a vector pointing along the line of nodes in the direction of the ascending node. The magnitude of n is of no consequence to us; we are only interested in its direction.
The third vector, e, is obtained from
Vector e points from the center of the Sun (focus of the orbit) toward perihelion with a magnitude exactly equal to the eccentricity of the orbit.
Now that we have h, n and e we can preceed rather easily to obtain the orbital elements. The semi-major axis, a, and the eccentricity, e, follow directly from r, v, and e, while all the remaining orbital elements are simply the angles between two vectors whose components are now known. If we know how to find the angle between two vectors the problem is solved. In general, the cosine of the angle, , between two vectors a and b is found by dividing the dot product of the two vectors by the product of their magnitudes.
Of course, being able to evaluate the cosine of an angle does not mean that we know the angle. We still have to decide whether the angle is smaller or greater than 180 degrees. The answer to this quadrant resolution problem must come from other information in the problem as we shall see.
We can outline the method of finding the orbital elements as follows:
- Calculate a and e,
- Since i is the angle between z and h,
- (Inclination is always less than 180o)
- Since is the angle between x and n,
- (If ny > 0 then is less than 180o)
- Since is the angle between n and e,
- (If ez > 0 then is less than 180o)
- Since o is the angle between e and r,
- (If r • v > 0 then o is less than 180o)
- Since uo is the angle between n and r,
- (If rz > 0 then uo is less than 180o)
- Calculate and o,
- ( and o are always less than 360o)
The angle , longitude of periapsis, is sometimes used in place of argument of periapsis. As a substitute for the time of periapsis passage, any of the following may be used to locate the spacecraft at a particular time, to, known as the 'epoch': o, true anomaly at epoch, uo, argument of latitude at epoch, or o, true longitude at epoch.
If there is no periapsis (circular orbit), then is undefined, and o = + uo. If there is no ascending node (equatorial orbit), then both and uo are undefined, and o = + o. If the orbit is both circular and equatorial, o is simply the true angle from x to ro, both of which are always defined.
The procedure outlined above describes a spacecraft in a solar orbit, but the method works equally well for satellites in Earth orbit, or around another planet or moon, where the position and velocity vectors are known in the geocentric-equatorial reference plane. Note, however, that it is customary for the geocentric-equatorial coordinate system to use unit vectors i, j and k instead of x, y and z as used in the heliocentric-ecliptic system.
Once the heliocentric transfer orbit has been selected, we next determine the spacecraft's velocity relative to Earth. The relative velocity, which we will define as the vector vs/p, is the difference between the spacecraft's heliocentric velocity, vs, and the planet's orbital velocity, vp (see Figure 5.10).
To determine the further parameters of the hyperbolic escape trajectory, please refer to the hyperbolic orbit as previously examined.
Arrival at the Target Planet
As before, the relative velocity vector and magnitude are calculated using equations (5.33) and (5.34).
If a dead center hit on the target planet is planned, then we solve the Gauss problem setting r2 equal to the position vector of the planet at arrival. This ensures that the target planet will be at the intercept point at the same time the spacecraft is there. It also means that the relative velocity vector, upon arrival at the target planet's sphere of influence, will be directed toward the center of the planet, resulting in a straight line hyperbolic approach trajectory.
If it is desired to fly by the planet instead of impacting it, then the transfer trajectory must be modified so that the spacecraft crosses the target planet's orbit ahead of or behind the planet. If the spacecraft crosses the planet's orbit a distance d from the planet, then the velocity vector vs/p, which represents the hyperbolic excess velocity on the approach hyperbola, is offset a distance b from the center of the target planet, as shown in Figure 5.12.
The sign of d is chosen depending on whether the spacecraft is to cross ahead of (positive) or behind (negative) the target planet. Assuming the target point lies within the same X-Y plane as the planet, the rectangular components of d are,
where rx and ry are scalar components of the planet's position vector.
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The angle is calculated as follows:
From the following we obtain the impact parameter, b
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Recalling that V∞ ≈ Vs/p, we calculate the hyperbola's semi-major axis and eccentricity as follows:
To calculate the remaining parameters of the hyperbolic approach trajectory, see the hyperbolic orbit.
For a close flyby, and understanding that the patched-conic method is only an approximate solution, it is generally adequate to ignore the miss distance, d, when solving the Gauss problem, assuming the position vector at arrival is equal to that of the planet. However, if the flyby distance is large, an improved Gauss solution is obtained by modifying the position vector to account for miss distance. If rx, ry and rz are the scalar components of the planet's position vector, the components of the target point, rx', ry' and rz', are as follows:
Compiled, edited and written in part by Robert A. Braeunig, 2012, 2013.
