If a spacecraft slingshots around a planet P (with escape velocity V) at an angle $\theta$, I understand that the resulting velocity is $${ v }_{ 2 }=({ v }_{ 1 }+2u)\sqrt { 1-\frac { 4u{ v }_{ 1 }(1-cos(\theta )) }{ ({ v }_{ 1 }+2u)^{ 2 } } } .$$
However, this equation does not involve the mass of the assisting planet, nor the distance/altitude from which the spacecraft must slingshot from. After reading the answer to To what extent could a single Triton flyby slow down a direct Hohmann transfer to Neptune for NOI?, I have a few questions:
- How is the formula for the bent angle and eccentricity derived?
- What exactly is the geometry behind the maneuver? More specifically, where does the hyperbola come from?
- How is the formula for the turning angle, $\delta =2\sin ^{ -1 }{ \left( \frac { 1 }{ 1+\frac { { r }_{ p }{ v }_{ \infty }^{ 2 } }{ \mu } } \right) } $ derived?
Please note that I'm a high school student with a knowledge of Calculus I & II, and that visual diagrams would be greatly helpful.
