Orbital Mechanics for Engineering Students

Cover
Orbital Mechanics for
Engineering Students
Copyright
Dedication
Preface
	Supplements to the text
	Acknowledgements
1
Dynamics of point masses
	Introduction
	Vectors
	Kinematics
	Mass, force, and Newtons law of gravitation
	Newtons law of motion
	Time derivatives of moving vectors
	Relative motion
	Numerical integration
		Runge-Kutta methods
		Heuns predictor-corrector method
		Runge-Kutta with variable step size
	Problems
	References
2
The two-body problem
	Introduction
	Equations of motion in an inertial frame
	Equations of relative motion
	Angular momentum and the orbit formulas
	The energy law
	Circular orbits (e=0)
	Elliptical orbits (01)
	Perifocal frame
	The Lagrange coefficients
	Circular restricted three-body problem
		Lagrange points
		Jacobi constant
	Problems
	References
3
Orbital position as a function of time
	Introduction
	Time since periapsis
	Circular orbits (e=0)
	Elliptical orbits (e<1)
	Parabolic trajectories (e=1)
	Hyperbolic trajectories (e>1)
	Universal variables
	Problems
	References
4
Orbits in three dimensions
	Introduction
	Geocentric right ascension-declination frame
	State vector and the geocentric equatorial frame
	Orbital elements and the state vector
	Coordinate transformation
	Transformation between geocentric equatorial and perifocal frames
	Effects of the earths oblateness
		Ground tracks
	Problems
	Reference
5
Preliminary orbit determination
	Introduction
	Gibbs method of orbit determination from three position vectors
	Lamberts problem
	Sidereal time
	Topocentric coordinate system
	Topocentric equatorial coordinate system
	Topocentric horizon coordinate system
	Orbit determination from angle and range measurements
	Angles-only preliminary orbit determination
	Gauss method of preliminary orbit determination
	Problems
	References
6
Orbital maneuvers
	Introduction
	Impulsive maneuvers
	Hohmann transfer
	Bielliptic Hohmann transfer
	Phasing maneuvers
	Non-Hohmann transfers with a common apse line
	Apse line rotation
	Chase maneuvers
	Plane change maneuvers
	Nonimpulsive orbital maneuvers
	Problems
	References
7
Relative motion and rendezvous
	Introduction
	Relative motion in orbit
	Linearization of the equations of relative motion in orbit
	Clohessy-Wiltshire equations
	Two-impulse rendezvous maneuvers
	Relative motion in close-proximity circular orbits
	Problems
	Reference
8
Interplanetary trajectories
	Introduction
	Interplanetary Hohmann transfers
	Rendezvous opportunities
	Sphere of influence
	Method of patched conics
	Planetary departure
	Sensitivity analysis
	Planetary rendezvous
	Planetary flyby
	Planetary ephemeris
	Non-Hohmann interplanetary trajectories
	Problems
	References
9
Lunar trajectories
	Introduction
	Coplanar patched conic lunar trajectories
	A simplified lunar ephemeris
	Patched conic lunar trajectories in three dimensions
	Lunar trajectories by numerical integration
	Problems
	References
10
Introduction to orbital perturbations
	Introduction
	Cowells method
	Enckes method
	Atmospheric drag
	Gravitational perturbations
	Variation of parameters
	Gauss' variational equations
		Variation of the specific angular momentum h
		Variation of the eccentricity e
		Variation of the true anomaly θ
		Variation of right ascension Omega
		Variation of the inclination i
		Variation of argument of periapsis ω
	Method of averaging
		Orbital-averaged angular momentum variation
		Orbital-averaged eccentricity variation
		Orbital-averaged true anomaly variation
		Orbital-averaged right ascension of ascending node variation
		Orbital-averaged inclination variation
		Orbital-averaged argument of perigee variation
	Solar radiation pressure
	Lunar gravity
	Solar gravity
	Problems
	References
11
Rigid body dynamics
	Introduction
	Kinematics
	Equations of translational motion
	Equations of rotational motion
	Moments of inertia
		Parallel axis theorem
	Euler equations
	Kinetic energy
	The spinning top
	Euler angles
	Yaw, pitch, and roll angles
	Quaternions
	Problems
	References
12
Spacecraft attitude dynamics
	Introduction
	Torque-free motion
	Stability of torque-free motion
	Dual-spin spacecraft
	Nutation damper
	Coning maneuver
	Attitude control thrusters
	Yo-yo despin mechanism
		Radial release
	Gyroscopic attitude control
	Gravity gradient stabilization
	Problems
	References
13
Rocket vehicle dynamics
	Introduction
	Equations of motion
	The thrust equation
	Rocket performance
	Restricted staging in field-free space
	Optimal staging
		Lagrange multiplier
	Problems
	References
APPENDIX A. Physical Data
APPENDIX B.
