Specific mechanical energy explained
| Specific mechanical energy |
| Unit: | J/kg, or m2/s2 |
| Symbols: | e, or ε |
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.
It is defined as:
=
k+
pwhere
k is the
specific kinetic energy
p it the
specific potential energyAstrodynamics
See main article: article and Specific orbital energy. In the gravitational two-body problem, the specific mechanical energy of one body
is given as:
[1] \begin{align}
\epsilon&=
-
=-
\left(1-e2\right)
=-
\end{align}
where
is the
orbital speed of the body; relative to
center of mass.
is the orbital distance between the body and center of mass;
is the
standard gravitational parameter of the bodies;
is the
specific relative angular momentum of the same body referenced
[2] 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;
is the
orbital eccentricity;
is the
semi-major axis of the body orbit.
The relations are used.[3] [4]
)
where
is the conic section
semi-latus rectum.
is distance at
periastron of the body from the center of mass.
v=\sqrt{\mu\left({2\over{r}}-{1\over{a}}\right)}
where
is the
standard gravitational parameter, G(m
1+m
2), often expressed as GM when one body is much larger than the other.
is the distance between the orbiting body and center of mass.
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:
where
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.
Notes and References
- Book: Bate, Mueller, White . Fundamentals Of Astrodynamics . 1971 . Dover . New York . 0-486-60061-0 . 16 . First.
- Book: Bate, Mueller, White . Fundamentals Of Astrodynamics . 1971 . Dover . New York . 0-486-60061-0 . 28–29 . First.
- Book: Bate, Mueller, White . Fundamentals Of Astrodynamics . 1971 . Dover . New York . 0-486-60061-0 . 28 . First.
- Book: Fundamental Planetary Sciences: physics, chemistry, and habitability . Lissauer . Jack J. . de Pater . Imke . 2019 . Cambridge University Press . 9781108411981 . New York, NY, USA . 29–31 .
