1. Apollo 18: They KeY to the Next Mission to the Moon
Jon Xia and Richard Zhu

2. Types of Transfer Orbits

3. Hohmann Transfer

4. Retrograde vs. Prograde Orbit

5. Models and Proof

6. Models and Proof

7. Models and Proof Circle Dynamics: Ellipse Dynamics:
𝑥 2 + 𝑦 2 = 𝑟 2 ∧𝑑 𝑥 2 +𝑑 𝑦 2 =1∧𝑣>0∧𝑟>0→ 𝑥 ′ =𝑟𝑣𝑑𝑥, 𝑦 ′ =𝑟𝑣𝑑𝑦,𝑑 𝑥 ′ =−𝑣𝑑𝑦,𝑑 𝑦 ′ =𝑣𝑑𝑥 ( 𝑥 2 + 𝑦 2 = 𝑟 2 ∧𝑑 𝑥 2 +𝑑 𝑦 2 =1)
Ellipse Dynamics:
𝑣 𝑥 2 + 𝑣 𝑦 2 =𝐺𝑀 2 𝑑 − 1 𝑎 ∧𝑑>0∧𝑎>0→
[ 𝑥 ′ = 𝑣 𝑥 , 𝑦 ′ = 𝑣 𝑦 , 𝑣 𝑥 ′ =− 𝑑 𝑥 𝐺𝑀 𝑑 3 , 𝑣 𝑦 ′ =− 𝑑 𝑦 𝐺𝑀 𝑑 3 &𝑥≤𝑎]
𝑣 𝑥 2 + 𝑣 𝑦 2 =𝐺𝑀 2 𝑑 − 1 𝑎