What is $v_{\infty}$?

So, you can calculate the trajectory of a celestial projectile by using this equation:

$$e=\frac{rv_{\infty}}{\mu},$$

where $$\mu$$ is the central body's Gravitational Parameter, so it is definable by $$\mu=GM$$. $$r$$ is the orbiting body's current distance from the sun. That leaves us with only one variable left to uncover; $$v_{\infty}$$ (the Hyperbolic Excess Velocity). If you do a bit of research, you get an equation defining it as:

$$v_{\infty}$$ definition

Where you have to know variables considering Earth, and an outside planet. But, what if we need to find out the Hyperbolic Excess Velocity of an outside planet, having nothing to do with the Earth or another outside planet?

In an answer to this question, I want a reliable explanation of how to calculate $$v_{\infty}$$, not referring to another form of the equation for eccentricity, isolating $$v_{\infty}$$. I also want an explanation of what the Hyperbolic Excess Velocity is, in its essence, and an example of how to calculate it.

• Thank you, AFG, for helping me by editing my post. – Questioner Apr 12 at 14:31

1 Answer

Speed $$v_\infty$$ is the speed of the body at the moment where one can neglect all gravitational interactions, i.e. at infinite distance or (which is usually the same) after/before infinite time. The quantity $$v_\infty$$ is well defined in 2-body problem where the second body has an infinitely larger mass and is at rest (or central field). (If it's not, then one can distinguish between $$v_{-\infty}$$ the speed at $$t=-\infty$$ and $$v_{+\infty}$$, speed at $$t=+\infty$$)

• Can you give me an equation defining it? – Questioner Apr 12 at 16:56
• What do you mean by equation? $v_\infty = |dr/dt|_{t=\pm\infty}$? If you mean, how can you relate $v_\infty$ to other quantities, it depends on a particular problem. Usually one needs to write energy and angular momentum conservation laws. – Vasily Mitch Apr 13 at 8:40