Bibliography
About the Book
Topics in Astrodynamics builds a mathematical foundation for understanding and analyzing artificial Earth satellite orbits, to includeEarth escape and flyby trajectories. Its chapters first deal with theclassical orbital elements, and then address the fundamental problemof satellite tracking: how to calculate ground traces and look angles, given the orbital elements of an artificial Earth satellite. Element set transformations and Gaussian orbit determination are then treated.
Orbital perturbations are dealt with via the topics of Cowell (numerical) orbit propagation, variation of parameters, and general perturbation theory. The final chapter addresses the fundamental problem of space surveillance: how to calculate an accurate state vector for the orbit of an artificial Earth satellite, given radar or optical observations (or some mix of both), and an initial estimate of the state vector at some epoch.
The book complements currently available works on celestial mechanics ('orbital mechanics applied to celestial bodies') by applying orbital mechanics to the approximately 10,000 artificial Earth satellites whoseorbital elements are to be found in the satellite catalog of NorthAmerican Aerospace Defense Command (NORAD) in Colorado Springs, Colorado U.S.A.
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The book's author states, 'Topics in Astrodynamics captures that which is worth passing along from what I myself have learned, worked out, and taught, both as an author and as a user of computer software for orbital analysts, over a space career that has spanned more than three decades.'
Topics in Astrodynamics was typeset as a 'standard LaTeX book' usingMacKichan Software's Scientific Word; see http://www.mackichan.com. Its 378 pages are sized at 8.5' by 11' and are bound between soft covers by means of a 1-1/8' diameter, 19-ring GBC plastic comb binding. See below for an actual photo of the book and a complete summary of the book's contents.
TOPICS IN ASTRODYNAMICS
Title Page i
Copyright Page ii
Dedication iii
Note on Typeset Manuscript iv
Preface v
Table of Contents ix
List of Figures xvii
Chapter 1. Introduction and Review 1
1.1 Scope 1
1.2 Review of Elementary Mechanics 2
1.2.1 Basic Definitions 2
1.2.2 Newton's Laws of Gravitation 3
1.2.3 Kepler's Laws 5
1.2.4 Work, Energy, and Conservative Forces 5
1.3 Review of the Conic Sections 8
1.3.1 Polar Transformations and Standard Form 8
1.3.2 Conic Sections and Conic Paths 9
1.4 Suggested Reading 12
Chapter 2. The Two-Body Problem 15
2.1 Equations of Relative Motion 15
2.2 Conservation Theorems 17
2.2.1 Conservation of Energy 17
2.2.2 Conservation of Angular Momentum 18
2.3 Solution of the Relative Equations 19
2.3.1 Proof of Kepler's First Law 21
2.3.2 Proof of Kepler's Second Law 25
2.3.3 Proof of Kepler's Third Law 26
2.4 The Flight Path Angle 28
2.5 Position in the Orbit Plane 29
2.5.1 Perifocal Coordinates and the Eccentric Anomaly 29
2.5.2 Kepler's Equation and the Mean Anomaly 32
2.5.3 Newton-Raphson Solution of Kepler's Equation 34
2.5.4 Orbital Position as a Function of Time 35
2.6 Useful Formulas for an Elliptical Orbit 35
2.7 Suggested Reading 37
Chapter 3. Celestial Sphere and ECI Coordinates 39
3.1 Need for an Inertial Reference Frame 39
3.2 The Celestial Sphere 4
3.3 The ECI Reference Frame 42
3.4 Celestial Coordinates and Transformations 43
3.5 Suggested Reading 44
Chapter 4. Rotation Matrices and Applications 45
4.1 Orthogonal Rotation 45
4.2 The EFG-to-ECI Transformation 48
4.3 The Euler Angle Transformation 51
4.4 Suggested Reading 53
Chapter 5. Orbital Elements & Orbit Propagation 55
5.1 Orbital Elements 55
5.2 Velocity in the Orbit Plane 58
5.3 Orbit Propagation 60
5.4 Summary Algorithm for Elliptical Orbit 61
5.5 Modification for an Orbit of Low Eccentricity 63
5.6 Suggested Reading 63
Chapter 6. Dynamical Time Conversion 65
6.1 Sidereal Time 66
6.2 Solar Time 68
6.3 Atomic Time vs. Universal Time 71
6.4 Newcomb's Formula 72
6.5 Suggested Reading 73
