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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 astrodynamics or celestial mechanics, an elliptic orbit or elliptical orbit is a Kepler orbit with an eccentricity of less than 1; this includes the special case of a circular orbit, with eccentricity equal to 0. In a stricter sense, it is a Kepler orbit with the eccentricity greater than 0 and less than 1 (thus excluding the circular orbit). In a wider sense, it is a Kepler orbit with negative energy. This includes the radial elliptic orbit, with eccentricity equal to 1.

In a gravitational two-body problem with negative energy, both bodies follow similar elliptic orbits with the same orbital period around their common barycenter. Also the relative position of one body with respect to the other follows an elliptic orbit.

Examples of elliptic orbits include Hohmann transfer orbits, Molniya orbits, and tundra orbits.

Velocity

Under standard assumptions, no other forces acting except two spherically symmetrical bodies m₁ and m₂,^[1] the orbital speed (${\displaystyle v\,}$) of one body traveling along an elliptic orbit can be computed from the vis-viva equation as:^[2]

${\displaystyle v={\sqrt {\mu \left({2 \over {r}}-{1 \over {a}}\right)}}}$

where:

• ${\displaystyle \mu \,}$ is the standard gravitational parameter, G(m₁+m₂), often expressed as GM when one body is much larger than the other.
• ${\displaystyle r\,}$ is the distance between the orbiting body and center of mass.
• ${\displaystyle a\,\!}$ is the length of the semi-major axis.

The velocity equation for a hyperbolic trajectory has either + ${\displaystyle {1 \over {a}}}$, or it is the same with the convention that in that case a is negative.

Orbital period

Under standard assumptions the orbital period (${\displaystyle T\,\!}$) of a body travelling along an elliptic orbit can be computed as:^[3]

${\displaystyle T=2\pi {\sqrt {a^{3} \over {\mu }}}}$

where:

• ${\displaystyle \mu }$ is the standard gravitational parameter.
• ${\displaystyle a\,\!}$ is the length of the semi-major axis.

Conclusions:

• The orbital period is equal to that for a circular orbit with the orbital radius equal to the semi-major axis (${\displaystyle a\,\!}$),
• For a given semi-major axis the orbital period does not depend on the eccentricity (See also: Kepler's third law).

Energy

Under standard assumptions, the specific orbital energy (${\displaystyle \epsilon }$) of an elliptic orbit is negative and the orbital energy conservation equation (the Vis-viva equation) for this orbit can take the form:^[4]

${\displaystyle {v^{2} \over {2}}-{\mu \over {r}}=-{\mu \over {2a}}=\epsilon <0}$

where:

• ${\displaystyle v\,}$ is the orbital speed of the orbiting body,
• ${\displaystyle r\,}$ is the distance of the orbiting body from the central body,
• ${\displaystyle a\,}$ is the length of the semi-major axis,
• ${\displaystyle \mu \,}$ is the standard gravitational parameter.

Conclusions:

• For a given semi-major axis the specific orbital energy is independent of the eccentricity.

Using the virial theorem to find:

• the time-average of the specific potential energy is equal to −2ε
  • the time-average of r⁻¹ is a⁻¹
• the time-average of the specific kinetic energy is equal to ε

Energy in terms of semi major axis

It can be helpful to know the energy in terms of the semi major axis (and the involved masses). The total energy of the orbit is given by

${\displaystyle E=-G{\frac {Mm}{2a}}}$,

where a is the semi major axis.

Derivation

Since gravity is a central force, the angular momentum is constant:

${\displaystyle {\dot {\mathbf {L} }}=\mathbf {r} \times \mathbf {F} =\mathbf {r} \times F(r)\mathbf {\hat {r}} =0}$

At the closest and furthest approaches, the angular momentum is perpendicular to the distance from the mass orbited, therefore:

${\displaystyle L=rp=rmv}$.

The total energy of the orbit is given by^[5]

${\displaystyle E={\frac {1}{2}}mv^{2}-G{\frac {Mm}{r}}}$.

Substituting for v, the equation becomes

${\displaystyle E={\frac {1}{2}}{\frac {L^{2}}{mr^{2}}}-G{\frac {Mm}{r}}}$.

This is true for r being the closest / furthest distance so two simultaneous equations are made, which when solved for E:

${\displaystyle E=-G{\frac {Mm}{r_{1}+r_{2}}}}$

Since ${\textstyle r_{1}=a+a\epsilon }$ and ${\displaystyle r_{2}=a-a\epsilon }$, where epsilon is the eccentricity of the orbit, the stated result is reached.

Flight path angle

The flight path angle is the angle between the orbiting body's velocity vector (equal to the vector tangent to the instantaneous orbit) and the local horizontal. Under standard assumptions of the conservation of angular momentum the flight path angle ${\displaystyle \phi }$ satisfies the equation:^[6]

${\displaystyle h\,=r\,v\,\cos \phi }$

where:

• ${\displaystyle h\,}$ is the specific relative angular momentum of the orbit,
• ${\displaystyle v\,}$ is the orbital speed of the orbiting body,
• ${\displaystyle r\,}$ is the radial distance of the orbiting body from the central body,
• ${\displaystyle \phi \,}$ is the flight path angle

$$Zdroj:https://en.wikipedia.org?pojem=Elliptic_orbit
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