View source for Specific orbital energy

{{Astrodynamics}}
In the [[gravitational two-body problem]], the '''specific orbital energy''' <math>\varepsilon</math> (or '''vis-viva energy''') of two [[orbiting body|orbiting bodies]] is the constant sum of their mutual [[potential energy]] (<math>\varepsilon_p</math>) and their total [[kinetic energy]] (<math>\varepsilon_k</math>), divided by the [[reduced mass]].<ref>{{Cite web |title=Specific energy |url=https://marspedia.org/Specific_energy |access-date=2022-08-12 |website=Marspedia |language=en}}</ref> According to the [[Vis-viva equation|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 state vectors|orbital distance]] between the bodies;
*<math>\mu = {G}(m_1 + m_2)</math> is the sum of the [[standard gravitational parameter]]s 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 [[eccentricity (orbit)|orbital eccentricity]];
*<math>a</math> is the [[semi-major axis]].

It is typically expressed in <math>\frac{\text{MJ}}{\text{kg}}</math> (mega[[joule]] 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 trajectory|parabolic orbit]]). For a [[hyperbolic trajectory|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 [[Specific relative angular momentum|conservation of specific angular momentum]] at one of the orbit's [[apsis|apsides]], simplifies to:<ref name="Bong Wie SVDC">{{cite book|last=Wie|first=Bong|title=Space Vehicle Dynamics and Control|publisher=[[American Institute of Aeronautics and Astronautics]] |location=Reston, Virginia|year=1998|series=AIAA Education Series|chapter=Orbital Dynamics|isbn=1-56347-261-9| page=[https://archive.org/details/spacevehicledyna00wieb_0/page/220 220]|chapter-url-access=registration| chapter-url=https://archive.org/details/spacevehicledyna00wieb_0/page/220}}</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.

{{math proof | proof =
For an elliptic orbit with [[specific angular momentum]] ''h'' given by
<math display="block">h^2 = \mu p = \mu a \left(1 - e^2\right)</math>
we use the general form of the specific orbital energy equation,
<math display="block">\varepsilon = \frac{v^2}{2} - \frac{\mu}{r}</math>
with the relation that the relative velocity at [[periapsis]] is
<math display="block"> v_p^2
 = {h^2 \over r_p^2}
 = {h^2 \over a^2(1 - e)^2}
 = {\mu a \left(1 - e^2\right) \over a^2(1 - e)^2}
 = {\mu \left(1 - e^2\right) \over a(1 - e)^2}
</math>
Thus our specific orbital energy equation becomes
<math display="block"> \varepsilon
 = \frac{\mu}{a} {\left[ {    1 - e^2 \over 2(1 - e)^2} - {1 \over 1 - e} \right]}
 = \frac{\mu}{a} {\left[ {(1 - e)(1 + e) \over 2(1 - e)^2} - {1 \over 1 - e} \right]}
 = \frac{\mu}{a} {\left[ {         1 + e \over 2(1 - e)}   - {2 \over 2(1 - e)} \right]}
 = \frac{\mu}{a} {\left[ {         e - 1 \over 2(1 - e)} \right]}
</math>
and finally with the last simplification we obtain:
<math display="block">\varepsilon = -{\mu \over 2a}</math>}}

For a [[parabolic orbit]] this equation simplifies to
<math display="block">\varepsilon = 0.</math>

For a [[hyperbolic trajectory]] this specific orbital energy is either given by
<math display="block">\varepsilon = {\mu \over 2a}.</math>

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 <math>C_3</math>) and is equal to the excess specific energy compared to that for a parabolic orbit.

It is related to the [[hyperbolic excess velocity]] <math>v_\infty</math> (the [[kinetic energy|orbital velocity]] at infinity) by
<math display="block">2\varepsilon = C_3 = v_\infty^2.</math>

It is relevant for interplanetary missions.

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

==Rate of change==

For an elliptic orbit the rate of change of the specific orbital energy with respect to a change in the semi-major axis is
<math display="block">\frac{\mu}{2a^2}</math>
where
*<math> \mu={G}(m_1 + m_2)</math> is the [[standard gravitational parameter]];
*<math>a\,\!</math> 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.

==Additional 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

<math display="block"> -\frac{\mu}{2a}+\frac{\mu}{R} = \frac{\mu(2a-R)}{2aR}</math>

The quantity <math>2a-R</math> 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 <math>a</math> just little more than <math>R</math> the additional specific energy is <math>(gR/2)</math>; which is the kinetic energy of the horizontal component of the velocity, i.e. <math display="inline">\frac{1}{2}V^2 = \frac{1}{2}gR</math>,  <math>V=\sqrt{gR}</math>.

==Examples==
<!-- This section is linked from [[Low Earth orbit]] -->

===ISS===
The [[International Space Station]] has an [[orbital period]] of 91.74 minutes (5504{{nbsp}}s), hence by [[Kepler's laws of planetary motion|Kepler's Third Law]] the semi-major axis of its orbit is 6,738{{nbsp}}km.{{cn|date=March 2022}}

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

The increase per meter would be 4.4{{nbsp}}J/kg; this rate corresponds to one half of the local gravity of 8.8{{nbsp}}m/s<sup>2</sup>.

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

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

The increase per meter would be 4.8{{nbsp}}J/kg; this rate corresponds to one half of the local gravity of 9.5{{nbsp}}m/s<sup>2</sup>. The speed is 7.8{{nbsp}}km/s, the net delta-v to reach this orbit is 8.0{{nbsp}}km/s.

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

===''Voyager 1''===
For ''[[Voyager 1]]'', with respect to the Sun:

*<math>\mu = GM</math> = 132,712,440,018&nbsp;km<sup>3</sup>⋅s<sup>−2</sup> is the [[standard gravitational parameter]] of the Sun
*''r'' = 17 [[1000000000 (number)|billion]] kilometers
*''v'' = 17.1&nbsp;km/s

Hence:
<math display="block">\varepsilon = \varepsilon_k + \varepsilon_p = \frac{v^2}{2} - \frac{\mu}{r} = \mathrm{146\,km^2 s^{-2}} - \mathrm{8\, km^2 s^{-2}} = \mathrm{138\,km^2 s^{-2}}</math>

Thus the hyperbolic excess velocity (the theoretical [[kinetic energy|orbital velocity]] at infinity) is given by
<math display="block">v_\infty = \mathrm{16.6\,km/s}</math>

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.

==Applying thrust==

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 <math> \mathbf{v} \cdot \mathbf{a}</math>: an amount <math>\mathbf{v} \cdot (\mathbf{a}-\mathbf{g})</math> for the kinetic energy and an amount <math>\mathbf{v} \cdot \mathbf{g}</math> for the potential energy.

The change of the specific energy of the rocket per unit change of delta-v is
<math display="block">\frac{\mathbf{v \cdot a}}{|\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.

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>

{{earth orbits}}

==See also==
*[[Tsiolkovsky rocket equation#Energy|Specific energy change of rockets]]
*[[Characteristic energy]] C3 (Double the specific orbital energy)

==References==
{{reflist}}

{{orbits}}
{{Voyager program}}

[[Category:Astrodynamics]]
[[Category:Orbits]]
[[Category:Physical quantities]]
[[Category:Mass-specific quantities]]