If a spacecraft slingshots around a planet P (with escape velocity $V$) at an angle $\theta$, I understand that the resulting velocity will be
$$v_{2}=(v_{1}+2u)\sqrt{1-\frac{4uv_{1}(1-\cos\theta)}{(v_{1}+2u)^{2}}}.$$
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, given by
$$\delta =2\sin^{-1}\left(\frac {1}{1+\dfrac{ {r}_{p}v_{\infty}^{2}}{\mu}}\right),$$
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.
