Specific mechanical energy

Specific mechanical energy is the mechanical energy of an object per unit of mass. Similar to mechanical energy, the specific mechanical energy of an object in an isolated system subject only to conservative forces will remain constant.

Specific mechanical energy
Common symbols	e, or ε
SI unit	J/kg, or m^2/s^2

It is defined as:

${\displaystyle {\begin{aligned}e&=e_{k}+e_{p}\end{aligned}}}$

where

• ${\displaystyle e_{k}}$ is the specific kinetic energy

• ${\displaystyle e_{p}}$ it the specific potential energy

Astrodynamics

Main article: Specific orbital energy

In the gravitational two-body problem, the specific mechanical energy ${\displaystyle \epsilon }$ is given as:

${\displaystyle {\begin{aligned}\epsilon &={\frac {v^{2}}{2}}-{\frac {\mu }{r}}=-{\frac {1}{2}}{\frac {\mu ^{2}}{h^{2}}}\left(1-e^{2}\right)=-{\frac {\mu }{2a}}\end{aligned}}}$

where

• ${\displaystyle v\,\!}$ is the relative orbital speed;
• ${\displaystyle r\,\!}$ is the orbital distance between the bodies;
• ${\displaystyle \mu ={G}(m_{1}+m_{2})\,\!}$ is the sum of the standard gravitational parameters of the bodies;
• ${\displaystyle h\,\!}$ is the specific relative angular momentum in the sense of relative angular momentum divided by the reduced mass;
• ${\displaystyle e\,\!}$ is the orbital eccentricity;
• ${\displaystyle a\,\!}$ is the semi-major axis.

Orbital Mechanics

When calculating the specific mechanical energy of a satellite in orbit around a celestial body, the mass of the satellite is assumed to be negligible:

${\displaystyle \mu =G(M+m)\approx GM}$

where ${\displaystyle M}$ is the mass of the celestial body.