Specific energy

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

Circular and ellipticla orbits

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

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

$\epsilon =0$

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

Hyperbolic orbits

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

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.