<p>If a spacecraft slingshots around a planet P (with escape velocity V) at an angle <span class="math-container">$\theta$</span>, I understand that the resulting velocity is <span class="math-container">$${ v }_{ 2 }=({ v }_{ 1 }+2u)\sqrt { 1-\frac { 4u{ v }_{ 1 }(1-cos(\theta )) }{ ({ v }_{ 1 }+2u)^{ 2 } }  } .$$</span></p>

<p>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 <a href="https://space.stackexchange.com/q/6504/">To what extent could a single Triton flyby slow down a direct Hohmann transfer to Neptune for NOI?</a>, I have a few questions: </p>

<ul>
<li>How is the formula for the bent angle and eccentricity derived?</li>
<li>What exactly is the geometry behind the maneuver? More specifically, where does the hyperbola come from?</li>
<li>How is the formula for the turning angle, <span class="math-container">$\delta =2\sin ^{ -1 }{ \left( \frac { 1 }{ 1+\frac { { r }_{ p }{ v }_{ \infty  }^{ 2 } }{ \mu  }  }  \right)  } $</span> derived?</li>
</ul>

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

CC BY-SA 4.0
2020-01-16T20:28:30.840Z
If you know the escape velocity you should be able to get the mass of the planet - see wikipedia for the formula,  $v_e = \sqrt{2GM/r} $  .  After that, if you can map your various "velocity" parameters from one equation to another for us, that might help.  Also keep in mind that the minimum altitude at least has to be exoatmospheric!
Carl Witthoft
CC BY-SA 4.0
2020-01-17T08:37:17.873Z
What is $u$ in the equation?
usernumber
CC BY-SA 4.0
2020-01-18T19:52:17.977Z
I see some similar stuff at [Wikipedia: Hyperbolic trajectory](https://en.wikipedia.org/wiki/Hyperbolic_trajectory) but would appreciate a link to the source of your first equation.
Mike G
<orbit><newtonian-gravity>
Gravitational Slingshot
DarkRunner
# Gravitational Slingshot

(Asked by: username_0 on 2020-01-15)

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?](https://space.stackexchange.com/q/6504/), 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.

**Tags:** orbit, newtonian-gravity

> **Comments:**
> **username_1**: If you know the escape velocity you should be able to get the mass of the planet - see wikipedia for the formula,  $v_e = \sqrt{2GM/r} $  .  After that, if you can map your various "velocity" parameters from one equation to another for us, that might help.  Also keep in mind that the minimum altitude at least has to be exoatmospheric!  
> **username_2**: What is $u$ in the equation?  
> **username_3**: I see some similar stuff at [Wikipedia: Hyperbolic trajectory](https://en.wikipedia.org/wiki/Hyperbolic_trajectory) but would appreciate a link to the source of your first equation.  

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## 0 answers

