Specific orbital energy

In the gravitational two-body problem, the specific orbital energy <math>\varepsilon</math> (or specific vis-viva energy) of two orbiting bodies is the constant quotient of their mechanical energy (the sum of their mutual potential energy, <math>\varepsilon_p</math>, and their kinetic energy, <math>\varepsilon_k</math>) to their reduced mass.<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref>

According to the orbital energy conservation equation (also referred to as vis-viva equation), it does not vary with time: <math display="block">\begin{align} \varepsilon &= \varepsilon_k + \varepsilon_p \\

          &=  \frac{v^2}{2} - \frac{\mu}{r}
           = -\frac{1}{2} \frac{\mu^2}{h^2} \left(1 - e^2\right)
           = -\frac{\mu}{2a}

\end{align}</math> where

• <math>v</math> is the relative orbital speed;
• <math>r</math> is the orbital distance between the bodies;
• <math>\mu = {G}(m_1 + m_2)</math> is the sum of the standard gravitational parameters of the bodies;
• <math>h</math> is the specific relative angular momentum in the sense of relative angular momentum divided by the reduced mass;
• <math>e</math> is the orbital eccentricity;
• <math>a</math> is the semi-major axis.

It is a kind of specific energy, typically expressed in units of <math>\frac{\text{MJ}}{\text{kg}}</math> (megajoule per kilogram) or <math>\frac{\text{km}^2}{\text{s}^2}</math> (squared kilometer per squared second). For an elliptic orbit the specific orbital energy is the negative of the additional energy required to accelerate a mass of one kilogram to escape velocity (parabolic orbit). For a hyperbolic orbit, it is equal to the excess energy compared to that of a parabolic orbit. In this case the specific orbital energy is also referred to as characteristic energy.

Equation forms for different orbits

For an elliptic orbit, the specific orbital energy equation, when combined with conservation of specific angular momentum at one of the orbit's apsides, simplifies to:<ref name="Bong Wie SVDC">Template:Cite book</ref>

<math display="block">\varepsilon = -\frac{\mu}{2a}</math> where

• <math>\mu = G\left(m_1 + m_2\right)</math> is the standard gravitational parameter;
• <math>a</math> is semi-major axis of the orbit.

Template:Math proof{|\mathbf{a}|}</math> which is |v| times the cosine of the angle between v and a.

Thus, when applying delta-v to increase specific orbital energy, this is done most efficiently if a is applied in the direction of v, and when |v| is large. If the angle between v and g is obtuse, for example in a launch and in a transfer to a higher orbit, this means applying the delta-v as early as possible and at full capacity. See also gravity drag. When passing by a celestial body it means applying thrust when nearest to the body. When gradually making an elliptic orbit larger, it means applying thrust each time when near the periapsis. Such maneuver is called an Oberth maneuver or powered flyby.

When applying delta-v to decrease specific orbital energy, this is done most efficiently if a is applied in the direction opposite to that of v, and again when |v| is large. If the angle between v and g is acute, for example in a landing (on a celestial body without atmosphere) and in a transfer to a circular orbit around a celestial body when arriving from outside, this means applying the delta-v as late as possible. When passing by a planet it means applying thrust when nearest to the planet. When gradually making an elliptic orbit smaller, it means applying thrust each time when near the periapsis.

If a is in the direction of v: <math display="block">\Delta \varepsilon = \int v\, d (\Delta v) = \int v\, a dt</math>

See also

• Specific energy change of rockets
• Characteristic energy C3 (Double the specific orbital energy)