Complete Solutions Manual - Interplanetary Astrodynamics, 1st Edition – David B. Spencer, Davide Conte – ISBN 9780367759704 (Chapters 2–6 Covered)

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Interplanetary
Astrodynamics 1st Edition
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SOLUTIONS
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MANUAL
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David B. Spencer
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Davide Conte
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Comprehensive Solutions Manual for
Instructors and Students
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© David B. Spencer & Davide Conte. All rights reserved.
Reproduction or distribution without permission is prohibited.
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9780367759704




© Medgeek

, Edrftgyihu jiuh
Solutions Manual — Interplanetary Astrodynamics, 1st Edition
— David B. Spencer and Davide Conte

Description
This solutions manual corresponds to the 1st edition of Interplanetary
Astrodynamics by David B. Spencer and Davide Conte. It follows the
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official textbook structure and provides detailed mathematical
derivations and computational procedures supporting advanced study
in astrodynamics, orbital mechanics, and interplanetary mission
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design. Coverage is limited to the chapters specified.

Table of Contents
Chapter 2: Kinematics, Dynamics, and Astrodynamics
Chapter 3: N-Body Problem
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Chapter 4: Coordinate Frames, Time, and Planetary Ephemerides
Chapter 5: Trajectory Design
Chapter 6: Navigation and Targeting
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© Medgeek

, Solutions Manual for Interplanetary Astrodynamics, 1e by David
Spencer, Davide Conte (Selective Chapters, 2-6)


Interplanetary Astrodynamics

Chapter 2 Problem Solutions

For all numerical problems, use 𝜇 = 398, 600 km3 /s2 as the gravitational parameter of the Earth.
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Problem 1
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Starting with the unperturbed two-body equations of motion, Equation (2.9), derive its state space
form in spherical coordinates.
Solution
Consider the Cartesian (𝑥, 𝑦, and 𝑧) formulation of the equations of motion for the two-body
problem:
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𝜇𝑥
𝑥̈ = −
𝑟3
𝜇𝑦
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𝑦̈ = − 3
𝑟
𝜇𝑧
𝑧̈ = − 3
𝑟

In order to convert between Cartesian and spherical coordinates, we use the following relationships
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𝑥 = 𝜌 sin 𝜙 cos 𝜃
𝑦 = 𝜌 sin 𝜙 sin 𝜃
𝑧 = 𝜌 cos 𝜙
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where 𝜌, 𝜙, and 𝜃 are the spherical coordinates.
Taking one time-derivative of the above equations for the 𝑥, 𝑦, and 𝑧 coordinates expressed in
terms of 𝜌, 𝜙, and 𝜃 gives
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𝑥̇ = 𝜌̇ cos 𝜃 sin 𝜙 + 𝜌𝜙̇ cos 𝜙 cos 𝜃 − 𝜌𝜃̇ sin 𝜙 sin 𝜃
𝑦̇ = 𝜌̇ sin 𝜙 sin 𝜃 + 𝜌𝜙̇ cos 𝜙 sin 𝜃 + 𝜌𝜃̇ cos 𝜃 sin 𝜃
𝑧̇ = 𝜌̇ cos 𝜙 − 𝜌𝜙̇ sin 𝜙
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, Taking another time-derivative:

𝑥̈ = 𝜌̈ cos 𝜃 sin 𝜙 − 𝜌𝜙̇ 2 cos 𝜃 sin 𝜙 − 𝜃̇ 2 cos 𝜃 sin 𝜙 + 𝜌𝜙̈ cos 𝜙 cos 𝜃+
̈ sin 𝜙 sin 𝜃 + 2𝜌̇ 𝜙̇ cos 𝜙 cos 𝜃 − 2𝜌̇ 𝜃̇ sin 𝜙 sin 𝜃 − 2𝜌𝜙̇ 𝜃̇ cos 𝜙 sin 𝜃
− 𝜃𝜌
𝑦̈ = 𝜌̈ sin 𝜙 sin 𝜃 − 𝜌𝜙̇ 2 sin 𝜙 sin 𝜃 − 𝜌𝜃̇ 2 sin 𝜙 sin 𝜃 + 𝜌𝜙̈ cos 𝜙 sin 𝜃+
+ 𝜌𝜃̈ cos 𝜃 sin 𝜙 + 2𝜌̇ 𝜙̇ cos 𝜙 sin 𝜃 + 2𝜌̇ 𝜃̇ cos 𝜃 sin 𝜙 + 2𝜌𝜃̇ 𝜙̇ cos 𝜙 cos 𝜃
𝑧̈ = 𝜌̈ cos 𝜙 − 2𝜌̇ 𝜙̇ sin 𝜙 − 𝜌𝜙̈ sin 𝜙 − 𝜌𝜙̇ 2 cos 𝜙
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Equating each 𝑥, 𝑦, and 𝑧 acceleration expressed in spherical coordinates with its respective
acceleration terms gives us the equations of motion for the two-body problem in terms of spherical
coordinates 𝜌, 𝜙, and 𝜃
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𝜌̈ cos 𝜃 sin 𝜙 − 𝜌𝜙̇ 2 cos 𝜃 sin 𝜙 − 𝜃̇ 2 cos 𝜃 sin 𝜙 + 𝜌𝜙̈ cos 𝜙 cos 𝜃+
̈ sin 𝜙 sin 𝜃 + 2𝜌̇ 𝜙̇ cos 𝜙 cos 𝜃 − 2𝜌̇ 𝜃̇ sin 𝜙 sin 𝜃 − 2𝜌𝜙̇ 𝜃̇ cos 𝜙 sin 𝜃+
− 𝜃𝜌
𝜇 sin 𝜙 cos 𝜃
+ =0
𝜌2
𝜌̈ sin 𝜙 sin 𝜃 − 𝜌𝜙̇ 2 sin 𝜙 sin 𝜃 − 𝜌𝜃̇ 2 sin 𝜙 sin 𝜃 + 𝜌𝜙̈ cos 𝜙 sin 𝜃+
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+ 𝜌𝜃̈ cos 𝜃 sin 𝜙 + 2𝜌̇ 𝜙̇ cos 𝜙 sin 𝜃 + 2𝜌̇ 𝜃̇ cos 𝜃 sin 𝜙 + 2𝜌𝜃̇ 𝜙̇ cos 𝜙 cos 𝜃
𝜇 sin 𝜙 sin 𝜃
+ =0
𝜌2
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𝜇 cos 𝜙
𝜌̈ cos 𝜙 − 2𝜌̇ 𝜙̇ sin 𝜙 − 𝜌𝜙̈ sin 𝜙 − 𝜌𝜙̇ 2 cos 𝜙 + =0
𝜌2
√
where we used the fact that 𝜌 = 𝑟 = 𝑥 2 + 𝑦 2 + 𝑧 2 .
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