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For elliptical orbits it can also be calculated from the periapsis and apoapsis since = and = (+), where a is the length of the semi-major axis. = + = / / + = + 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.
Halley's calculations enabled the comet's earlier appearances to be found in the historical record. The following table sets out the astronomical designations for every apparition of Halley's Comet from 240 BC, the earliest documented sighting. [7] [167] In the designations, "1P/" refers to Halley's Comet; the first periodic comet discovered.
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 (), For a given semi-major axis the orbital period does not depend on the eccentricity (See also: Kepler's third law).
Also shown are: semi-major axis a, semi-minor axis b and semi-latus rectum p; center of ellipse and its two foci marked by large dots. For θ = 0°, r = r min and for θ = 180°, r = r max. Mathematically, an ellipse can be represented by the formula: = + ,
is the length of the semi-major axis. Conclusions: The orbital period is equal to that for a circular orbit with the orbit radius equal to the semi-major axis (), For a given semi-major axis the orbital period does not depend on the eccentricity (See also: Kepler's third law).
The semi-major axis is known if the mean motion and the gravitational mass are known. [ 2 ] [ 3 ] It is also quite common to see either the mean anomaly ( M ) or the mean longitude ( L ) expressed directly, without either M 0 or L 0 as intermediary steps, as a polynomial function with respect to time.
a is the orbit's semi-major axis; G is the gravitational constant, M is the mass of the more massive body. For all ellipses with a given semi-major axis the orbital period is the same, regardless of eccentricity. Inversely, for calculating the distance where a body has to orbit in order to have a given orbital period T:
Note that the semi-major axis is proportional to the 2/3 power of the orbital period. For example, planets in a 2:3 orbital resonance (such as plutinos relative to Neptune) will vary in distance by (2/3) 2/3 = −23.69% and +31.04% relative to one another. 2 Ceres and Pluto are dwarf planets rather than major planets.