Specific orbital energy

In the gravitational two-body problem, the specific orbital energy ${\displaystyle \varepsilon }${\displaystyle \varepsilon } (or specific vis-viva energy) of two orbiting bodies is the constant quotient of their mechanical energy (the sum of their mutual potential energy, ${\displaystyle \varepsilon _{p}}${\displaystyle \varepsilon _{p}}, and their kinetic energy, ${\displaystyle \varepsilon _{k}}${\displaystyle \varepsilon _{k}}) to their reduced mass.^[1]

According to the orbital energy conservation equation (also referred to as vis-viva equation), it does not vary with time: $${\displaystyle {\begin{aligned}\varepsilon &=\varepsilon _{k}+\varepsilon _{p}\\&={\frac {v^{2}}{2}}-{\frac {\mu }{r}}=-{\frac {1}{2}}{\frac {\mu ^{2}}{h^{2}}}\left(1-e^{2}\right)=-{\frac {\mu }{2a}}\end{aligned}}}$${\displaystyle {\begin{aligned}\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{aligned}}} where

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

It is a kind of specific energy, typically expressed in units of ${\displaystyle {\frac {\text{MJ}}{\text{kg}}}}${\displaystyle {\frac {\text{MJ}}{\text{kg}}}} (megajoule per kilogram) or ${\displaystyle {\frac {{\text{km}}^{2}}{{\text{s}}^{2}}}}${\displaystyle {\frac {{\text{km}}^{2}}{{\text{s}}^{2}}}} (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.

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:^[2]

$${\displaystyle \varepsilon =-{\frac {\mu }{2a}}}$${\displaystyle \varepsilon =-{\frac {\mu }{2a}}} where

• ${\displaystyle \mu =G\left(m_{1}+m_{2}\right)}${\displaystyle \mu =G\left(m_{1}+m_{2}\right)} is the standard gravitational parameter;
• ${\displaystyle a}${\displaystyle a} is semi-major axis of the orbit.

For a parabolic orbit this equation simplifies to $${\displaystyle \varepsilon =0.}$${\displaystyle \varepsilon =0.}

For a hyperbolic trajectory this specific orbital energy is either given by $${\displaystyle \varepsilon ={\mu \over 2a}.}$${\displaystyle \varepsilon ={\mu \over 2a}.}

or the same as for an ellipse, depending on the convention for the sign of a.

In this case the specific orbital energy is also referred to as characteristic energy (or ${\displaystyle C_{3}}${\displaystyle C_{3}}) and is equal to the excess specific energy compared to that for a parabolic orbit.

It is related to the hyperbolic excess velocity ${\displaystyle v_{\infty }}${\displaystyle v_{\infty }} (the orbital velocity at infinity) by $${\displaystyle 2\varepsilon =C_{3}=v_{\infty }^{2}.}$${\displaystyle 2\varepsilon =C_{3}=v_{\infty }^{2}.}

It is relevant for interplanetary missions.

Thus, if orbital position vector (${\displaystyle \mathbf {r} }${\displaystyle \mathbf {r} }) and orbital velocity vector (${\displaystyle \mathbf {v} }${\displaystyle \mathbf {v} }) are known at one position, and ${\displaystyle \mu }${\displaystyle \mu } is known, then the energy can be computed and from that, for any other position, the orbital speed.

For an elliptic orbit the rate of change of the specific orbital energy with respect to a change in the semi-major axis is $${\displaystyle {\frac {\mu }{2a^{2}}}}$${\displaystyle {\frac {\mu }{2a^{2}}}} where

• ${\displaystyle \mu ={G}(m_{1}+m_{2})}${\displaystyle \mu ={G}(m_{1}+m_{2})} is the standard gravitational parameter;
• ${\displaystyle a,円\!}${\displaystyle a,円\!} is semi-major axis of the orbit.

In the case of circular orbits, this rate is one half of the gravitation at the orbit. This corresponds to the fact that for such orbits the total energy is one half of the potential energy, because the kinetic energy is minus one half of the potential energy.

If the central body has radius R, then the additional specific energy of an elliptic orbit compared to being stationary at the surface is

$${\displaystyle -{\frac {\mu }{2a}}+{\frac {\mu }{R}}={\frac {\mu (2a-R)}{2aR}}}$${\displaystyle -{\frac {\mu }{2a}}+{\frac {\mu }{R}}={\frac {\mu (2a-R)}{2aR}}}

The quantity ${\displaystyle 2a-R}${\displaystyle 2a-R} is the height the ellipse extends above the surface, plus the periapsis distance (the distance the ellipse extends beyond the center of the Earth). For the Earth and ${\displaystyle a}${\displaystyle a} just little more than ${\displaystyle R}${\displaystyle R} the additional specific energy is ${\displaystyle (gR/2)}${\displaystyle (gR/2)}; which is the kinetic energy of the horizontal component of the velocity, i.e. ${\textstyle {\frac {1}{2}}V^{2}={\frac {1}{2}}gR}${\textstyle {\frac {1}{2}}V^{2}={\frac {1}{2}}gR}, ${\displaystyle V={\sqrt {gR}}}${\displaystyle V={\sqrt {gR}}}.