A Road Map
APPENDIX C.
Numerical Integration of the N-Body Equations of Motion
APPENDIX D.
MATLAB Scripts
	Introduction
	Chapter 1: Dynamics of Point Masses
		Algorithm 1.1: Numerical integration by Runge-Kutta methods RK1, RK2, RK3, or RK4
			Function file rkf1_4.m
			Function file: Example_1_18.m
		Algorithm 1.2: Numerical integration by Heuns predictor-corrector method
			Function file: heun.m
			Function file: Example_1_19.m
		Algorithm 1.3: Numerical integration of a system of first-order differential equations by the Runge-Kutta-Fehlberg 4 ...
			Function file: rkf45.m
			Function file: Example_1_20.m
	Chapter 2: The Two-body Problem
		Algorithm 2.1: Numerical solution of the two-body problem relative to an inertial frame
			Function file: twobody3d.m
		Algorithm 2.2: Numerical solution of the two-body relative motion problem
			Function file: orbit.m
		Calculation of the Lagrange f and g functions and their time derivatives in terms of change in true anomaly
			Function file: f_and_g_ta.m
			Function file: fDot_and_gDot_ta.m
		Algorithm 2.3: Calculate the state vector from the initial state vector and the change in true anomaly
			Function file: rv_from_r0v0_ta.m
			Script file: Example_2_13.m
			Output from Example_2_13.m
		Algorithm 2.4: Find the root of a function using the bisection method
			Function file: bisect.m
			Function file: Example_2_16.m
			Output from Example_2_16.m
		MATLAB solution of Example 2.18
			Function file: Example_2_18.m
			Output from Example_2_18.m
	Chapter 3: Orbital Position as a Function of Time
		Algorithm 3.1: Solution of Keplers equation by Newtons method
			Function file: kepler_E.m
			Script file: Example_3_02.m
			Output from Example_3_02.m
		Algorithm 3.2: Solution of Keplers equation for the hyperbola using Newtons method
			Function file: kepler_H.m
			Script file: Example_3_05.m
			Output from Example_3_05.m
		Calculation of the Stumpff functions S(z) and C(z)
			Function file: stumpS.m
			Function file: stumpC.m
		Algorithm 3.3: Solution of the universal Keplers equation using Newtons method
			Function file: kepler_U.m
			Script file: Example_3_06.m
			Output from Example_3_06.m
		Calculation of the Lagrange coefficients f and g and their time derivatives in terms of change in univeral anomaly
			Function file: f_and_g.m
			Function file: fDot_and_gDot.m
		Algorithm 3.4: Calculation of the state vector given the initial state vector and the time lapse Deltat
			Function file: rv_from_r0v0.m
			Script file: Example_3_07.m
			Output from Example_3_07
	Chapter 4: Orbits in Three Dimensions
		Algorithm 4.1: Obtain the right ascension and declination from the position vector
			Function file: ra_and_dec_from_r.m
			Script file: Example_4_01.m
			Output from Example_4_01.m
		Algorithm 4.2: Calculation of the orbital elements from the state vector
			Function file: coe_from_sv.m
			Script file: Example_4_03.m
			Output from Example_4_03
		Calculation of arctan (y/x) to lie in the range 0ú to 360
			Function file: atan2d_0_360.m
		Algorithm 4.3: Obtain the classical Euler angle sequence from a direction cosine matrix
			Function file: dcm_to_euler.m
		Algorithm 4.4: Obtain the yaw, pitch, and roll angles from a direction cosine matrix
			Function file: dcm_to_ypr.m
		Algorithm 4.5: Calculation of the state vector from the orbital elements
			Function file: sv_from_coe.m
			Script file: Example_4_07.m
			Output from Example_4_05
		Algorithm 4.6: Calculate the ground track of a satellite from its orbital elements
			[B] Function file: ground_track.m
	Chapter 5: Preliminary Orbit Determination
		Algorithm 5.1: Gibbs method of preliminary orbit determination
			Function file: gibbs.m
			Script file: Example_5_01.m
			Output from Example_5_01
		Algorithm 5.2: Solution of Lamberts problem
			Function file: lambert.m
			Script file: Example_5_02.m
			Output from Example_5_02
		Calculation of Julian day number at 0 hr UT
			Function file: J0.m
			Script file: Example_5_04.m