Chapter 7. Ground Traces and Look Angles 75
7.1 The Figure of the Earth 76
7.2 Geocentric and Geodetic Latitude 78
7.3 Subpoint Latitude and Height 80
7.4 East Longitude 83
7.5 Look Angles and Slant Range 85
7.6 Suggested Reading 87
Chapter 8. Element Set Transformations 89
8.1 Cartesian-to-Classical Transformation 90
8.1.1 Calculation of a, e, and M 90
8.1.2 Calculation of i, Omega, and omega 92
8.2 Nodal Orbital Elements 94
8.2.1 Transformations Involving Nodal Elements 94
8.2.2 Orbit Propagation Using Nodal Elements 95
8.2.3 Summary Algorithm 100
8.3 Equinoctial Orbital Elements 101
8.3.1 Transformations Involving Equinoctial Elements 102
8.3.2 Orbit Propagation Using Equinoctial Elements 103
8.3.3 Summary Algorithm 109
8.4 Summary 110
8.5 Suggested Reading 112
Chapter 9. Gaussian Orbit Determination 113
9.1 Closed-Form f and g Series 115
9.2 Derivation of Gauss's Method 116
9.2.1 Area Ratio of Sector to Triangle 118
9.2.2 The First Equation of Gauss 119
9.2.3 The Second Equation of Gauss 122
9.2.4 Iteration for E2 - E1 and Solution for a 124
9.3 Summary Algorithm for Gauss's Method 126
9.4 Applications of Gauss's Method 127
9.4.1 Artificial Earth Satellite Orbit Determination 127
9.4.2 Interpolation on Ephemerides 128
9.4.3 Determination of an Avoidance Trajectory 128
9.5 Critique of Gauss's Method 129
9.6 Suggested Reading 130
Chapter 10. Cowell Propagation 133
10.1 Classification of Perturbative Accelerations 136
10.2 Conservative Accelerations 137
10.2.1 Earth's Gravity 137
10.2.2 Sun, Moon, and Major Planet Gravity 140
10.3 Non-Conservative Accelerations 141
10.3.1 Solar Radiation Pressure 141
10.3.2 Atmospheric Drag 144
10.4 Numerical Propagation 145
10.4.1 Reduction of Order 146
10.4.2 Runge-Kutta Numerical Integration 147
10.4.3 Application to the Cowell Problem 148
10.5 Summary 150
10.6 Suggested Reading 150
Chapter 11. Variation of Parameters 153
11.1 Lagrange's Planetary Equations 155
11.1.1 Lagrange's Brackets 157
11.1.2 Lagrange's Brackets for the Classical Elements 158
11.1.3 Substitution of M for M0 164
11.2 Transformation to Other Variables 165
11.3 Gauss's Form of Lagrange's Equations 167
11.4 VOP for Earth's Equatorial Bulge 170
11.5 VOP for Atmospheric Drag 172
11.6 Numerical Integration 175
11.7 Concluding Remarks 176
11.8 Suggested Reading 177
Chapter 12. General Perturbation Theory 179
12.1 Kozai's Method 181
12.2 First-Order, Secular Perturbation Theory 185
12.3 Chebotarev's Method for Small e 187
12.4 Modeling the Drag Acceleration 188
12.4.1 Secular Changes in a and e 188
12.4.2 Two Key Assumptions 191
12.5 Orbit Propagation with Mean Elements 192
12.6 Calculation of Time Elapsed Since Epoch 196
12.7 Concluding Remarks 198
12.8 Suggested Reading 199
Chapter 13. Launch Profiles and Nominals 201
13.1 Calculating Launch Nominal Elements 202
13.1.1 Computation of Omega and M at Injection 203
13.1.2 Computation of a-bar, Given rp or Hp 208
13.1.3 Computation of i and DI from AzI and Converse 209
13.1.4 The Case Where omega is not Specified 210
13.2 Moving Epoch to Revolution Zero 210
13.2.1 Purpose of Moving Epoch 210
13.2.2 Propagation of Mean Elements 211
13.2.3 Computation of n-bar and Delta-tI 212
13.3 The January 1.0 UTC Liftoff Convention 214
13.3.1 When a Cooperative Launch is Delayed 214
13.3.2 Non-Cooperative Launch Assessment 216
13.4 Polar Orbiter Launch Practice 217
13.5 Hypothetical NPOESS Launch Example 219
13.6 Orbital Maneuvers 224
13.6.1 One-Impulse Maneuvers 224
13.6.2 Multiple-Impulse Maneuvers 225
13.6.3 Application of the Hohmann Transfer 228
13.7 Geostationary Launch Practice 228
13.8 Hypothetical GOES Launch Example 231
13.9 Suggested Reading 235
Chapter 14. Escape and Flyby Trajectories 237
14.1 Uniform Path Mechanics 238
14.1.1 Stumpff's c-Functions 239
14.1.2 Conic Elements 249
14.1.3 Uniform Propagation of Conic Elements 250
14.1.4 Kepler's Equation Revisited 260
14.1.5 Propagation of Position and Velocity 263
14.2 Gaussian Orbit Determination 268
14.3 Goodyear's State Transition Matrix 272
14.4 Suggested Reading 275
