# Orbital Mechanics for Engineering Students

## 3rd Edition

**Author:**Howard Curtis

**eBook ISBN:**9780080977485

**Hardcover ISBN:**9780080977478

**Imprint:**Butterworth-Heinemann

**Published Date:**5th October 2013

**Page Count:**768

**View all volumes in this series:**Aerospace Engineering

## Table of Contents

Dedication

Preface

Supplements to the text

Acknowledgements

Chapter 1. Dynamics of Point Masses

Abstract

1.1 Introduction

1.2 Vectors

1.3 Kinematics

1.4 Mass, force, and Newton’s law of gravitation

1.5 Newton’s law of motion

1.6 Time derivatives of moving vectors

1.7 Relative motion

1.8 Numerical integration

Problems

Section 1.3

Section 1.4

Section 1.5

Section 1.6

Section 1.7

Section 1.8

Chapter 2. The Two-Body Problem

Abstract

2.1 Introduction

2.2 Equations of motion in an inertial frame

2.3 Equations of relative motion

2.4 Angular momentum and the orbit formulas

2.5 The energy law

2.6 Circular orbits (*e* = 0)

2.7 Elliptical orbits (0 < *e* < 1)

2.8 Parabolic trajectories (*e* = 1)

2.9 Hyperbolic trajectories (*e* > 1)

2.10 Perifocal frame

2.11 The Lagrange coefficients

2.12 Restricted three-body problem

Problems

Section 2.2

Section 2.3

Section 2.4

Section 2.5

Section 2.6

Section 2.7

Section 2.8

Section 2.9

Section 2.11

Section 2.12

Chapter 3. Orbital Position as a Function of Time

Abstract

3.1 Introduction

3.2 Time since periapsis

3.3 Circular orbits (*e* = 0)

3.4 Elliptical orbits (*e* < 1)

3.5 Parabolic trajectories (*e* = 1)

3.6 Hyperbolic trajectories (*e* > 1)

3.7 Universal variables

Problems

Section 3.4

Section 3.5

Section 3.6

Section 3.7

Chapter 4. Orbits in Three Dimensions

Abstract

4.1 Introduction

4.2 Geocentric right ascension–declination frame

4.3 State vector and the geocentric equatorial frame

4.4 Orbital elements and the state vector

4.5 Coordinate transformation

4.6 Transformation between geocentric equatorial and perifocal frames

4.7 Effects of the earth’s oblateness

4.8 Ground tracks

Problems

Section 4.4

Section 4.5

Section 4.6

Section 4.7

Section 4.8

Chapter 5. Preliminary Orbit Determination

Abstract

5.1 Introduction

5.2 Gibbs method of orbit determination from three position vectors

5.3 Lambert's problem

5.4 Sidereal time

5.5 Topocentric coordinate system

5.6 Topocentric equatorial coordinate system

5.7 Topocentric horizon coordinate system

5.8 Orbit determination from angle and range measurements

5.9 Angles-only preliminary orbit determination

5.10 Gauss method of preliminary orbit determination

Problems

Section 5.3

Section 5.4

Section 5.8

Section 5.10

Chapter 6. Orbital Maneuvers

Abstract

6.1 Introduction

6.2 Impulsive maneuvers

6.3 Hohmann transfer

6.4 Bi-elliptic Hohmann transfer

6.5 Phasing maneuvers

6.6 Non-Hohmann transfers with a common apse line

6.7 Apse line rotation

6.8 Chase maneuvers

6.9 Plane change maneuvers

6.10 Nonimpulsive orbital maneuvers

Problems

Section 6.3

Section 6.4

Section 6.5

Section 6.6

Section 6.7

Section 6.8

Section 6.9

Section 6.10

Chapter 7. Relative Motion and Rendezvous

Abstract

7.1 Introduction

7.2 Relative motion in orbit

7.3 Linearization of the equations of relative motion in orbit

7.4 Clohessy–Wiltshire equations

7.5 Two-impulse rendezvous maneuvers

7.6 Relative motion in close-proximity circular orbits

Problems

Section 7.3