			Output from Example_5_04
		Algorithm 5.3: Calculation of local sidereal time
			Function file: LST.m
			Script file: Example_5_06.m
			Output from Example_5_06
		Algorithm 5.4: Calculation of the state vector from measurements of range, angular position, and their rates
			Function file: rv_from_observe.m
			Script file: Example_5_10.m
			Output from Example_5_10
		Algorithms 5.5 and 5.6: Gauss method of preliminary orbit determination with iterative improvement
			Function file: gauss.m
			Script file: Example_5_11.m
			Output from Example_5_11
	Chapter 6: Orbital Maneuvers
		Calculate the state vector after a finite time, constant thrust delta-v maneuver
			Function file: integrate_thrust.m
	Chapter 7: Relative Motion and Rendezvous
		Algorithm 7.1: Find the position, velocity, and acceleration of B relative to As LVLH frame
			Function file: rva_relative.m
			Script file: Example_7_01.m
			Output from Example_7_01.m
		Plot the position of one spacecraft relative to another
			Script file: Example_7_02.m
		Solution of the linearized equations of relative motion with an elliptical reference orbit
			Function file: Example_7_03.m
	Chapter 8: Interplanetary Trajectories
		Convert the numerical designation of a month or a planet into its name
			Function file: month_planet_names.m
		Algorithm 8.1: Calculation of the heliocentric state vector of a planet at a given epoch
			Function file: planet_elements_and_sv.m
			Script file: Example_8_07.m
			[Output from Example_8_07
		Algorithm 8.2: Calculation of the spacecraft trajectory from planet 1 to planet 2
			Function file: interplanetary.m
			Script file: Example_8_08.m
			Output from Example_8_08
	Chapter 9: Lunar Trajectories
		Lunar state vector vs. time
			Function file: simpsons_lunar_ephemeris.m
		Numerical calculation of lunar trajectory
			Script File: Example_9_03.m
			Output from Example_9_03.m
	Chapter 10: Introduction to Orbital Perturbations
		US Standard Atmosphere 1976
			Function file: atmosphere.m
		Time for orbit decay using Cowells method
			Function file: Example_10_01.m
		J2 perturbation of an orbit using Enckes method
			Function file: Example_10_02.m
		Example 10.6: Using Gauss variational equations to assess J2 effect on orbital elements
			Function file: Example_10_06.m
		Algorithm 10.2: Calculate the geocentric position of the sun at a given epoch
			Function file: solar_position.m
		Algorithm 10.3: Determine whether or not a satellite is in earths shadow
			Function file: los.m
		Example 10.9: Use Gauss variational equations to determine the effect of solar radiation pressure on an earth satel ...
			Function file: Example_10_09.m
		Algorithm 10.4: Calculate the geocentric position of the moon at a given epoch
			Function file: lunar_position.m
		Example 10.11: Use Gauss variational equations to determine the effect of lunar gravity on an earth satellites orbi ...
			Function file: Example_10_11.m
		Example 10.12: Use Gauss variational equations to determine the effect of solar gravity on an earth satellites orbi ...
			Function file: Example_10_12.m
	Chapter 11: Rigid Body Dynamics
		Algorithm 11.1: Calculate the direction cosine matrix from the quaternion
			Function file: dcm_from_q.m
		Algorithm 11.2: Calculate the quaternion from the direction cosine matrix
			Function file: q_from_dcm.m
		Quaternion vector rotation operation (Eq. 11.160)
			Function file: quat_rotate.m
		Example 11.26: Solution of the spinning top problem
			Function file: Example_11_23.m
	Chapter 12: Spacecraft Attitude Dynamics
	Chapter 13: Rocket Vehicle Dynamics
		Example 13.3: Calculation of a gravity turn trajectory
			Function file: Example_13_03.m
APPENDIX E.
Gravitational Potential of a Sphere
APPENDIX F.
Computing the Difference Between Nearly Equal Numbers
	Reference
APPENDIX G.
Direction Cosine Matrix in Terms of the Unit Quaternion
Index
	A
	B
	C
	D
	E
	F
	G
	H
	I
	J
	K
	L
	M
	N
	O
	P
	Q
	R
	S
	T
	U
	V
	W
	Y
	Z
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