Orbital eccentricity

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

• Circular orbit: e = 0
• Elliptic orbit: 0 < e < 1
• Parabolic trajectory: e = 1
• Hyperbolic trajectory: e > 1

The eccentricity e is given by^[1]

$${\displaystyle e={\sqrt {1+{\frac {2EL^{2}}{m_{\text{red}}\,\alpha ^{2}}}}}}$$

where E is the total

, and ${\displaystyle \alpha }$ the coefficient of the inverse-square law

$${\displaystyle F={\frac {\alpha }{r^{2}}}}$$

${\displaystyle \alpha }$

or in the case of a gravitational force:^[2]: 24

$${\displaystyle e={\sqrt {1+{\frac {2\varepsilon h^{2}}{\mu ^{2}}}}}}$$

where ε is the

For values of e from 0 to 1 the orbit's shape is an increasingly elongated (or flatter) ellipse; for values of e from 1 to infinity the orbit is a hyperbola branch making a total turn of 2 arccsc(e), decreasing from 180 to 0 degrees. Here, the total turn is analogous to turning number, but for open curves (an angle covered by velocity vector). The limit case between an ellipse and a hyperbola, when e equals 1, is parabola.

Radial trajectories are classified as elliptic, parabolic, or hyperbolic based on the energy of the orbit, not the eccentricity. Radial orbits have zero angular momentum and hence eccentricity equal to one. Keeping the energy constant and reducing the angular momentum, elliptic, parabolic, and hyperbolic orbits each tend to the corresponding type of radial trajectory while e tends to 1 (or in the parabolic case, remains 1).

For a repulsive force only the hyperbolic trajectory, including the radial version, is applicable.

For elliptical orbits, a simple proof shows that ${\displaystyle \arcsin(e)}$ yields the projection angle of a perfect circle to an ellipse of eccentricity e. For example, to view the eccentricity of the planet Mercury (e = 0.2056), one must simply calculate the inverse sine to find the projection angle of 11.86 degrees. Then, tilting any circular object by that angle, the apparent ellipse of that object projected to the viewer's eye will be of the same eccentricity.

Etymology

The word "eccentricity" comes from

The eccentricity of an orbit can be calculated from the orbital state vectors as the magnitude of the eccentricity vector:

$${\displaystyle e=\left|\mathbf {e} \right|}$$

• e is the eccentricity vector ("Hamilton's vector").^{[2]: 25, 62–63}

For

${\displaystyle r_{\text{p}}=a\,(1-e)}$

${\displaystyle r_{\text{a}}=a\,(1+e)\,,}$

]

$${\displaystyle {\begin{aligned}e&={\frac {r_{\text{a}}-r_{\text{p}}}{r_{\text{a}}+r_{\text{p}}}}\\\,\\&={\frac {r_{\text{a}}/r_{\text{p}}-1}{r_{\text{a}}/r_{\text{p}}+1}}\\\,\\&=1-{\frac {2}{\;{\frac {r_{\text{a}}}{r_{\text{p}}}}+1\;}}\end{aligned}}}$$

where:

• r_a is the radius at
  apoapsis (also "apofocus", "aphelion", "apogee"), i.e., the farthest distance of the orbit to the center of mass of the system, which is a focus
  of the ellipse.
• r_p is the radius at
  periapsis
  (or "perifocus" etc.), the closest distance.

The eccentricity of an elliptical orbit can also be used to obtain the ratio of the

periapsis

radius:

$${\displaystyle {\frac {r_{\text{a}}}{r_{\text{p}}}}={\frac {\,a\,(1+e)\,}{\,a\,(1-e)\,}}={\frac {1+e}{1-e}}}$$

For Earth, orbital eccentricity e ≈ 0.01671,

periapsis

is perihelion, relative to the Sun.

For Earth's annual orbit path, the ratio of longest radius (r_a) / shortest radius (r_p) is ${\displaystyle {\frac {\,r_{\text{a}}\,}{r_{\text{p}}}}={\frac {\,1+e\,}{1-e}}{\text{ ≈ 1.03399 .}}}$

Examples

The eccentricity of Earth's orbit is currently about 0.0167; its orbit is nearly circular. Venus and Neptune have even lower eccentricities. Over hundreds of thousands of years, the eccentricity of the Earth's orbit varies from nearly 0.0034 to almost 0.058 as a result of gravitational attractions among the planets.^[3]

The table lists the values for all planets and dwarf planets, and selected asteroids, comets, and moons.

Most of the Solar System's asteroids have orbital eccentricities between 0 and 0.35 with an average value of 0.17.^[4] Their comparatively high eccentricities are probably due to the influence of Jupiter and to past collisions.

The

ʻOumuamua is the first interstellar object found passing through the Solar System. Its orbital eccentricity of 1.20 indicates that ʻOumuamua has never been gravitationally bound to the Sun. It was discovered 0.2 AU (30000000 km; 19000000 mi) from Earth and is roughly 200 meters in diameter. It has an interstellar speed (velocity at infinity) of 26.33 km/s (58900 mph).

Mean average

The mean eccentricity of an object is the average eccentricity as a result of perturbations over a given time period. Neptune currently has an instant (current epoch) eccentricity of 0.0113,^[11] but from 1800 to 2050 has a mean eccentricity of 0.00859.^[12]

Climatic effect

Orbital mechanics require that the duration of the seasons be proportional to the area of Earth's orbit swept between the

Apsidal precession also slowly changes the place in Earth's orbit where the solstices and equinoxes occur. This is a slow change in the orbit of Earth, not the axis of rotation, which is referred to as axial precession. The climatic effects of this change are part of the Milankovitch cycles. Over the next 10000 years, the northern hemisphere winters will become gradually longer and summers will become shorter. Any cooling effect in one hemisphere is balanced by warming in the other, and any overall change will be counteracted by the fact that the eccentricity of Earth's orbit will be almost halved.^[15] This will reduce the mean orbital radius and raise temperatures in both hemispheres closer to the mid-interglacial peak.

Exoplanets

Of the many

• Equation of time

Footnotes

References