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# Hyperbolic trajectory

In astrodynamics or celestial mechanics a hyperbolic trajectory is an orbit with the eccentricity greater than 1. Under standard assumptions a body traveling along this trajectory will coast to infinity, arriving there with hyperbolic excess velocity relative to the central body. Similarly to parabolic trajectory all hyperbolic trajectories are also escape trajectories. Specific energy of hyperbolic trajectory orbit is positive.

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## Hyperbolic excess velocity

Under standard assumptions the body traveling along along hyperbolic trajectory will attain in infinity an orbital velocity called hyperbolic excess velocity ($v_\infty\,\!$) that can be computed as:

$v_\infty=\sqrt{\mu\over{a}}\,\!$

where:

The hyperbolic excess velocity is related to the specific orbital energy or characteristic energy by

$2\epsilon=2C_3=v_{\infty}^2\,\!$

## Velocity

Under standard assumptions the orbital velocity ($v\,$) of a body traveling along hyperbolic trajctory can be computed as:

$v=\sqrt{2\mu\left({1\over{r}}+{1\over{2a}}\right)}$

where:

Under standard assumptions, at any position in the orbit the following relation holds for orbital velocity ($v\,$), local escape velocity(${v_{esc}}\,$) and hyperbolic excess velocity ($v_\infty\,\!$):

$v^2={v_{esc}}^2+{v_\infty}^2$

Note that this means that a relatively small extra delta-v above that needed to accelerate to the escape speed, results in a relatively large speed at infinity.

## Energy

Under standard assumptions, specific orbital energy ($\epsilon\,$) of parabolic trajectory is greater than zero and the orbital energy conservation equation for this kind of trajectory takes form:

$\epsilon={v^2\over2}-{\mu\over{r}}={\mu\over{2a}}$

where: