Solutions Manual for Interplanetary Astrodynamics, 1e by David
Spencer, Davide Conte (Selective Chapters, 2-6)
hapter 2 Problem Solutions
For all numerical problems, use π = 398, 600 km3/s2 as the gravitational parame
...
Solutions Manual for Interplanetary Astrodynamics, 1e by David
Spencer, Davide Conte (Selective Chapters, 2-6)
hapter 2 Problem Solutions
For all numerical problems, use π = 398, 600 km3/s2 as the gravitational parameter ofthe Earth.
Problem 1
Starting with the unperturbed two-body equations of motion, Equation (2.9), derive its state space
form in spherical coordinates.
Solution
Consider the Cartesian (π₯, π¦, and π§) formulation of the equations of motion for the two-body
problem:
ππ₯
π₯ = β
π
3
ππ¦
π¦ = β
π
3
ππ§
π§ = β
π
3
In order to convert between Cartesian and spherical coordinates, we use the following relationships
π₯ = π sinπcos π
π¦ = π sinπsin π
π§ = π cosπ
where π, π, and π are the spherical coordinates.
Taking one time-derivative of the above equations for the π₯, π¦, and π§ coordinates expressed in
terms of π, π, and π gives
π₯ = π cosπ sinπ + ππ cosπcosπ β ππ sinπsinπ
π¦ = π sinπsin π + ππ cosπsin π + ππ cos π sinπ
π§ = π cosπ β ππ sinπ
Test & Solution
Taking another time-derivative:
π₯ = π cos π sin π β ππ 2 cos π sin π β π 2 cos π sin π + ππ cos π cos π+
β π π sin π sin π + 2π π cos π cos π β 2π π sin π sin π β 2ππ π cos π sin π
π¦ = π sin π sin π β ππ 2 sin π sin π β ππ 2 sin π sin π + ππ cos π sin π+
+ ππ cos π sin π + 2π π cos π sin π + 2π π cos π sin π + 2ππ π cos π cos π
π§ = π cosπ β 2π π sinπ β ππ sinπ β ππ 2 cosοΏ½
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