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

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

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

$\epsilon =0$

Since orbital mechanics only concerns itself with changes in orbital energy, the zero could be chosen arbitrarily. It is computationally most convenient to choose the value at escape velocity (i.e. parabolic orbit). This choice makes the semi-major axis inversely proportional to the specific energy and if the mass does not change also to the total orbital energy.

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.