# Hyperbolic trajectory A hyperbolic trajectory is depicted in the bottom-right quadrant of this diagram, where the gravitational potential well of the central mass shows potential energy, and the kinetic energy of the hyperbolic trajectory is shown in red. The height of the kinetic energy decreases as the speed decreases and distance increases according to Kepler's laws. The part of the kinetic energy that remains above zero total energy is that associated with the hyperbolic excess velocity.

In astrodynamics or celestial mechanics a hyperbolic trajectory is the trajectory of any object around a central body with more than enough velocity to escape the central object's gravitational pull. The name derives from the fact that according to Newtonian theory such an orbit has the shape of a hyperbola. In more technical terms this can be expressed by the condition that the orbital eccentricity is greater than one.

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. The specific energy of a hyperbolic trajectory orbit is positive.

Planetary flybys, used for gravitational slingshots, can be described within the planet's sphere of influence using hyperbolic trajectories.

## Hyperbolic excess velocity

Under standard assumptions the body traveling 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

$2\epsilon =C_{3}=v_{\infty }^{2}\,\!$ $C_{3}$ is commonly used in planning interplanetary missions.

## Velocity

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

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

$v^{2}={v_{esc}}^{2}+{v_{\infty }}^{2}$ 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. For example, at a place where escape speed is 11.2 km/s, the addition of 0.4 km/s yields a hyperbolic excess speed of 3.02 km/s.

${\sqrt {11.6^{2}-11.2^{2}}}=3.02$ ## Angle between approach and departure

Let the angle between approach and departure (between asymptotes) be $2\theta \,$ .

$\theta =\cos ^{-1}(1/e)\,$ and $e=1/\cos \theta \,$ where:

## Distance of closest approach

The distance of closest approach, also called the periapse distance and the focal distance, is given by $-a(e-1)\,$ ## Energy

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

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