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

Hyperbolic excess velocity

See also: Characteristic energy

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

$v\_\backslash infty=\backslash sqrt\{\backslash mu\backslash over\{a\}\}\backslash ,\backslash !$

where:

• $\backslash mu\backslash ,\backslash !$ is standard gravitational parameter,
• $a\backslash ,\backslash !$ is length of semi-major axis of orbit\'s hyperbola.

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

$2\backslash epsilon=C\_3=v\_\{\backslash infty\}^2\backslash ,\backslash !$

Velocity

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

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

where:

• $\backslash mu\backslash ,$ is standard gravitational parameter,
• $r\backslash ,$ is radial distance of orbiting body from central body,
• $a\backslash ,\backslash !$ is length of semi-major axis, always a negative number for hyperbolas.

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

$v^2=\{v\_\{esc\}\}^2+\{v\_\backslash infty\}^2$

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

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

where:

• $v\backslash ,$ is orbital velocity of orbiting body,
• $r\backslash ,$ is radial distance of orbiting body from central body,
• $a\backslash ,$ is length of semi-major axis,
• $\backslash mu\backslash ,$ is standard gravitational parameter.

See also

• Orbit
• Orbital equation
• List of orbits

External links

• http://homepage.mac.com/sjbradshaw/msc/traject.html
• http://www.go.ednet.ns.ca/~larry/orbits/ellipse.html

This article is licensed under the GNU Free Documentation License. It uses material from Wikipedia