Chapter 15. Differential Correction 277
15.1 Batch Least Squares 277
15.1.1 Optical Residuals and Partials 282
15.1.2 Radar Residuals and Partials 285
15.1.3 The H Matrix 290
15.1.4 Summary Algorithm 294
15.1.5 HTWH Matrix Accumulation 296
15.2 Variant Orbit Partials 297
15.3 Escape Trajectory Example 299
15.4 State Space Analysis 306
15.4.1 Batch Filter for Two-Body Trajectory 306
15.4.2 Batch Filter for Perturbed Trajectory 308
15.4.3 Batch DC vs. Batch Filter 310
15.4.4 Statistical Orbit Determination 311
15.5 Suggested Reading 312
Appendix A. Astrodynamic Notation 313
A.1 Chapter 1 - Introduction and Review 314
A.2 Chapter 2 - The Two-Body Problem 315
A.3 Chapter 3 - Celestial Sphere and ECI Coordinates 316
A.4 Chapter 4 - Rotation Matrices and Applications 316
A.5 Chapter 5 - Orbital Elements and Orbit Propagation 316
A.6 Chapter 6 - Dynamical Time Conversion 317
A.7 Chapter 7 - Ground Traces and Look Angles 318
A.8 Chapter 8 - Element Set Transformations 319
A.9 Chapter 9 - Gaussian Orbit Determination 319
A.10 Chapter 10 - Cowell Propagation 320
A.11 Chapter 11 - Variation of Parameters 321
A.12 Chapter 12 - General Perturbation Theory 321
A.13 Chapter 13 - Launch Profiles 322
A.14 Chapter 14 - Escape and Flyby 322
A.15 Chapter 15 - Differential Correction 323
A.16 References 323
Appendix B. Astrodynamic Constants 325
B.1 Canonical Units 327
B.2 Precession and Nutation 328
B.3 References 329
Appendix C. Spherical Trigonometry 331
C.1 Spherical Law of Sines 333
C.2 Spherical Law of Cosines for Sides 333
C.3 Spherical Law of Cosines for Angles 334
C.4 Napier's Rules 334
C.5 Earth Satellite Injection 335
C.6 Azimuth Direction from a Point 336
C.7 Radio Wave Propagation 339
C.8 Suggested Reading 339
Appendix D. Chebotarev's Method 341
D.1 Lagrange's Equations for Small e 341
D.2 The Disturbing Potential for Small e 342
D.3 First-Order Perturbations 344
D.3.1 Mean Argument of Latitude 346
D.3.2 Secular and Periodic Updating 348
D.4 Orbit Propagation Procedure 348
D.4.1 Preliminary Calculations 348
D.4.2 Convert to Nodal Elements 349
D.4.3 Update for Secular Perturbations 350
D.4.4 Update for Periodic Perturbations 351
D.4.5 Transform to Position and Velocity 352
D.5 Suggested Reading 352
Index 353
About the Author
Basic Astrodynamics Formulas For Beginners
Roger L. Mansfield is a space professional with more than 30 years of military, industrial, and academic experience. He began his space career as an orbital analyst for the Defense Meteorological Satellite Program (DMSP) in August 1967, when he wasassigned to the 4000th Support Group at Offutt Air Force Base, Nebraska. (Offutt AFB is now the home of Headquarters U.S. Strategic Command.)
As principal engineerfor space surveillance applications at Ford Aerospace and at Loral Command & Control Systems, Mr. Mansfield led efforts to develop algorithms and software for the 427M Space Surveillance Center (1976-1981) and for the Space Defense Operations Center (1982-1996) in Air Force Space Command's Cheyenne Mountain Air Force Station. As assistant professor at CU-Colorado Springs, hetaught astrodynamics and numerical methods to graduate space engineers working for Lockheed Martin Astronautics at the Waterton Canyon facility near Denver, Colorado.
Mr. Mansfield's personal webpage athttp://mathcadwork.astroger.com/ describes just a few of the Mathcad worksheets he has constructed since 1997 to solve problems in the mechanics of Earth orbital, escape, flyby, and interplanetary trajectories. His freely downloadable Mathcad worksheets provide live,graphical examples of many of the algorithms and procedures in his book. And the worksheets employ familiar mathematical notation, not ASCII program code.
With 'Nicolaus Copernicus' at AGI's 15th Annual Monte Carlo Night, April 2015
How to Purchase the Book
The book's intended audience has been: military and civilian members of the U.S. Air Force; other U.S. governmental departments and agencies dealing with space; the U.S. space industry; professors and students of space engineering.