If a spacecraft slingshots around a planet P (with escape velocity $V$) at an angle $\theta$, I understand that the resulting velocity will be


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

$$\delta =2\sin^{-1}\left(\frac {1}{1+\dfrac{ {r}_{p}v_{\infty}^{2}}{\mu}}\right),$$


Please note that I'm a high school student with a knowledge of Calculus I & II, and that visual diagrams would be greatly helpful.



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