View source for Specific mechanical energy

{{more citations needed|date=April 2022}}
{{Infobox physical quantity
|bgcolour={default}
|name   = Specific mechanical energy
|image  =
|caption=
|unit   =  J/kg, or m<sup>2</sup>/s<sup>2</sup>
|symbols = e, or ε
|derivations = 
}}

'''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 force]]s will remain constant.

It is defined as:

<math>\epsilon</math>= <math>\epsilon</math><sub>k</sub>+<math>\epsilon</math><sub>p</sub>

where

* <math>\epsilon</math><sub>k</sub> is the [[specific kinetic energy]]
* <math>\epsilon</math><sub>p</sub> it the [[specific potential energy]]

== Astrodynamics ==
{{Main article|Specific orbital energy}}In the [[gravitational two-body problem]], the specific mechanical energy of one body <math>\epsilon</math> is given as:<ref>{{cite book |last1=Bate, Mueller, White |title=Fundamentals Of Astrodynamics |date=1971 |publisher=Dover |location=New York |isbn=0-486-60061-0 |page=16 |edition=First}}</ref>

<math>\begin{align}
  \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{align}</math>

where

* <math>v\,\!</math> is the [[orbital speed]] of the body; relative to [[center of mass]].
* <math>r\,\!</math> is the [[Orbital position vector|orbital distance]] between the body and center of mass;
* <math>\mu = {G}(m_1 + m_2)\,\!</math> is the [[standard gravitational parameter]] of the bodies;
* <math>h\,\!</math> is the [[specific relative angular momentum]] of the same body referenced<ref>{{cite book |last1=Bate, Mueller, White |title=Fundamentals Of Astrodynamics |date=1971 |publisher=Dover |location=New York |isbn=0-486-60061-0 |pages=28–29 |edition=First}}</ref> to the center of mass. In other context h is used in the sense of a total for two bodies expressed as [[relative angular momentum]] of the system divided by the reduced mass, giving the same result for a central force problem;
* <math>e\,\!</math> is the [[Eccentricity (orbit)|orbital eccentricity]];
* <math>a\,\!</math> is the [[semi-major axis]] of the body orbit.

The relations are used.<ref>{{cite book |last1=Bate, Mueller, White |title=Fundamentals Of Astrodynamics |date=1971 |publisher=Dover |location=New York |isbn=0-486-60061-0 |page=28 |edition=First}}</ref><ref name="lissauer2019">{{Cite book |title=Fundamental Planetary Sciences: physics, chemistry, and habitability |last1=Lissauer |first1=Jack J. |last2=de Pater |first2=Imke |year=2019 |publisher=Cambridge University Press |isbn=9781108411981 |location=New York, NY, USA |pages=29–31 }}</ref>

<math>p= \frac{h^2}{\mu}</math> <math>= a(1-{e^2}</math>) <math> = r_{p}</math><math>(1+e)</math>

where

* <math>p\,\!</math> is the conic section [[Conic section|semi-latus rectum]].
* <math>r_p\,\!</math> is distance at [https://en.wiktionary.org/wiki/periastron periastron] of the body from the center of mass.

<math>v = \sqrt{\mu\left({2\over{r}} - {1\over{a}}\right)}</math>

where
*<math>\mu\,</math> is the [[standard gravitational parameter]], G(m<sub>1</sub>+m<sub>2</sub>), often expressed as GM when one body is much larger than the other.
*<math>r\,</math> is the distance between the orbiting body and center of mass.
*<math>a\,\!</math> is the length of 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:

<math>\mu = G(M + m) \approx GM</math>

where <math>M</math> is the mass of the celestial body. When GM is used the center of mass is at the center of M.
When bodies cannot accurately be described as point masses in the equations, other math is required and a difference may be required between center of mass and center of gravity.
In star systems of more than one planet, a planet orbit differs slightly from ideal with corrections applied for the other planets.

==References==
{{Reflist}}

{{DEFAULTSORT:Specific Mechanical Energy}}
[[Category:Energy (physics)]]


{{Physics-stub}}