The International Space Station has an orbital period of 91.74 minutes (5504 s), hence by Kepler's Third Law the semi-major axis of its orbit is 6,738 km.^{[citation needed ]}

The specific orbital energy associated with this orbit is −29.6 MJ/kg: the potential energy is −59.2 MJ/kg, and the kinetic energy 29.6 MJ/kg. Compared with the potential energy at the surface, which is −62.6 MJ/kg., the extra potential energy is 3.4 MJ/kg, and the total extra energy is 33.0 MJ/kg. The average speed is 7.7 km/s, the net delta-v to reach this orbit is 8.1 km/s (the actual delta-v is typically 1.5–2.0 km/s more for atmospheric drag and gravity drag).

The increase per meter would be 4.4 J/kg; this rate corresponds to one half of the local gravity of 8.8 m/s².

For an altitude of 100 km (radius is 6471 km):

The energy is −30.8 MJ/kg: the potential energy is −61.6 MJ/kg, and the kinetic energy 30.8 MJ/kg. Compare with the potential energy at the surface, which is −62.6 MJ/kg. The extra potential energy is 1.0 MJ/kg, the total extra energy is 31.8 MJ/kg.

The increase per meter would be 4.8 J/kg; this rate corresponds to one half of the local gravity of 9.5 m/s². The speed is 7.8 km/s, the net delta-v to reach this orbit is 8.0 km/s.

Taking into account the rotation of the Earth, the delta-v is up to 0.46 km/s less (starting at the equator and going east) or more (if going west).

For Voyager 1 , with respect to the Sun:

• ${\displaystyle \mu =GM}${\displaystyle \mu =GM} = 132,712,440,018 km³⋅s⁻² is the standard gravitational parameter of the Sun
• r = 17 billion kilometers
• v = 17.1 km/s

Hence: $${\displaystyle \varepsilon =\varepsilon _{k}+\varepsilon _{p}={\frac {v^{2}}{2}}-{\frac {\mu }{r}}=\mathrm {146,円km^{2}s^{-2}} -\mathrm {8,円km^{2}s^{-2}} =\mathrm {138,円km^{2}s^{-2}} }$${\displaystyle \varepsilon =\varepsilon _{k}+\varepsilon _{p}={\frac {v^{2}}{2}}-{\frac {\mu }{r}}=\mathrm {146,円km^{2}s^{-2}} -\mathrm {8,円km^{2}s^{-2}} =\mathrm {138,円km^{2}s^{-2}} }

Thus the hyperbolic excess velocity (the theoretical orbital velocity at infinity) is given by $${\displaystyle v_{\infty }=\mathrm {16.6,円km/s} }$${\displaystyle v_{\infty }=\mathrm {16.6,円km/s} }

However, Voyager 1 does not have enough velocity to leave the Milky Way. The computed speed applies far away from the Sun, but at such a position that the potential energy with respect to the Milky Way as a whole has changed negligibly, and only if there is no strong interaction with celestial bodies other than the Sun.

Assume:

• a is the acceleration due to thrust (the time-rate at which delta-v is spent)
• g is the gravitational field strength
• v is the velocity of the rocket

Then the time-rate of change of the specific energy of the rocket is ${\displaystyle \mathbf {v} \cdot \mathbf {a} }${\displaystyle \mathbf {v} \cdot \mathbf {a} }: an amount ${\displaystyle \mathbf {v} \cdot (\mathbf {a} -\mathbf {g} )}${\displaystyle \mathbf {v} \cdot (\mathbf {a} -\mathbf {g} )} for the kinetic energy and an amount ${\displaystyle \mathbf {v} \cdot \mathbf {g} }${\displaystyle \mathbf {v} \cdot \mathbf {g} } for the potential energy.

The change of the specific energy of the rocket per unit change of delta-v is $${\displaystyle {\frac {\mathbf {v\cdot a} }{|\mathbf {a} |}}}$${\displaystyle {\frac {\mathbf {v\cdot a} }{|\mathbf {a} |}}} 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: $${\displaystyle \Delta \varepsilon =\int v,円d(\Delta v)=\int v,円adt}$${\displaystyle \Delta \varepsilon =\int v,円d(\Delta v)=\int v,円adt}

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

• Astrodynamics
• Orbits
• Physical quantities
• Mass-specific quantities

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