Specific energy

In orbital mechanics, specific energy (symbol $ϵ$) is the total orbital energy per unit mass of an orbiting body.

Circular and ellipticla orbits

$ϵ=-\frac{\mu }{2a}=\frac{V^2}{2}-\frac{\mu }{r}<0$

where $a<0$ is the semi-major axis, $\mu$ is the gravitational parameter for the body being orbited, $r$ is the distance to the body being orbited at some point in time and $V$ is velocity at that time. This relationship comes about because $\frac{V^2}{2}$ is the kinetic energy and $-\frac{\mu }{2a}$ the potential energy of the system.

Parabolic orbits

$ϵ=0$

• to do: a more in-depth explanation of why orbital energy is defined in such a way that $ϵ$ is relative to the parabolic orbit.

Hyperbolic orbits

$ϵ=-\frac{\mu }{2a}=\frac{V^2}{2}-\frac{\mu }{r}>0$

where $a<0$ is the semi-transverse axis, $\mu$ is the gravitational parameter for the body being orbited, $r$ is distance to the body being orbited at some point in time and $V$ is velocity at that time.