Section 7.4

Section 7.5

Section 7.6

Chapter 8. Interplanetary Trajectories

Abstract

8.1 Introduction

8.2 Interplanetary Hohmann transfers

8.3 Rendezvous opportunities

8.4 Sphere of influence

8.5 Method of patched conics

8.6 Planetary departure

8.7 Sensitivity analysis

8.8 Planetary rendezvous

8.9 Planetary flyby

8.10 Planetary ephemeris

8.11 Non-Hohmann interplanetary trajectories

Problems

Section 8.3

Section 8.4

Section 8.6

Section 8.7

Section 8.8

Section 8.9

Section 8.10

Section 8.11

Chapter 9. Rigid Body Dynamics

Abstract

9.1 Introduction

9.2 Kinematics

9.3 Equations of translational motion

9.4 Equations of rotational motion

9.5 Moments of inertia

9.6 Euler's equations

9.7 Kinetic energy

9.8 The spinning top

9.9 Euler angles

9.10 Yaw, pitch, and roll angles

9.11 Quaternions

Problems

Section 9.5

Section 9.7

Section 9.8

Section 9.9

Chapter 10. Satellite Attitude Dynamics

Abstract

10.1 Introduction

10.2 Torque-free motion

10.3 Stability of torque-free motion

10.4 Dual-spin spacecraft

10.5 Nutation damper

10.6 Coning maneuver

10.7 Attitude control thrusters

10.8 Yo-yo despin mechanism

10.9 Gyroscopic attitude control

10.10 Gravity-gradient stabilization

Problems

Section 10.3

Section 10.4

Section 10.6

Section 10.7

Section 10.8

Section 10.9

Section 10.10

Chapter 11. Rocket Vehicle Dynamics

Abstract

11.1 Introduction

11.2 Equations of motion

11.3 The thrust equation

11.4 Rocket performance

11.5 Restricted staging in field-free space

11.6 Optimal staging

Problems

Section 11.5

Section 11.6

Chapter 12. Introduction to Orbital Perturbations

Abstract

12.1 Introduction

12.2 Cowell’s method

12.3 Encke’s method

12.4 Atmospheric drag

12.5 Gravitational perturbations

12.6 Variation of parameters

12.7 Gauss variational equations

12.8 Method of averaging

12.9 Solar radiation pressure

12.10 Lunar gravity

12.11 Solar gravity

Problems

Section 12.3

Section 12.4

Section 12.5

Section 12.6

Section 12.7

Section 12.8

Section 12.9

Section 12.10

Section 12.11

Appendix A. Physical Data

Appendix B. A Road Map

Appendix C. Numerical Integration of the *n*-Body Equations of Motion

Appendix E. Gravitational Potential of a Sphere

Appendix F. Computing the Difference Between Nearly Equal Numbers

References and Further Reading

Index

Appendix D. MATLAB Scripts

D.1 Introduction

Chapter 1

D.3 Algorithm 1.2: Numerical integration by Heun’s predictor-corrector method

Chapter 2

D.6 Algorithm 2.2: Numerical solution of the two-body relative motion problem

D.7 Calculation of the Lagrange *f* and *g* functions and their time derivatives in terms of change in true anomaly

D.8 Algorithm 2.3: Calculate the state vector from the initial state vector and the change in true anomaly

D.9 Algorithm 2.4: Find the root of a function using the bisection method

D.10 MATLAB solution of Example 2.18

Chapter 3

D.12 Algorithm 3.2: Solution of Kepler’s equation for the hyperbola using Newton’s method

D.13 Calculation of the Stumpff functions S(z) and C(z)

D.14 Algorithm 3.3: Solution of the universal Kepler’s equation using Newton’s method

D.15 Calculation of the Lagrange coefficients *f* and *g* and their time derivatives in terms of change in universal anomaly

D.16 Algorithm 3.4: Calculation of the state vector given the initial state vector and the time lapse *Δt*

