This professional course covers basic orbital mechanics and includes an introduction to the physics of cislunar space, including the Lagrange stability points around the earth and moon. The course also includes historical information about the development of classical orbital mechanics and modern astrodynamics. MIT OpenCourseWare is a web-based publication of virtually all MIT course content. OCW is open and available to the world and is a permanent MIT activity. Topics in Astrodynamics builds a mathematical foundation for understanding and analyzing artificial Earth satellite orbits, to include Earth escape and flyby trajectories. Its chapters first deal with the classical orbital elements, and then address the fundamental problem of satellite tracking: how to calculate ground traces and look angles. SpaceDSL is a astrodynamics simulation library. This library is Written by C. The purpose is to provide an open framework for astronaut dynamics enthusiasts, and more freely to achieve astrodynamics simulation. The project is open under the MIT protocol, and it is also for freer purposes. The Ulysses Orbit: Classical Orbital Elements. For most applications, each phase of the Ulysses orbit is well represented by a special mathematical expression called a conic function (i.e., an ellipse or a hyperbola).
Postal and E-Mail AddressesAbout 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.
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
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.
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The Binomial Theorem: Formulas (page 1 of 2)
Sections: The formulas, Worked examples
The Binomial Theorem is a quick way (okay, it's a less slow way) of expanding (or multiplying out) a binomial expression that has been raised to some (generally inconveniently large) power. For instance, the expression (3x – 2)10 would be very painful to multiply out by hand. Thankfully, somebody figured out a formula for this expansion, and we can plug the binomial 3x – 2 and the power 10 into that formula to get that expanded (multiplied-out) form.
The formal expression of the Binomial Theorem is as follows:
Copyright © Elizabeth Stapel 1999-2009 All Rights Reserved
Basic Astrodynamics Formulas Pdf
Yeah, I know; that formula never helped me much, either. And it doesn't help that different texts use different notations to mean the same thing. The parenthetical bit above has these equivalents:
Recall that the factorial notation 'n!' means ' the product of all the whole numbers between 1 and n', so, for instance, 6! = 1×2×3×4×5×6. Then the notation '10C7' (often pronounced as 'ten, choose seven') means:
Many calculators can evaluate this 'n choose m' notation for you. Just look for a key that looks like 'nCm' or 'nCr', or for a similar item on the 'Prob' or 'Math' menu, or check your owner's manual under 'probability' or 'combinations'. | |
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Keep going, always adding pairs of numbers from the previous row..
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As you might imagine, drawing Pascal's Triangle every time you have to expand a binomial would be a rather long process, especially if the binomial has a large exponent on it. People have done a lot of studies on Pascal's Triangle, but in practical terms, it's probably best to just use your calculator to find nCr, rather than using the Triangle. The Triangle is cute, I suppose, but it's not terribly helpful in this context, being more time-consuming than anything else. For instance, on a test, do you want to evaluate '10C7' by calculating eleven rows of the Triangle, or by pushing four buttons on your calculator?
I could never remember the formula for the Binomial Theorem, so instead, I just learned how it worked. I noticed that the powers on each term in the expansion always added up to whatever n was, and that the terms counted up from zero to n. Returning to our intial example of (3x – 2)10, the powers on every term of the expansion will add up to 10, and the powers on the terms will increment by counting up from zero to 10:
(3x – 2)10 = 10C0 (3x)10–0(–2)0 + 10C1 (3x)10–1(–2)1 + 10C2 (3x)10–2(–2)2
+ 10C3 (3x)10–3(–2)3 + 10C4 (3x)10–4(–2)4 + 10C5 (3x)10–5(–2)5
+ 10C6 (3x)10–6(–2)6 + 10C7 (3x)10–7(–2)7 + 10C8 (3x)10–8(–2)8
+ 10C9 (3x)10–9(–2)9 + 10C10 (3x)10–10(–2)10
Note how the highlighted counter number counts up from zero to 10, with the factors on the ends of each term having the counter number, and the factor in the middle having the counter number subtracted from 10. This pattern is all you really need to know about the Binomial Theorem; this pattern is how it works.
Your first step, given a binomial to expand, should be to plug it into the Theorem, just like I did above. Don't try to do too many steps at once. Only after you've set up your binomial in the Theorem's pattern should you start to simplify the terms. The Binomial Theorem works best as a 'plug-n-chug' process, but you should plug in first; chug later. I've done my 'plugging' above; now 'chugging' gives me:
(1)(59049)x10(1) + (10)(19683)x9(–2) + (45)(6561)x8(4) + (120)(2187)x7(–8)
+ (210)(729)x6(16) + (252)(243)x5(–32) + (210)(81)x4(64)
+ (120)(27)x3(–128) + (45)(9)x2(256) + (10)(3)x(–512) + (1)(1)(1)(1024)
= 59049x10 – 393660x9 + 1180980x8 – 2099520x7 + 2449440x6 – 1959552x5
+ 1088640x4 – 414720x3 + 103680x2 – 15360x + 1024
As painful as the Binomial-Theorem process is, it's still easier than trying to multiply this stuff out by hand. So don't let the Formula put you off. It's just another thing to memorize, so memorize it, at least for the next test. The biggest source of errors in the Binomial Theorem (other than forgetting the Theorem) is the simplification process. Don't try to do it in your head, or try to do too many steps at once. Write things out nice and clearly, as I did above, so you have a better chance of getting the right answer. (And it would be good to do a bunch of practice problems, so the process is fairly automatic by the time you hit the next test.)
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Cite this article as: | Stapel, Elizabeth. 'The Binomial Theorem: Formulas.' Purplemath. Available from |