Part of a series on
Astrodynamics
Orbital mechanics
Orbital elements Apsis Argument of periapsis Eccentricity Inclination Mean anomaly Orbital nodes Semi-major axis True anomaly
Types of two-body orbits by eccentricity Circular orbit Elliptic orbit Transfer orbit (Hohmann transfer orbit Bi-elliptic transfer orbit) Parabolic orbit Hyperbolic orbit Radial orbit Decaying orbit
Equations Dynamical friction Escape velocity Kepler's equation Kepler's laws of planetary motion Orbital period Orbital velocity Surface gravity Specific orbital energy Vis-viva equation
Celestial mechanics
Gravitational influences Barycenter Hill sphere Perturbations Sphere of influence
N-body orbits Lagrangian points (Halo orbits) Lissajous orbits Lyapunov orbits
Engineering and efficiency
Preflight engineering Mass ratio Payload fraction Propellant mass fraction Tsiolkovsky rocket equation
Efficiency measures Gravity assist Oberth effect
Propulsive maneuvers Orbital maneuver Orbit insertion
v t e

In the gravitational two-body problem, the specific orbital energy ${\displaystyle \varepsilon }$ (or vis-viva energy) of two orbiting bodies is the constant sum of their mutual potential energy (${\displaystyle \varepsilon _{p}}$) and their total kinetic energy (${\displaystyle \varepsilon _{k}}$), divided by the reduced mass.^[1] According to the orbital energy conservation equation (also referred to as vis-viva equation), it does not vary with time:

$${\displaystyle {\begin{aligned}\varepsilon &=\varepsilon _{k}+\varepsilon _{p}\\&={\frac {v^{2}}{2}}-{\frac {\mu }{r}}=-{\frac {1}{2}}{\frac {\mu ^{2}}{h^{2}}}\left(1-e^{2}\right)=-{\frac {\mu }{2a}}\end{aligned}}}$$

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

It is typically expressed in ${\displaystyle {\frac {\text{MJ}}{\text{kg}}}}$ (megajoule per kilogram) or ${\displaystyle {\frac {{\text{km}}^{2}}{{\text{s}}^{2}}}}$ (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.

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:^[2]

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

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

Proof

For an elliptic orbit with specific angular momentum h given by

$${\displaystyle h^{2}=\mu p=\mu a\left(1-e^{2}\right)}$$

we use the general form of the specific orbital energy equation,

$${\displaystyle \varepsilon ={\frac {v^{2}}{2}}-{\frac {\mu }{r}}}$$

with the relation that the relative velocity at periapsis is

$${\displaystyle v_{p}^{2}={h^{2} \over r_{p}^{2}}={h^{2} \over a^{2}(1-e)^{2}}={\mu a\left(1-e^{2}\right) \over a^{2}(1-e)^{2}}={\mu \left(1-e^{2}\right) \over a(1-e)^{2}}}$$

Thus our specific orbital energy equation becomes

$${\displaystyle \varepsilon ={\frac {\mu }{a}}{\left}={\frac {\mu }{a}}{\left}={\frac {\mu }{a}}{\left}={\frac {\mu }{a}}{\left}}$$

Zdroj:https://en.wikipedia.org?pojem=Specific_orbital_energy
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