Chapter 4

D.18 Algorithm 4.2: Calculation of the orbital elements from the state vector

D.19 Calculation of tan–1 (*y*/*x*) to lie in the range 0 to 360°

D.20 Algorithm 4.3: Obtain the classical Euler angle sequence from a direction cosine matrix

D.21 Algorithm 4.4: Obtain the yaw, pitch, and roll angles from a direction cosine matrix

D.22 Algorithm 4.5: Calculation of the state vector from the orbital elements

D.23 Algorithm 4.6 Calculate the ground track of a satellite from its orbital elements

Chapter 5

D.25 Algorithm 5.2: Solution of Lambert’s problem

D.26 Calculation of Julian day number at 0 hr UT

D.27 Algorithm 5.3: Calculation of local sidereal time

D.28 Algorithm 5.4: Calculation of the state vector from measurements of range, angular position, and their rates

D.29 Algorithms 5.5 and 5.6: Gauss method of preliminary orbit determination with iterative improvement

Chapter 6

Chapter 7

D.32 Plot the position of one spacecraft relative to another

D.33 Solution of the linearized equations of relative motion with an elliptical reference orbit

Chapter 8

D.35 Algorithm 8.1: Calculation of the heliocentric state vector of a planet at a given epoch

D.36 Algorithm 8.2: Calculation of the spacecraft trajectory from planet 1 to planet 2

Chapter 9

D.38 Algorithm 9.2: Calculate the quaternion from the direction cosine matrix

D.39 Example 9.23: Solution of the spinning top problem

Chapter 11

Chapter 12

D.43 *J*2 perturbation of an orbit using Encke’s method

D.44 Example 12.6: Using Gauss variational equations to assess *J*2 effect on orbital elements

D.45 Algorithm 12.2: Calculate the geocentric position of the sun at a given epoch

D.46 Algorithm 12.3: Determine whether or not a satellite is in earth’s shadow

D.47 Example 12.9: Use the Gauss variational equations to determine the effect of solar radiation pressure on an earth satellite’s orbital parameters

D.48 Algorithm 12.4: Calculate the geocentric position of the moon at a given epoch

D.49 Example 12.11: Use the Gauss variational equations to determine the effect of lunar gravity on an earth satellite’s orbital parameters

D.50 Example 12.12: Use the Gauss variational equations to determine the effect of solar gravity on an earth satellite’s orbital parameters

## Description

Written by Howard Curtis, Professor of Aerospace Engineering at Embry-Riddle University, *Orbital Mechanics for Engineering Students* is a crucial text for students of aerospace engineering. Now in its 3e, the book has been brought up-to-date with new topics, key terms, homework exercises, and fully worked examples. Highly illustrated and fully supported with downloadable MATLAB algorithms for project and practical work, this book provides all the tools needed to fully understand the subject.

## Key Features

- New chapter on orbital perturbations
- New and revised examples and homework problems
- Increased coverage of attitude dynamics, including new MATLAB algorithms and examples

## Readership

Undergraduate students in aerospace, astronautical, mechanical engineering, and engineering physics; related professional aerospace and space engineering fields.

## Details

- No. of pages:
- 768

- Language:
- English

- Copyright:
- © Butterworth-Heinemann 2013

- Published:
- 5th October 2013

- Imprint:
- Butterworth-Heinemann

- eBook ISBN:
- 9780080977485

- Hardcover ISBN:
- 9780080977478

## Ratings and Reviews

## About the Author

### Howard Curtis

Professor Curtis is former professor and department chair of Aerospace Engineering at Embry-Riddle Aeronautical University. He is a licensed professional engineer and is the author of two textbooks (Orbital Mechanics 3e, Elsevier 2013, and Fundamentals of Aircraft Structural Analysis, McGraw Hill 1997). His research specialties include continuum mechanics, structures, dynamics, and orbital mechanics.

### Affiliations and Expertise

Professor Emeritus, Aerospace Engineering, Embry-Riddle Aeronautical University, Florida, USA